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Uncollapsing the wavefunction by undoing quantum measurementsAndrew N. Jordan a; Alexander N. Korotkov ba Department of Physics and Astronomy, University of Rochester, Rochester, New York, USA b

Department of Electrical Engineering, University of California, Riverside, CA, USA

First published on: 12 February 2010

To cite this Article Jordan, Andrew N. and Korotkov, Alexander N.(2010) 'Uncollapsing the wavefunction by undoingquantum measurements', Contemporary Physics, 51: 2, 125 — 147, First published on: 12 February 2010 (iFirst)To link to this Article: DOI: 10.1080/00107510903385292URL: http://dx.doi.org/10.1080/00107510903385292

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Uncollapsing the wavefunction by undoing quantum measurements

Andrew N. Jordana* and Alexander N. Korotkovb

aDepartment of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA; bDepartment of ElectricalEngineering, University of California, Riverside, CA 92521, USA

(Received 17 June 2009; final version received 2 October 2009)

We review and expand on recent advances in theory and experiments concerning the problem of wavefunctionuncollapse: given an unknown state that has been disturbed by a generalised measurement, restore the state to itsinitial configuration. We describe how this is probabilistically possible with a subsequent measurement that involveserasing the information extracted about the state in the first measurement. The general theory of abstractmeasurements is discussed, focusing on quantum information aspects of the problem, in addition to investigating avariety of specific physical situations and explicit measurement strategies. Several systems are considered in detail:the quantum double dot charge qubit measured by a quantum point contact (with and without Hamiltoniandynamics), the superconducting phase qubit monitored by a SQUID detector, and an arbitrary number of entangledcharge qubits. Furthermore, uncollapse strategies for the quantum dot electron spin qubit, and the opticalpolarisation qubit are also reviewed. For each of these systems the physics of the continuous measurement process,the strategy required to ideally uncollapse the wavefunction, as well as the statistical features associated with themeasurement are discussed. We also summarise the recent experimental realisation of two of these systems, the phasequbit and the polarisation qubit.

Keywords: wavefunction uncollapse; measurement reversal; quantum Bayesianism

1. Introduction

The irreversibility of quantum measurement is anaxiomatic property of textbook quantum mechanics[1]. In his famous article Law without law, JohnWheelerexpresses the idea with poetic flare: ‘We are dealingwith [a quantum] event that makes itself known byan irreversible act of amplification, by an indeliblerecord, an act of registration’ [2]. However, it has beengradually recognised that the textbook treatment of aninstantaneous wavefunction collapse is really a veryspecial case of what is in general a dynamical process –continuous quantum measurement. Continuous mea-surements do not project the system immediately intoan eigenstate of the observable, but describe a pro-cess whereby the collapse happens over a period of time[3–13]. The fact that continuous measurement is adynamical process with projective measurement as aspecial case, leads us to ask whether the irreversibilityof quantum measurement is also a special case. Thepurpose of this paper is to review and expand on recentdevelopments in this area of research, showing that it ispossible to undo a quantum measurement, therebyuncollapsing the wavefunction, and to describe thisphysics in detail for both the abstract and concretephysical realisations.

This paper follows our earlier work on the subject[14,15], as well as other papers investigating similar

questions [16,17]. Wavefunction uncollapse teaches usseveral things about the fundamentals of quantummechanics. First, there is a notion that wavefunctionrepresents many possibilities, but that reality iscreated by measurement. The fact that the effects ofmeasurement can be undone suggests that this ideais flawed, or at least too simplistic. If you createreality with quantum measurements, does undoingthem erase the reality you created? Secondly, thereis a wide-spread belief that quantum measurementis nothing more than a decoherence process. Thissuggests that the superposition never really collapses;it only appears to collapse. What actually happens,according to this idea, is that all the informationabout the system disperses into the environment:when a quantum system interacts with a classicalmeasuring device, it becomes irreversibly entangledwith all the particles that make up the measuringdevice and its surroundings. The uncollapse of thewavefunction demonstrates that decoherence theorycannot be the whole story, because a true decoherenceprocess is irreversible [18]. The perspective we take inthis paper further advances the ‘quantum Bayesian’point of view, where the quantum state is nothingmore than a reflection of our information aboutthe system. When we receive more information aboutthe system, the state changes or collapses not because

*Corresponding author. Email: [email protected]

Contemporary Physics

Vol. 51, No. 2, March–April 2010, 125–147

ISSN 0010-7514 print/ISSN 1366-5812 online

� 2010 Taylor & FrancisDOI: 10.1080/00107510903385292

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of any mysterious forces, but simply as a result ofBayesian updating.

We note that there are many approaches to thenon-projective (continuous, weak, generalised, etc.)quantum measurement, which are essentially equiva-lent, in spite of quite different formalisms andterminology. To name just a few, let us mention mathe-matical POVM-type approaches [3,4] (POVM standsfor positive operator-valued measure), method ofrestricted path integrals [5,6], analysis of quantumjumps in atomic shelving [7], quantum trajectories[8], quantum-state diffusion [9], Monte-Carlo wave-function method [10], weak values [11], quantumfiltering [12], and the Bayesian formalism for weakmeasurements in solid-state systems [13]. In this paperwe will use the Bayesian and POVM-type formalismsbecause they are most suitable for the physical setups ofour main interest.

Our work is indirectly related to the ‘quantumeraser’ of Scully and Drühl [19]. There, the which-pathinformation of a particle is encoded in the quantumstate of an atom, resulting in a destruction of inter-ference fringe visibility. On the other hand, if thewhich-path information of the particle is erased, theinterference fringes are restored. Both the Scullyproposal and our proposal erase information. How-ever, there is an important difference. In order for theuncollapsing procedure to work, we have to erase theinformation that was already extracted classically. Inthe ‘quantum eraser’, only potentially extractableinformation is erased.

Rather than begin with the abstract idea ofuncollapse, we first introduce the concept with actualmeasurement processes in specific physical contexts.This brings to mind the saying of Asher Peres:‘Quantum phenomena do not occur in a Hilbertspace. They occur in a laboratory’ [20]. FollowingPeres’ dictum, we discuss measurement processes ina variety of solid state systems, where there hasbeen remarkable experimental progress in recentyears. Quantum coherence has been demonstrated tooccur in a controllable fashion in systems such assemiconductor quantum dots and superconductingJosephson junctions. We will discuss the physics ofmeasurement in these systems, as well as concretestrategies for uncollapsing the wavefunction. It shouldbe stressed that two of these proposals (a super-conducting phase qubit and optical polarisationqubit) have now been implemented in the laboratory[21,22], providing conclusive demonstration of wave-function uncollapse. It is still too early to predictwhether uncollapse can eventually become a reallyuseful tool or it will remain only as a surprising andeducating curiosity [18]. One possibly useful applica-tion is suppression of decoherence by the uncollapse

procedure [23], which will be briefly discussed in ourpaper.

The paper is organised as follows. We introducethe general subject with a preliminary discussion inSection 2. We then give some specific examples inSections 3 (the superconducting phase qubit) and 4(the double quantum dot qubit, monitored by aquantum point contact). With these physical imple-mentations, we discuss the uncollapsing strategies andresults. In Section 5 we give a general treatment of thephysics, using the POVM-based formalism. This isdone both for pure states (Section 5.1) and mixedstates (Section 5.2). In Section 5.3 we discuss theinterpretation of wavefunction uncollapse, and what ittells us about quantum information. These generalresults are applied to the examples given previously inSection 5.4. In Section 6 we further apply the generalresults to the case of the finite-Hamiltonian qubitundergoing the uncollapse process. In Section 7, wegeneralise to the case of many charge qubits, anddiscuss an explicit procedure to undo any generalisedmeasurement. We discuss recent developments in thetheory and experiments of measurement reversal inSection 8 and conclude in Section 9. The Appendixcontains results about the statistics of the waiting timedistribution in the charge qubit example.

2. Preliminary discussion

Our goal is to restore an initial quantum statedisturbed by measurement. However, it is importantto discuss what exactly we mean by that. For example,if we start with a known pure state jcini and perform atextbook projective measurement, then it is trivial torestore the initial state: since we also know the post-measurement wavefunction jcmi, we just need to applya unitary operation which transfers jcmi into jcini. Ifwe start with a known mixed state, then its restorationafter a projective measurement is a little moreinvolved;1 however, such a procedure still can beeasily analysed using standard quantum mechanics andclassical probability theory.

In this paper we consider a different, non-trivialsituation: we assume that an arbitrary initial state isunknown to us, and we still want to restore it afterthe measurement disturbance. To make this idea moreprecise, we consider a contest between the uncollapseproponent Plato, and an uncollapse skeptic Socrates.Socrates prepares a quantum system in any state helikes, but it is unknown to Plato. Socrates sendsthe state to Plato, who makes some measurement onthe system, verified by the arbiter Aristotle. Plato thentries to undo the measurement. If Plato judges that theattempt succeeded, the system is returned to Socrates,with the claim that it is the original state. Socrates is

126 A.N. Jordan and A.N. Korotkov


then allowed to try and find a contradiction in anyway he likes, with the whole process monitored byAristotle. If a contradiction can be found, then he canclaim to refute the uncollapse claim, but in the absenceof contradiction, the uncollapse claim stands. IfSocrates would like to try again to find a contradiction,or if Plato judges that his undoing attempt wasunsuccessful (and does not return a state), thenSocrates prepares a new (still unknown to Plato) state,and the competition continues. If Socrates cannotfind a contradiction after many rounds of the com-petition, then Plato will win the contest, and will havesuccessfully demonstrated the uncollapsing of thequantum state.

A slightly different but equivalent situation iswhen we know the initial state, but our uncollapsingprocedure must be independent of the initial state(so we can pretend that it is unknown), and there-fore the uncollapsing should restore any initial statein the same way. This formulation is most appro-priate for a real experiment demonstrating theuncollapsing. Finally, we may consider the moregeneral case where the measured system is entangledwith another system, and we wish to restore the initialstate of the compound system without any access toits second part.

The traditional statement of irreversibility of aquantum measurement can be traced to the fact that itmay be described as a mathematical projection.Projection is a many-to-one mapping in the Hilbertspace, and therefore the same post-measurement stategenerally corresponds to (infinitely) many initialstates.2 It is therefore impossible to undo a projectivemeasurement.

However, the situation is different for a general[4] (POVM-type) measurement, which typically cor-responds to a one-to-one mapping jcini!jcmi in theHilbert space of wavefunctions (in this paper weconsider only ‘ideal’ measurements which do notintroduce extra decoherence). In this case the post-measurement wavefunction jcmi can still be asso-ciated with the unique initial state jcini, and awell-defined inverse mapping exists mathematically.This makes the uncollapsing possible in principle.Since the inverse mapping is typically non-unitary,it cannot be realised as an evolution with a suit-able Hamiltonian. However, it can be realised usinganother POVM-type measurement with a specific(‘lucky’) result.

Rather than giving the most general, abstract casefirst, we start with a couple of specific examples ofrealistic systems where such a measurement either hasbeen done, or could be done in the near future. Oncethe reader has the basic idea, we will proceed togeneralise the process and develop its characteristics.

3. Example 1: the phase qubit

We first give a simple example of erasing informationand uncollapsing the wavefunction for the case of asuperconducting phase qubit [24]. The system iscomprised of a superconducting loop interrupted byone Josephson junction (Figure 1(a)), which iscontrolled by an external flux fe in the loop. Twoqubit states j0i and j1i (Figure 1(b)) correspond totwo lowest states in the quantum well for thepotential energy V (f), where f is the superconduct-ing phase difference across the junction. Transitionsbetween the levels j0i and j1i (Rabi oscillations) canbe induced by applying microwave pulses that areresonant with the energy level difference. Because ofthe energy difference, the two basis states acquire aphase difference between them, which linearly in-creases in time. However, this can be eliminated bygoing into the rotating frame; we always assume it inthis section.

The qubit is measured by lowering the barrier(which depends on fe), so that the upper state j1itunnels into the continuum with the rate �, while thestate j0i does not tunnel out. The tunnelling event issensed by a two-junction detector SQUID inductivelycoupled to the qubit (Figure 1(a)).

3.1. Partial collapse

For sufficiently long tunnelling time t, �t � 1,the measurement is a (partially destructive) projectivemeasurement [24]: the system is destroyed if thetunnelling occurs, while if there is no record oftunnelling, then the state is projected onto the lowerstate j0i. This measurement technique is remarkablein the fact that the wavefunction is collapsed ifnothing happens. A more subtle situation arises if thebarrier is raised after a finite time t * �71 [25].The system is still destroyed if tunnelling happens,while in the case of no tunnelling (which we referto as a null-result measurement) the state is partiallycollapsed because the inference from the measure-ment is ambiguous. This is due to the fact that

Figure 1. (a) Schematic of a phase qubit controlled by anexternal flux fe and inductively coupled to the detectorSQUID. (b) Energy profile V(f) with quantised levelsrepresenting qubit states. The tunnelling event through thebarrier is sensed by the SQUID.

Contemporary Physics 127


classically a null-result measurement could be be-cause the system is in the lower energy state (andthus would not ever tunnel), or because it is in theupper energy level (and we simply didn’t give itenough time).

The conditional probability for decaying if thequbit is in state j1i is given by 1 7 exp (7�t). Givensome initial density matrix rin, we can interpret thediagonal density matrix elements (rin,00 and rin,11)as classical probabilities of being in states j0i orj1i. We can then invoke the Bayes rule [26,27] toupdate the probabilities,3 given the data of ‘nothinghappened’. It is not possible to use a simple generalrule for updating the off-diagonal element r01; how-ever, analysis of the full dynamics in the extendedHilbert space [25,28] gives a simple result as well, sothat the qubit density matrix changes after the null-result measurement as

r00ðtÞ ¼rin;00

rin;00 þ rin;11exp ð��tÞ; ð1Þ

r11ðtÞ ¼rin;11exp ð��tÞ

rin;00 þ rin;11exp ð��tÞ; ð2Þ

r01ðtÞ ¼rin;01exp ½ijðtÞ�exp ð��t=2Þrin;00 þ rin;11exp ð��tÞ

: ð3Þ

Such a measurement is ideal and does not decoherethe qubit. Notice that in a simple model [10,28]there is no relative phase j that is added during themeasurement process, if one uses the rotating frame.In the real experiment [25], however, the energydifference between the states j0i and j1i changes inthe process of measurement because it is affectedby the changing external flux fe, and therefore evenin the (constantly) rotating frame the phase j is non-zero.

Actually, in the experiment [25] the situation iseven more complex because the tunnelling rategradually changes in time; also, instead of controllingthe measurement time t, it is much easier to controlthe tunnelling rate. As a result, the measurementshould be characterised by the overall strengthpt ¼ 1� exp ½�

R t0 �ðt

0Þ dt0�. Nevertheless, for simpli-city, we use here the physically transparent languageof Equations (1), (2) and (3) with exp(7�t) understoodas 1 7 pt.

Up to such changes of notation, the coherentnon-unitary evolution ((1), (2), (3)) has been experi-mentally verified in [25] using tomography of the post-measurement state. The state tomography consistedof three types of rotations of the qubit Bloch sphere,followed by complete (projective) measurement. Inthe experiment it was not possible to select only the

null-result cases, because it was not possible todistinguish if a tunnelling event happened duringmeasurement or during tomography. However, asimple trick of comparing the protocols with andwithout tomography made it possible to separate thenull-result cases.

3.2. Uncollapsing

We will now describe how to undo the statedisturbance ((1), (2), (3)) caused by the partial collapseresulting from the null-result measurement. The un-doing of this measurement consists of three steps [14]:(i) exchange the amplitudes for the states j0i and j1iby application of a microwave p-pulse, (ii) performanother measurement by lowering the barrier, identicalto the first measurement, (iii) apply a second p -pulse.If the tunnelling did not happen during the secondmeasurement, then the information about the initialqubit state is cancelled (both basis states have equallikelihood for two null-result measurements). Corre-spondingly, according to Equations (1), (2) and (3)(which are applied for the second time with exchangedindices 0 $ 1), any initial qubit state is fully restored.An added benefit to this strategy is that the phase j isalso cancelled automatically; the physics of this phasecancellation is the same as in the spin-echo techniquefor qubits.

It is easy to mistake the above pulse-sequence assimply the well known spin-echo technique alone. Westress that this is not the case: spin-echo deterministi-cally reverses an unknown unitary transformation(arising, for example, from a slowly varying magneticfield) without gaining or losing any information aboutwhich state the system is in. Our strategy is probabil-istic and requires erasing the classical information thatone extracts from the system to begin with. It is a(probabilistic) reversal of a known non-unitary trans-formation – and therefore quite different from spinecho.

While both measurements should give null resultsfor the success of our procedure, let us define theuncollapsing success probability PS as the null-resultprobability of only the second measurement, assumingthat the first measurement already gave the null result.Such a definition originates from our intention tocharacterise the probability of undoing the firstmeasurement. (For the joint probability of two nullresults we will later use notation P̃S). To calculate PS,let us start with the initial qubit state rin. Then after thefirst null-result measurement the qubit state is given byEquations (1), (2) and (3), and after the p-pulsethe occupation of the upper level is ~r11 ¼ rin;00=½rin;00 þ rin;11 exp ð��tÞ�. The success probability issimply the probability that the second tunnelling



will not occur, PS ¼ ~r00 þ ~r11exp ð��tÞ ¼ 1� ~r11ð1�exp ð��tÞÞ, which can be expressed as

PS ¼e��t

rin;00 þ e��trin;11: ð4Þ

Notice that if �t ¼ 0 (i.e. the measurement has notstarted), then the success probability is 1 because thestate never changed, and consequently does not needto be changed back. This is unsurprising. In the otherlimit, given a sufficiently long time of no tunnelling,�t � 1, the qubit state j0i is indicated with highconfidence after the first null-result measurement, andconsequently the uncollapse success probability be-comes very small, recovering the traditional statementof irreversibility for a projective measurement. Westress that the possibility of uncollapsing as well as ourformalism requires a quantum-limited detector, i.e. onethat introduces no additional dephasing to the system.For such a detector measuring a pure state, the stateremains pure throughout the partial collapse, and theuncollapse. We also note that if the qubit is entangledwith other qubits, the uncollapsing restores the state ofthe whole system.

Uncollapsing of the phase qubit state has recentlybeen experimentally realised by Nadav Katz andcolleagues in the lab of John Martinis, at UC SantaBarbara [21]. The experimental protocol was slightlyshorter than that described above: it was missing thesecond p-pulse, so the uncollapsed state was actuallythe p-rotation of the initial state. Shortening of theprotocol helped in decreasing the duration of the pulsesequence, which was about 45 ns, including the statetomography. Since the qubit energy relaxation anddephasing times were significantly longer, T1 ¼ 450 nsand T�2 ¼ 350 ns, the simple theory described abovewas sufficiently accurate. The same trick as for thepartial-collapse experiment [25] was used to separatetunnelling events during the first, second, and tomo-graphy measurements, because the detector SQUIDwas too slow to distinguish them directly.

The uncollapse procedure should restore any initialstate. However, instead of examining all initial states tocheck this fact, it is sufficient to choose four initialstates with linearly independent density matrices anduse the linearity of quantum operations [4]. In theexperiment [21] the uncollapse procedure was appliedto the initial states (j0i þ j1i)/21/2, (j0i7 ij1i)/21/2, j0i,and j1i, and then the results were expressed via thelanguage of the quantum process tomography [4](QPT).4 The experimental [21] QPT fidelity of theuncollapsing procedure was above 70% for pt 5 0.6.A significant decrease of the uncollapsing fidelity forlarger measurement strength pt, especially for pt 4 0.8,was due to finite T1 time and the fact that the

null-result selection preferentially selects the caseswith energy relaxation events, so that the procedureshould no longer work well when 1 7 pt becomescomparable to the probability of energy relaxation. (Itis interesting that in some range of parameters, thesame procedure preferentially selects the cases withoutenergy relaxation events, thus effectively suppressingqubit decoherence [23] – see Section 8.3.)

Exact uncollapsing requires an ideal detection,which does not decohere a quantum state; Equations(1), (2) and (3) correspond to such an ideal detection.However, if various decoherence mechanisms aretaken into account [28], then only imperfect uncollap-sing is possible. The theory of imperfect uncollapsing isa subject of further research.

4. Example 2: double-quantum-dot charge qubit

The next example we consider is illustrated in Figure 2:a charge qubit made of a double-quantum-dot (DQD),populated by a single electron, which is measuredcontinuously by a symmetric quantum point contact(QPC). This setup has been extensively studied inearlier papers, both theoretically [29] and experimen-tally [30]. In contrast to the previous example, we willdenote the basis states of the DQD charge qubit as j1iand j2i (the electron being in one dot, or the other), tobe consistent with previous papers on this setup. Themeasurement is characterised by the average currentsI1 and I2 corresponding to the states j1i and j2i, and bythe shot noise spectral density SI.

5 We treat theadditive detector shot noise as a Gaussian, white,stochastic process, and assume the detector is in theweakly responding regime, jDIj � I0, where DI ¼ I1 7I2 and I0 ¼ (I1 þ I2)/2, with QPC voltage much larger

Figure 2. Illustration of the uncollapsing procedure for thecharge qubit. The slanted lines indicate the deterministicoutput of the detector in the absence of noise, if the qubit isin state j1i or j2i. The initial measurement yields the result r0.The detector is again turned on, hoping that at some futuretime the measurement result r(t) ¼ r0 þ ru(t) crosses theorigin, at which time the detector is turned off, successfullyerasing the information obtained in the first measurement,and restoring the initial qubit state.



than all other energy scales, so that the measurementprocess can be described by the quantum Bayesianformalism [13].

4.1. Measurement dynamics for a non-evolving qubit

We begin for simplicity with the assumption that thereis no qubit Hamiltonian evolution, so that the qubitstate evolves due to the measurement only (this canalso be effectively done using ‘kicked’ quantumnondemolition (QND) measurements [31]). As wasshown in [13], at low temperature the QPC is an idealquantum detector (which does not decohere themeasured qubit), so that the evolution of the qubitdensity matrix r due to continuous measurementpreserves the ‘murity’ [32] M:r12/(r11r22)1/2 whilethe diagonal matrix elements evolve according to theclassical Bayes rule [26,27].3 We define the electricalcurrent through the QPC averaged in a time t as�IðtÞ ¼ ½

R t0 Iðt

0Þ dt0�=t, and the quantum Bayesian equa-tions read [13]

r11ðtÞ ¼r11ð0ÞP1ð�IÞ

r11ð0ÞP1ð�IÞ þ r22ð0ÞP2ð�IÞ; ð5Þ

r22ðtÞ ¼r22ð0ÞP2ð�IÞ

r11ð0ÞP1ð�IÞ þ r22ð0ÞP2ð�IÞ; ð6Þ

M ¼ r12=ðr11r22Þ1=2 ¼ const; ð7Þ

where the conditional (Gaussian) probability densitiesof a current �I realisation, given that the qubit is in j1i,j2i are

P1;2ð�IÞ ¼ ðt=pSIÞ1=2

� exp ½�ð�I� I1;2Þ2t=SI�: ð8Þ

Equations (5) and (6) may be simplified by noting

r11ðtÞr22ðtÞ

¼ r11ð0Þr22ð0Þ

exp ½2rðtÞ�; ð9Þ

where we define the dimensionless measurement resultas

rðtÞ ¼ tDISI½�IðtÞ � I0�

¼ DISI

Z t0

½Iðt0Þ � I0� dt0: ð10Þ

Notice that r(t) is closely related to the total chargepassed through the QPC, and therefore r(t) accumu-lates in time. For times much longer than the‘measurement time’ [33] TM ¼ 2SI/(DI)2 (the time scalerequired to obtain a signal-to-noise ratio of 1), the

average current �I tends to either I1 or I2 because theprobability density P(�I) of a particular �I is

Pð�IÞ ¼Xi¼1;2

rii ð0ÞPið�IÞ; ð11Þ

and Pi (�I)! d(�I–Ii) for t/TM!? Therefore, r(t) tendsto +?, continuously collapsing the state to either j1i(for r ! ?) or j2i (for r ! –?). Importantly, for thespecial case when the initial state is pure, the stateremains pure during the entire process. Notice that thedensity matrix Equations (5), (6) and (7) formallycoincide with Equations (1), (2) and (3) in the phasequbit example, if the qubit states are renumbered, thephase j is neglected, and �t is replaced with 2r.

4.2. Uncollapsing for the charge qubit

In order to describe how to uncollapse the charge qubitstate, we note that if r(t) ¼ 0 at some moment t, thenthe qubit state becomes exactly the same as it wasinitially, r(t) ¼ r(0), as follows from Equations (9) and(7). This of course must be the case if t ¼ 0, i.e. beforethe measurement began, but is equally valid for somelater time. To see why this is so from the informationalpoint of view, we note that in the absence of noise,the measurement result from states j1i, j2i wouldsimply be r1,2(t) ¼ +t/TM. With the noise present, themeasurement outcome r(t) ¼ 0 splits in half thedifference between states j1i and j2i. Such an outcomecorresponds to an equal statistical likelihood of thestates j1i and j2i, and therefore provides no informa-tion about the state of the qubit.

Suppose the outcome of a measurement is r0,partially collapsing the qubit state toward either statej1i (if r0 4 0), or state j2i (if r0 5 0). The previous ‘noinformation’ observation suggests the following strat-egy for uncollapsing [14]: continue measuring, with thehope that after some time t the stochastic result ofthe second measurement ru(t) becomes equal to 7r0,so the total result r(t) ¼ r0 þ ru(t) is zero, and thereforethe initial qubit state is fully restored. If this happens,the measuring device is immediately switched offand the uncollapsing procedure is successful (Figure 2).However, r(t) may never cross the origin, and then theuncollapsing attempt fails. The probability of successPS is therefore the probability that r(t) crosses theorigin (at least once) after it starts from r0.

This strategy requires the observation of a parti-cular measurement result that may never materialise.The strategy shifts the randomness to the amount oftime that needs to elapse in order to find the desiredmeasurement result. Of course, in a given realisationthe measurement result could take on the desired valuemultiple times, so we will take as our strategy to turnoff the detector the first time the measurement result



takes on r ¼ 0. In the classical stochastic physics this isknown as a first passage process [34], the theory ofwhich is well developed and is detailed in Appendix 1.

In order to analyse the uncollapsing strategyperformance, in particular to find the success prob-ability PS, it is important to notice that the off-diagonal elements of the qubit density matrix r do notcome into play when we consider the detector outputI(t) (this is true only in the case of zero or QND-eliminated qubit Hamiltonian; in Section 6 we willgeneralise to the finite qubit Hamiltonian case). As aresult, the quantum problem can be exactly reduced toa classical problem by substituting r11(t) and r22(t)with classical probabilities, evolving in the course ofmeasurement according to the classical Bayes rule,while evolution of the off-diagonal elements r12 ¼ r�21can be found automatically from the murity conserva-tion law (7). We therefore model the qubit by aclassical bit with probability p1 ¼ r11(0) of beingprepared in state ‘1’ and probability p2 ¼ r22(0) ofbeing in state ‘2’. If the bit is in state ‘1’, thedimensionless measurement result r(t) evolves as arandom walk with diffusion coefficient D ¼ (DI)2/4SI ¼ 1/2TM and drift velocity v1 ¼ (DI)2/2SI ¼1/TM (see Equations (8) and (10)). For the bit state‘2’ the random walk of r(t) has the same diffusioncoefficient but the opposite drift velocity v2 ¼ 7(DI)2/2SI. We are given the fact that the first part of themeasurement had the result r0 (i.e. we select onlysuch realisations). We need to analyse the stochasticbehaviour of the total measurement result r(t) duringthe second part of measurement, with most attentionto the crossing of the zero line r(t) ¼ 0 (for conve-nience we shift t ¼ 0 to the beginning of the secondmeasurement, so that r(0) ¼ r0).

Let us find the probability PS of such a crossing.Here we obtain it in a simple way, while in Appendix 1we reproduce the result in a more complicated way,which also allows us to analyse the statistics of thewaiting time. For definiteness take r0 4 0 (this will beextended later). Then the result r(t) will necessarilycross 0 if the bit is actually in the state ‘2’, because inthis case r(t ¼ ?) ¼ 7? while r(t ¼ 0) ¼ r0 4 0.The probability of being in the state ‘2’ is ~p2 ¼ p2 exp(7r0)/(p1 exp (r0) þ p2 exp (7r0)) from (9), whichdiffers from p2 because of the Bayesian update.

3 Ifthe bit is in state ‘1’ (this happens with probability~p1 ¼ 1 7 ~p2), then r(t ¼ ?) ¼ þ? and the crossingof 0 may never happen; however, it is still possible withsome probability PC, which depends on r0, and also onD and v1. To find PC, let us consider an infinitesimaltime step dt and model the diffusion by discrete jumpsin r of magnitude Dr ¼ + (2D dt)1/2. After a step dt,the coordinate will then shift to one of two positions,r ¼ r+, where r+ ¼ r0 þ v1dt + (2D dt)1/2. Each of

these new coordinates will have its own probabilityof eventually crossing the origin, PC(r+). Becausethe diffusive dynamics is generated by choosing eitherrþ or r7 with equal weighting, it follows in the limitdt ! 0 that

PCðr0Þ ¼X�

1

2PCðr�Þ: ð12Þ

Expanding PC (r+) in this relation in a Taylor series,we find from the linear in dt term that

D @2r0PC ¼ �v1 @r0PC; ð13Þ

where @r0 and @2r0

denote the first and secondderivatives with respect to r0. Taking into accountthat PC ¼ 1 for r0 ¼ 0 and PC ¼ 0 for r0 ¼ ?, theabove differential equation may be easily solved tofind PC ¼ exp (7v1r0/D) ¼ exp (72r0). Now collect-ing the probabilities of the zero line crossing for bothbit states, we find

PS ¼ ~p1PC þ ~p2 ¼ exp ð�r0Þ=ðp1 exp ðr0Þ þ p2 exp ð�r0ÞÞ: ð14Þ

The derivation for r0 5 0 is similar and leads to theextra factor exp (2r0), so that the crossing probabilityin both cases can be written as PS ¼ exp (7jr0j)/(p1 exp (r0) þ p2 exp (7r0)).

Thus, using the trick of reducing the quantumdynamics to the classical problem, we have foundthe probability of successful uncollapsing for a DQDcharge qubit with no Hamiltonian evolution [14]:

PS ¼exp ð�jr0jÞ

exp ðr0Þrin;11 þ exp ð�r0Þrin;22; ð15Þ

where rin characterises the qubit state before the firstmeasurement.

This result formally corresponds to Equation (4)with substitution �t ! 2r0 (for positive r0), and itsphysical meaning is also similar. When the firstmeasurement result indicates either qubit state withgood confidence (jr0 j � 1), the probability of successPS given by Equation (15) becomes very small,eventually becoming PS ¼ 0 for a projective measure-ment, realised for r0 ¼ +?. In the other limit ofr0 ¼ 0, the success probability is unity because no timeneeds to elapse – the state is already undisturbed.

While the above method allows us to find thesuccess probability PS, there are other importantcharacteristics of the uncollapsing strategy, which wehave not discussed, such as the mean waiting timerequired to uncollapse the state. This and other charac-teristics of the strategy can be calculated through the



more powerful methods of first-passage theory. Thismethod and the results obtained from it are discussedin Appendix 1.

5. General formalism

The previous two examples were fairly straightforwardto analyse because the unitary dynamics was effectivelyturned off, and only the measurement dynamics neededto be accounted for. In general the situation is morecomplicated. For example, if the DQD qubit had afinite Hamiltonian while the QPC was on, there wouldbe both measurement dynamics and Hamiltoniandynamics. This situation leads us to discuss first thegeneral formalism in its most abstract form beforeturning our attention to this more complicated example.

5.1. Formalism for wavefunctions

Let us first consider a pure initial state jcini, andpostpone a generalisation to mixed states until Section5.2. In the formalism of a general (ideal) quantummeasurement [4] which transfers pure states into purestates, the measurement with result m is associatedwith the linear Kraus operator Mm, so that theprobability of result m is

PmðjciniÞ ¼ jjMmjcini jj2; ð16Þ

where jj. . . jj denotes the norm of the state, and the(conditioned) state after measurement is

jcmi ¼Mmjcini

½PmðjciniÞ�1=2; ð17Þ

where the denominator makes jcmi properly normal-ised. (Very often people prefer to omit this denomi-nator and work with non-normalised states; this makesthe mapping linear.) The operators Em ¼MymMm(called POVM elements [4]) are Hermitian and positivesemidefinite by construction; these operators mustobey the completeness relation SmEm ¼ 1, whichensures that the total probability of all measurementresults is unity. A measurement operator Mm canalways be written as

Mm ¼ UmE1=2m ; ð18Þ

where Um is a unitary operator (an important specialcase is when Mm ¼ E1=2m ; this corresponds to the‘quantum Bayes theorem’ [6,35]).

Now let us discuss wavefunction uncollapse in thisgeneral and abstract context [14]. The state disturbancerule (17) is typically a nonunitary one-to-one map inthe Hilbert space. To undo the measurement withknown result m, we have to realise a physical process

corresponding to the nonunitary inverse operatorM�1m ,multiplied by an arbitrary constant (which is notimportant because of the normalisation). This can beaccomplished with another measurement, possiblytogether with unitary operations. As shown below,we can realise measurement undoing if the secondmeasurement realises a Krauss operator of the form

L ¼ CULE�1=2m VL; ð19Þ

where UL and VL are any unitary operators, and C isan unspecified constant that will be discussed later.(The operator L also has a decomposition of the formUE1/2, but with a different POVM element E.) Theuncollapse then consists of three steps: first, the unitary

operator VyLUym is applied to reverse the unitary part of

Mm and prepare for the second measurement. Next the

measurement operator L is applied. Finally the unitary

operator UyL is applied to reverse the remaining unitary

part of L. We can now see the effect of the uncollapsingoperation on the state jcmi by applying Equation (17)and the unitaries to find

jcfi ¼UyLLVyLUymjcmi

jjUyLLVyLUymjcmijj

¼ jcini; ð20Þ

thus restoring the original state, because UyLLVyL

UymMm ¼ C, which is removed by the normalisation.

(The phase of C is not important, since it affects onlythe overall phase of the wavefunction.)

However, in order for the operator L to be physi-cally realisable, the operator L{L must belong toanother complete set of POVM elements, and thereforeall its eigenvalues must not exceed unity (otherwisesome states will be assigned probabilities that areabove unity; notice that the eigenvalues are non-

negative automatically). Since LyL ¼ jCj2VyLE�1m VL, itseigenvalues are directly related to the eigenvalues p

ðmÞi

of the operator Em. Expressing Em ¼P

ipðmÞi jiihij,

where the eigenvectors jii form an orthonormal basis,the eigenvectors of L{L are obviously V

yLjiE, and the

corresponding eigenvalues are jCj2=pðmÞi . Since all theseeigenvalues must not exceed 1, we find the followinginequality on jCj2,

jCj2 mini

pðmÞi ¼ minPm; ð21Þ

where min Pm is the probability of the result m,minimised over all possible states jcini in the Hilbertspace. The equality of min Pm to mini p

ðmÞi follows from

Equation (16).In general, the uncollapse cannot be accomplished

deterministically, since we rely on a measurement with



a specific result, corresponding to the operator L.We can calculate the uncollapse success probabilityPS from Equation (16), with Mm ! L and cini !VyLUymjcmi,

PS ¼ jjLVyLUymjcmijj

2

¼ Cjcini½PmðjciniÞ�

1=2

��

��2

¼ jCj2

PmðjciniÞ: ð22Þ

Now using the bound (21) for jCj2, we find the boundfor the success probability of uncollapsing after thefirst measurement with result m [14]:

PS minPmPmðjciniÞ

; ð23Þ

where the denominator is the probability of the resultm for a given initial state, while the numerator is thisprobability minimised over all possible initial states.

The bound (23) is one of the most importantresults (notice a similar result in [16]) and deservesdiscussion. First, this bound is exact in the sense that itis achievable by an optimal uncollapsing procedure.This is because the uncollapsing operator with jCj ¼(min Pm)

1/2 is still a physically allowed operator. As wewill see later, the upper bound (23) is achievable in realexperimental setups (in particular, in the examplesdiscussed in the two previous sections). However, non-optimal uncollapsing procedures, especially involving asequence of measurements, can lead to smaller successprobabilities (an example of non-optimal uncollapsinghas been discussed in [36]; another example will bediscussed at the end of Section 6). An analysis of theprocedures with an arbitrary sequence of measure-ments and unitary operations is similar to the above:the corresponding measurement and unitary operatorsshould simply be multiplied.

Notice that the success probability (22) and theinequality (23) depends on the initial state, which isunknown to the person performing the uncollapsing(Plato, see description in Section 2). Therefore, thesuccess probability PS can be calculated by the manwho knows what the initial state is (Socrates), whilePlato can only estimate PS; for example, he cancalculate the worst-case scenario (the minimum of PSover the accessible Hilbert space) or can calculatethe average of PS over all possible initial states (thisprocedure will be discussed in the next subsection).

Recalling the fact that it is not possible to undo afully collapsed state due to the nature of projectivemeasurement, the uncollapsing probability PS shoulddecrease with increasing strength of the first measure-ment. Qualitatively, a stronger measurement is onethat tends to a projection, as the uncertainty in the

measurement decreases. Mathematically, this meansthat some eigenvalues of Em become closer to 0. As aconsequence, min Pm becomes smaller (see Equation(21)), therefore lowering the upper bound for PS. Fora projective measurement PS ¼ min Pm ¼ 0, thusmaking the state uncollapse impossible.

It is interesting to discuss the case when the initialstate jcini is known to belong to a certain subspace ofthe Hilbert space, and we therefore wish to restorestates only in this subspace. In this case, the calculationof min Pm should be limited to this subspace, whichmay increase the bound (23) for the success probabilityPS. A trivial example of such a situation is when theinitial state jcini is known to Plato. Then it is notnecessary to minimise Pm over all possible initial statesin Equation (23), because the set of possible statesconsists of only one (known) state, thus allowinguncollapsing with 100% probability. This is exactly thecase discussed at the beginning of Section 2.

5.2. Formalism for density matrices

So far we have dealt only with pure states; however, itis very simple to generalise the uncollapsing formalismto include density matrices. In this case the initialdensity matrix rin is transformed by the first measure-ment into the state [4]

rm ¼MmrinM

ym

Pm; ð24Þ

where the probability Pm of the measurement resultm is

PmðrinÞ ¼ TrðMymMmrinÞ: ð25Þ

Using the uncollapsing procedure previously discussedand using the same measurement operator L given byEquation (19), we find that the uncollapsed state

rf ¼UyLLVyLUymrmUmV

yLLyUL

TrðLyLVLUymrmUmV

yLÞ¼ rin ð26Þ

coincides with the initial state. The uncollapsingsuccess probability PS is equal to the denominator inEquation (26), and satisfies the relation

PS ¼ jCj2=PmðrinÞ ð27Þ

(as in Equation (22)). The constant jCj2 is still limitedby the inequality (21), and therefore the probability ofsuccess has the upper bound [14]

PS minPmPmðrinÞ

; ð28Þ



which is the same as the bound (23), except for the newnotation in the denominator, which reminds us of thepossibly mixed initial state. The minimisation of Pm inthe numerator should now be performed over the spaceof all possible initial mixed states; however, the resultobviously coincides with the minimisation over thepure states only. Similar to the case discussed inSection 5.1, the inequality (28) is the exact bound; it isachieved by an optimal uncollapsing procedure, whichmaximises jCj.

If the initial state is pure, then the formalism of thissubsection is trivially equivalent to the formalism ofSection 5.1. It becomes more general in the case whenthe ‘actual’ initial state is mixed; for example, thishappens when the initial state has been in contact withan unmonitored environment or Socrates prepares astate by a blind random choice from a set of purestates. A more interesting case for the result (28) iswhen the measured system is entangled with anothersystem, which does not evolve by itself. Then theformalism can be applied to the compound system;however, the measurement probability Pm dependsonly on the reduced initial density matrix, traced overthe entangled second part. Therefore, in the entangledbipartite case the uncollapsing procedure restores thestate of the whole system, while the success probabilityPS is given by Equation (28) with rin being the reduceddensity matrix.

Another advantage of Equation (28) in comparisonwith Equation (23) can be seen by referring back to thepreliminary discussion with the contest between Platoand Socrates. In the derivation of both results theinitial state is the ‘actual’ initial state, which is knownto Socrates, but typically unknown to Plato. However,as we will prove below, Equation (28) can still be usedby Plato in a somewhat different sense: with rin beingunderstood as an averaged density matrix representinga distribution of possible initial states. In this case,Equation (28) gives the uncollapsing probabilityaveraged over this distribution. For example, if Platoknew that Socrates’ strategy is to prepare one of twopossible (nonorthogonal) states jc1i, jc2i, with prob-abilities P and 1� P, then he could find the averageuncollapsing probability in two ways. The first methodis that he could simply average the uncollapsingprobabilities of the two states (also taking into accountthe information acquired in the first measurement, seebelow). Alternatively, he could recall that the randomstate preparation described above is equivalent toconsidering the initial density matrix rin ¼ Pjc1i c1jþhð1� PÞjc2i c2jh , and then apply the result (28) to thisdensity matrix.6 In this way, in the absence of anyinformation, Plato could estimate his typical successrate by calculating (28) for a fully mixed state,invoking the principle of indifference [27].

In the general case the above statement, that bothways of computing the averaged uncollapsing prob-ability are equivalent, can be proven both logically andexplicitly. For the logical proof we notice that Plato’sjudgment of successful uncollapsing does not dependon whether or not Socrates knows the randomlypicked state; therefore, the average probability of thecases judged to be successful should be the same inboth situations (whether or not Socrates knows whatthe state is). Now let us also prove this statementexplicitly, thus checking that our formalism is self-consistent. Suppose the initial state is prepared bySocrates by choosing randomly from a set of initialstates r(k) with probabilities Pk (the most naturalcase is when initial states are pure, r(k) ¼ jcki hckj;however, this is not necessary). Then the bound forthe average probability of uncollapsing success P

ðavÞS is

the average of the bounds (28):

PðavÞS

Xk

minPm

PmðrðkÞÞP0k: ð29Þ

Notice, however, that P0k is the posterior probabilitydistribution given the result m, which is different fromPk. We may now invoke the classical Bayes rule3[26,27] to relate the posterior P0k to the prior Pk andthe conditional probability Pm(r

(k)) to have result mgiven state k, so that

P0k ¼PmðrðkÞÞPkP

~kPmðrð~kÞÞP ~k

: ð30Þ

Substituting Equation (30) into Equation (29) andusing

PkPk ¼ 1 in the numerator, we obtain

PðavÞS

minPmPk

PmðrðkÞÞPk¼ minPm

PmðrðavÞÞ; ð31Þ

where rðavÞ ¼P

krðkÞPk is the averaged initial state.This ends the proof that Equation (28) can be used foran unknown initial state, with rin being understood asthe average of all possible initial states.

5.3. Uncollapsing probability, information, and anirreversibility measure

We defined the success probability PS as a probabilityto uncollapse the post-measurement state rm. We nowwish to start counting the overall success probabilityP̃S from the time before the first measurement, so thatP̃S is the probability of the pair of events: measurementwith result m and then successful uncollapsing. UsingEquations (27) and (28) we easily find the relation

~PS ¼ PmðrinÞPS ¼ jCj2 ð32Þ



and the upper bound

~PS minPm: ð33Þ

Notice that P̃S is independent of the initial state. Whilethis property may seem somewhat surprising, we willsee later why it is rather obvious.

If we now wish to consider all possible results of thefirst measurement, and perform different uncollapsingprocedures for each measurement result, then the totalprobability of uncollapsing ~PtotalS is bounded as

~PtotalS X

mminPm; ð34Þ

and is also independent of the initial state. The bounds(33) and (34) are exact and reachable by optimaluncollapsing procedures. The bound (34) indicatesthat 1–Sm min Pm can be used as a measure of irrever-sibility (collapse strength) due to the measurementoperation.

Now let us discuss the relationship between theuncollapsing procedure and our knowledge of theinitial state. While uncollapsing is possible even if weknow nothing about the initial state of the system, atfirst glance it seems like we gain some knowledge aboutthe initial state in the process. This leads to thefollowing interesting paradox, initially consideredby Royer [37]. By doing both a measurement andunmeasurement, one can seemingly learn somethingabout the initial state without disturbing it. Then byrepeating the measurement þ unmeasurement processmany times, even though the probability of such anevent rapidly decreases to zero, the successful eventwould lead to essentially perfect knowledge of theinitial state, leaving the state itself perfectly intact! Onecould then violate a host of known results, such as theno-cloning theorem.

The resolution of the paradox lies in the fact thatthe pair of measurement and unmeasurement actuallybrings exactly zero information. Uncollapsing the statecan only occur when the information in the secondmeasurement exactly contradicts the informationgained in the first measurement, thus nullifying it.This can happen in weak quantum measurementsbecause there is uncertainty about the system in themeasurement result. It is to the extent that thisambiguity exists that it is possible to undo the weakmeasurement. Let us examine this in more detail.

We learn something about a pre-measurement statewhen the measurement result depends on the state. Themeasurement with result m brings some informationabout rin because the probability Pm(rin) dependson the initial state rin. The ability to successfully un-collapse the state also brings some information aboutthe initial state because the uncollapsing probability

PS ¼ jCj2/Pm(rin) also depends on rin. However,the collapse–uncollapse probability P̃S of observingboth the result m followed by a successful uncollapse isindependent of rin – see Equation (32). Therefore, thecombined effect of partial collapse and uncollapsebrings no information about the initial state.

More quantitatively, we can use the same frame-work as at the end of Section 5.2 in order to track theinformation gain during the procedure. Suppose Platoassigns an initial distribution Pk of possible initialstates as a statistical prior, to be updated as moreinformation comes in. The measurement with result mbrings in this information, so Plato updates his prior tothe posterior distribution P0k (see Equation (30)).Calculating in a similar way the distribution P00k afterthe pair of the measurement and unmeasurementresults, we find

P00k ¼PSðrðkÞÞP

0

kP~kPSðrð

~kÞÞP0~k¼

~PSðrðkÞÞPkP~k~PSðrðkÞÞP ~k

; ð35Þ

where PS(r) and P̃S(r) denote, respectively, the prob-abilities of uncollapsing (28) and a combined collapse–uncollapse pair (32) for the initial state r. However, aswe have already stressed, P̃S(r) is independent ofthe initial state r, and therefore cancels out of theexpression (35). This fact (and the normalisation of theprior {Pk}) restores the initial prior distribution,P00k ¼ Pk, and therefore Plato has learned nothing,thus avoiding the paradox. Reversing the logic, in orderto avoid the paradox, P̃S must be independent of theinitial state, as found in Equations (32) and (33).

5.4. Revisiting the previous uncollapse strategies

With these general and abstract results in hand, we cannow revisit the two examples discussed in Sections 3and 4. Notice though that the principle drawback ofthe general results is that while they give the idealsuccess probability, they do not provide the strategy asto how achieve the upper bound.

The measurement situation in the case of the phasequbit (Section 3.1) may be described with a two-outcome POVM, with elements En and Ey, where ndenotes the null result, and y denotes the affirmative(tunnelling) result. The POVM elements, given in thej0i, j1i basis are

En¼1 0

0 expð��tÞ

� �; Ey¼

0 0

0 1�expð��tÞ

� �; ð36Þ

with the obvious completeness relation En þ Ey ¼ 1.It is interesting to notice that while En ¼ M

ynMn

corresponds to a Kraus operator Mn (see below), nomeaningful Kraus operator My can be introduced for



the POVM element Ey, because in the case of atunnelling event the system leaves its two-dimensionalHilbert space and becomes incoherent (so that asingle Kraus operator cannot be introduced even inthe extended Hilbert space). However, this is notimportant for us because we are interested in the null-result case only. The null-result Kraus operator Mncan be found by comparing the phase qubit dynami-cal Equations (1), (2) and (3) with the generalstate disturbance rule (24); this gives Mn ¼ diag{1,exp (7�t/2) exp (7ij)}.

It is easy to find that the uncollapsing strategy ofSection 3.2 based on two null-result measurements isoptimal in the sense that its success probability PSreaches the upper bound (28). The numerator of (28)is the smallest eigenvalue of En, which is exp (7�t),while the denominator is Tr(Enrin) ¼ rin,00 þexp (7�t)rin,11. Therefore, the bound (28) coincideswith Equation (4), thus confirming the optimality ofthe analysed uncollapsing procedure.

The overall success probability P̃S, which is thejoint probability of two null results is

~PS ¼ ðrin;00 þ exp ð��Þtrin;11ÞPS ¼ exp ð��tÞ: ð37Þ

As expected (see Section 5.3) this probability does notdepend on the initial state. It is easy to see that it alsocoincides with the upper bound (33) for P̃S.

While the analysed double-null-result uncollapsingstrategy is optimal, an example of a non-optimaluncollapsing for a phase qubit was considered in [36].It was shown that if the measurement process isperformed simultaneously with Rabi oscillations, thenin the null-result case the initial state is periodicallyrestored. The non-optimality of uncollapsing for sucha procedure is due to measurement of an evolvingqubit, which corresponds to a sequence of manymeasurements; a similar reason for the non-optimalitywill be discussed at the end of Section 6.

Now let us turn to the DQD charge qubit exampleof Section 4. First, the qubit measurement dynamics[(5), (6) and (7)] can be related to the general POVM-type measurement formalism in the following way(similar to [32]). For a fixed time t the measurementresult m can be associated with the averaged QPCcurrent �I (or, equivalently, with the dimensionlessquantity r). The Kraus operator Mm in the measure-ment basis j1i and j2i then should be chosen asMm ¼ diag{[P1(�I)]1/2,[P2(�I)]1/2} in order to reproduceEquations (5), (6) and (7). Notice that Pm in Equation(25) now describes the probability density of the result�I instead of probability, because the measurementresult becomes a continuous variable.

We can now check to see if the ‘wait and stop’uncollapsing strategy of Section 4.2 is the optimal one

by comparing the general upper bound (28) for thesuccess probability PS with the result (15). Substitutingthe probabilities in the bound (28) with probabilitydensities, we find

PS min fP1ð�IÞ;P2ð�IÞg

P1ð�IÞrin;11 þ P2ð�IÞrin;22; ð38Þ

where �I corresponds to the measurement result r0. Thisbound coincides with Equation (15) because P1(�I)/P2(�I) ¼ exp(2r0), which proves the optimality of thediscussed ‘wait and stop’ strategy. (Surely, there arenumerous non-optimal uncollapsing strategies; forexample, by stopping the measurement after thesecond crossing of the origin by r(t).)

It is also instructive to not specify the result of thefirst measurement, but to find the total probability~PtotalS (see Equation (34)) that the initial qubit state canbe restored after a measurement for time t. This isgiven by averaging PS in Equation (15) over the resultsr0 with the corresponding weights (11). This averagingis technically easier using the form of Equation (38)and gives

~PtotalS ¼Z

d�IminfP1ð�IÞ;P2ð�IÞg¼ 1� erft

2TM

� �1=2 !;

ð39Þ

which depends only on the ‘strength’ t/TM ¼ t(DI)2/2SI of the first measurement, but not on the initialstate, as expected from the discussion in Section 5.3.Notice that the result (39) reaches the upper bound(34) because (15) reaches the upper bound (28).

6. Evolving charge qubit

Armed with the general uncollapsing formalismof Section 5, we now turn to the case of thefinite-Hamiltonian DQD measured by QPC, byincluding internal evolution of the qubit via a qubitHamiltonian,

HQB ¼ �ðe=2Þsz þHsx; ð40Þ

where e is the energy asymmetry between the quantumdot levels, and H is the tunnel coupling between thedots. In this case Equations (5)–(7) are no longer validand should be replaced by the Bayesian equations [13](in Stratonovich form [38])

_r11¼� _r22¼�2HImr12þr11r222DISI½IðtÞ� I0�; ð41Þ

_r12 ¼ ier12 þ iHðr11 � r22Þ � ðr11 � r22Þ

� DISI½IðtÞ � I0� r12; ð42Þ



where I(t) is the QPC current,

IðtÞ ¼ r11ðtÞI1 þ r22ðtÞI2 þ xðtÞ; ð43Þ

containing white noise x(t) with spectral density SI,and we use �h ¼ 1. These evolution equations arenonlinear and not very simple to deal with. To discussthe undoing of a continuous measurement, it is moreconvenient to use a non-normalised density matrix s,which has an advantage of dealing with linearequations.

We rewrite Equations (41)–(42) in the form

r ¼ s=Trs; ð44Þ

_s11 ¼ �2H Ims12 � s111

SI½IðtÞ � I1�2; ð45Þ

_s22 ¼ 2H Ims12 � s221

SI½IðtÞ � I2�2; ð46Þ

_s12 ¼ ies12 þ iHðs11 � s22Þ

� s12½IðtÞ � I0�2

SIþ ðDIÞ

2

4SI

( ); ð47Þ

so that s(0) ¼ r(0), while the ratio s(t)/r(t) decreaseswith time and is equal to the normalised probabilitydensity of the corresponding realisation of the detectoroutput I(t0), 0 t0 t.7 In the language of generalquantum measurement this formulation correspondsto omitting the denominator in Equation (24). Noticethat we still consider an ideal detector, so an initiallypure state remains pure, js12 j2 ¼ s11 s22.7

A casual inspection of Equations (45)–(47) showsthat they are seemingly not well defined because theterms [I(t) – I0,1,2]

2 contain the term x(t)2 ¼ ? (fromthe relation hx(t)x(0)i ¼ (SI/2)d(t)). This divergence isartificial because there will always be a small correla-tion time T of the noise and/or a finite detectorbandwidth B (corresponding to T ¼ 1/4B), so therewill be a large but finite constant C ¼ hx(t)2i ¼ SI/4Tcontained in terms of the form [I(t) 7 I0,1,2]

2. It iseasy to see that Equations (45)–(47) do not changeif we subtract the same constant from these terms[I(t) 7 I0,1,2]

2! [I(t) 7 I0,1,2]2 7 C. This can be shownby considering another unnormalised density matrixZ ¼ s exp (t/T). Writing the linear Bayesian equations(45)–(47) in the form _sij ¼ fij½s�, the equations trans-form to _Zij ¼ fij½Z exp ð�t=TÞ� exp ðt=TÞ þ Zij=T underthe change of variables. The unnormalised Bayesianequations are linear in the density matrix elementssij, so the exponential factors cancel out. The newequations are thus the same as the old ones with aconstant C ¼ SI/4T subtracted from the [I(t) 7 I0,1,2]2terms. The unspecified constant T in the density matrix

transformation may be chosen to be the short corre-lation time T discussed above, thus cancelling thelarge term and making Equations (45)–(47) welldefined. The only price to be paid for this transforma-tion is an altered normalisation, that will cancel in thenormalised density matrix (44).

For a particular realisation of the detector outputI(t0), 0 t0 t, Equations (45)–(47) define a linearmap s(0)! s (t), corresponding to a particular Krausoperator Mm (which, therefore, can be denoted asM{I}). For the uncollapsing we have to realise the map,corresponding to the inverse Kraus operator CM�1fIg(see Section 5.1). It is obvious that in contrast to thecase of the non-evolving qubit discussed in Section 4,this cannot be done by simply continuing themeasurement and waiting for a specific result. Thereason is that now the map is characterised by six realparameters (eight parameters for a linear operatorCM�1fIg with neglected overall phase and normalisa-tion), instead of one parameter for the non-evolvingcase (see Equations (7) and (9)). We will discuss a littlelater how the six-parameter uncollapsing procedurecan be realised explicitly. Before that we discuss how tofind the operator M{I} in a more straightforward way,from Equations (45)–(47).

Let us consider only the evolution of (unnorma-lised) pure states jc(t)i ¼ a(t)j1i þ b(t)j2i, so thats ¼ jci hcj. Then Equations (45)–(47) can be rewrittenas

_a ¼ þi e2a� iHb� a 1

2SI½IðtÞ � I1�2; ð48Þ

_b ¼ �i e2b� iHa� b 1

2SI½IðtÞ � I2�2; ð49Þ

where the infinite part of I2 (t) can be cancelled in thesame way as discussed above. The linearity of theseequations guarantees that for any given realisation ofI(t), it is sufficient to solve (48) and (49) for the initialstates j1i and j2i in order to find the solution for anarbitrary initial state of the qubit. Defining v1 ¼a1(t)j1i þ b1(t)j2i as the solution of (48) and (49) forinitial state j1i, and v2 ¼ a2(t)j1i þ b2(t)j2i as thesolution of (48) and (49) for initial state j2i, we canwrite the solution for an arbitrary initial state jcini ¼jc(0)i ¼ aj1i þ bj2i as jc(t)i ¼ av1 þ bv2. Therefore,the Kraus operator M{I} for a given realisation of I(t),in the j1i, j2i basis is

MfIg ¼a1ðtÞ a2ðtÞb1ðtÞ b2ðtÞ

� �: ð50Þ

For the uncollapsing we need to apply the Krausoperator CM�1fIg, which maps the state v1 onto Cj1i andthe state v2 onto Cj2i. The reason why we need a non-unitary transformation is that the vectors v1 and v2 are



in general non-orthogonal and have different norms.Geometrically, such a transformation can be doneby using a relative shrinking or stretching of twoorthogonal axes (found for a given M{I}), which wouldmake v1 and v2 orthogonal and equal in norm,followed by a unitary transformation (this wouldcorrespond to the decomposition of the form UE1/2 –see Section 5.1). However, for a practical realisation ofuncollapsing it is most natural to use the shrinking orstretching of the axes j1i and j2i via a continuousQND measurement with the QPC in the way con-sidered above for a non-evolving qubit. In this case theuncollapsing procedure can be done in three steps (seeSection 5.1): unitary evolution V, continuous QNDmeasurement (where the qubit Hamiltonian is turnedoff, e ¼ H ¼ 0) described by a diagonal matrix L, anda final unitary operation U (in the notation of Section

5.1,V corresponds toVyLUym, andU corresponds toU

yL).

These operators should satisfy

ULV ¼ CM�1fIg; ð51Þ

and it is easy to find U, L, and V explicitly byrecognising Equation (51) as a singular value decom-position of the operator CM�1fIg (recall here that L isdiagonal; also notice that the standard form for thesingular value decomposition is slightly different, withV denoted as V{).

To find L explicitly, we notice that

C�2MyfIgMfIg ¼ UL

�2Uy; ð52Þ

which is simply the diagonalisation of C�2MyfIgMfIg.

Therefore,

L¼C l�1=2� 0

0 l�1=2þ

!or L¼C l

�1=2þ 0

0 l�1=2�

!; ð53Þ

where

l� ¼jjv1jj2 þ jjv2jj2

2

� jjv1jj2 � jjv2jj2

2

!2þjv1 v�2j

2

24

351=2

ð54Þ

are the eigenvalues of the operator MyfIgMfIg and the

vectors vi are defined above Equation (50). TheCauchy–Schwartz inequality, jv1 v�2j

2 jjv1jj2jjv2jj2,guarantees the non-negativity of l7. (The notationv1 v2 is used for the inner product hv2jv1i of v2 and v1.)

To find U, we use Equation (52) again and see thatthe columns of U are composed of the eigenvectors ofMyfIgMfIg (the sequence of columns depends on the

choice in Equation (53)). Finally, V is given by

V ¼ Uy CL�1M�1fIg. For brevity we will not show thematrices U and V explicitly.

In the physical realisation of the uncollapsingprocedure the measurement step L can be performedin exactly the same way as in Section 4.2. ComparingEquation (53) with Equations (9) and (7), we see thatthe continuous measurement by the QPC should bestopped when the dimensionless measurement resultr(t) reaches the value

r1¼ lnðlþ=l�Þ1=2> 0 or r2¼ lnðl�=lþÞ1=2< 0; ð55Þ

for the first and second choice in Equation (53),respectively (the choice should be made beforehand,since it determines operation V). As previously men-tioned, the constant C is not important here becausethe physical state is always normalised. The procedurefails if the desired result is not reached during thecontinuous measurement.

The unitary operations V and U can be practicallyrealised in three substeps each: z-rotation on the Blochsphere by applying non-zero energy asymmetry e forsome time, y-rotation by applying non-zero tunnellingH, and then one more z-rotation. However, the lastz-rotation of V and the first z-rotation of U are simplyadded to each other (since L does not change therelative phase of the state components or, equivalently,the azimuth angle on the Bloch sphere). The corre-sponding trivial degree of freedom can be eliminated,for example, by realising the operation V in only twosubsteps, without the second z-rotation.

Let us count the number of real parameters,characterising the uncollapsing procedure. Since Vand U together provide 2 6 3 – 1 ¼ 5 parameters, andthe desired result r in the measurement step adds onemore parameter, the overall number of parameters is 6.As expected, this is exactly the needed number ofparameters characterising an arbitrary Kraus operatorfor the qubit (neglecting normalisation and overallphase). Let us also mention the fact from linear algebrathat the singular value decomposition (51) is uniquein the non-degenerate case (lþ 4 l 7 4 0), up to thepermutation of singular values (corresponding to thechoice in Equation (53)) and arbitrary phase factors incolumns of U, with compensating changes in V (thiscorresponds to the previously discussed compensationof the z-rotations).

Now let us discuss the probability PS of thesuccessful uncollapsing. From the general theory ofSection 5 (Equation (23)), it is bounded from aboveby a fraction, PS minP{I}/P{I}(jcini), in which thedenominator is the probability density of the givenrealisation I(t) for the initial state jcini ¼ aj1i þ bj2i(with jaj2 þ jbj2 ¼ 1), while the numerator is thisprobability minimised over all initial states. So, the



denominator is given by the squared norm of the finalstate jc(t)i ¼ av1 þ bv2,

PfIgðjciniÞ ¼ jjav1 þ bv2jj2; ð56Þ

while the numerator is given by minimising (56) overall normalised initial states. It is easy to see that thisminimum is equal to the minimum eigenvalue l7of the operator M

yfIgMfIg, given by Equation (54);

therefore,

PS l�

jjav1 þ bv2jj2: ð57Þ

Converting this result into the language of densitymatrices and simultaneously generalising it to anarbitrary initial state rin, we obtain the bound

PS

jjv1jj2þjjv2jj22 �

jjv1jj2�jjv2jj22

� �2þ jv1 v�2j

2

� �1=2rin;11jjv1jj

2þrin;22jjv2jj2þ2Re½rin;12 v1 v�2�

; ð58Þ

in which the numerator is the explicit expression (54)for l7. It is easy to check that this result reduces tothe bound (38) in the non-evolving case, in whichv1 ¼ ([P1(�I)]1/2,0)T and v2 ¼ (0,[P2(�I)1/2)T.

The uncollapsing procedure discussed in thissection is optimal in the sense that it correspondsto the upper bound of Equation (58). To prove thisstatement, instead of calculating PS explicitly, let ususe the fact (see Section 5.3) that the product PS P{I}cannot depend on the initial state. Therefore, it issufficient to prove the optimality of PS only for oneinitial state. Let us choose the state jcini that is theeigenvector of M

yfIgMfIg, corresponding to the eigen-

value l–. Then after the first measurement (operatorM{I}) and the unitary operation V it is transformedinto one of the basis states (j1i or j2i for the first orsecond choice in (53), respectively). Recall that for thenon-evolving (QND) measurement case, r(t) ! ? forthe initial state j1i, while r(t) ! 7? for the initialstate j2i. The crossing thresholds (55) indicate thatthe measurement L is always successful because r(t)necessarily crosses the desired value (which is positivefor j1i and negative for j2i, as discussed above).Therefore, the uncollapsing success probability forthis special state is 100%, that is equal to the upperbound (58). As mentioned above, the optimality ofthe procedure for this special state also proves theoptimality for any initial state.

Obviously, an uncollapsing procedure can also benon-optimal. As an example, let us consider a proce-dure which realises the desired mapping {v1, v2} !{Cj1i,Cj2i} using two measurements instead of one.The goal of the first measurement is to map {v1, v2}

into an orthogonal pair of vectors, while the goal of thesecond measurement is to equalise their norms,keeping them orthogonal. The first goal can beachieved by stretching/shrinking of the Hilbert spacealong any axis u of the form v1 þ cv2 (v1v2)/jv1v2j withan arbitrary positive real number c (it is easy tovisualise this procedure of making two vectorsorthogonal by assuming the space of real vectors, forwhich the axis u is geometrically in between v1 and v2;the same geometrical idea works for complex vectors).We recall that measurement for a non-evolving qubit(Section 4.1) stretches (squeezes) the j1i axis, whilesqueezing (stretching) the j2i axis. Therefore, the firstgoal can be achieved by a unitary operation whichrotates u into j1i, followed by a continuous measure-ment (with a QPC) of a non-evolving qubit, to bestopped when the mapped vectors become orthogonal.After the vectors {v1, v2} are transformed into anorthogonal pair by the (successful) first measurement,the second part of the procedure should stretch/shrinkthe 2D Hilbert space along the resulting vectors tomake them equal in norm. This can be done similarly,by a unitary rotation and partial measurement of anon-evolving qubit. Finally, another unitary operationcan be used to map the resulting pair of vectors into{Cj1i, Cj2i}, thus completing the uncollapsing proce-dure. Notice that both measurements are performed inthe ‘wait and stop’ manner, and both measurementsshould be successful to realise the uncollapsing. Whilethe successfully uncollapsed state is still perfect in thisprocedure, the probability of success is lower than thebound (58). To prove this non-optimality, let us againuse the initial eigenstate jcini, which corresponds toeigenvalue l7, so that the bound (58) is 100%. Thenthe probability of success for the first measurement isin general less than 100% (it is 100% only for onespecific axis discussed previously, while here weconsider a range of possible axes by allowing c tovary). Thus, the success probability is less than 100%for this special state, and therefore PS is below thebound (58) for any initial state.

7. General procedure for entangled charge qubits

Let us present an explicit procedure [14] which can beused in principle to undo an arbitrary measurementMm of any number N of entangled qubits with maxi-mum probability of success PS. For simplicity weconsider double-quantum-dot charge qubits andassume that any unitary transformation can be usedin the procedure. If the operator Mm was produced bya one-qubit measurement, and other entangled qubitswere not experiencing a Hamiltonian evolution, thenthe formalism of Section 4.1 is essentially unchanged[40], and uncollapsing of the measured qubit leads to



the restoration of the whole entangled state. If theoperator Mm was produced by a one-qubit measure-ment, while other qubits were evolving in a unitaryway but not interacting with the measured qubit, thenuncollapsing is also easy: we should uncollapse themeasured qubit in the usual way (Section 4.2) andshould apply the inverse unitary transformation for theother qubits. In this section, however, we do notconsider these simple special cases; the goal is to undoan arbitrary Kraus operator Mm.

Let us decompose Mm asMm ¼ UmE1=2m (see Equa-tion (18)). Reversing the unitary operation Um can bedone in the regular Hamiltonian way, so the nontrivialpart is undoing the E

1=2m operator. We recall the

diagonalisation of Em is given by Em ¼P

i pðmÞi jiihij

with vectors jii forming an orthonormal basis. Asdiscussed in Section 5.1, for the optimal uncollapsingwhich maximises the success probability, we have toperform a procedure corresponding to the measure-ment operator ~L ¼ ðmin jpðmÞj Þ

1=2 E�1=2m which is also

diagonal in the basis jii with correspondingmatrix elements ~Lii ¼ hij ~Ljii ¼ ½ðmin jpðmÞj Þ=p

ðmÞi �

1=2,all of which are between 0 and 1. Given N qubits, iranges from 1 to 2N. Notice that L is obviouslyHermitian.

Our procedure is to realise ~L with a sequence ofnull-result measurements and unitary operations.Shown in Figure 3 is an illustration of the physicalset-up that is used for the measurements: a QPC(tunnel junction) capacitively coupled to N non-evolving DQD charge qubits. We assume that theQPC is tuned to a highly nonlinear regime, for whichno electron can tunnel across the QPC barrier onexperimentally relevant time-scales unless all qubits arein the state j1i. We name this multi-qubit statej1i:j1,1, . . . ,1i. Such a regime is possible because ofthe exponential dependence of the tunnelling rate onQPC barrier height, while the barrier height dependslinearly on the states of the coupled qubits. Of course,this regime is not quite realistic; however, we discussthe procedure in principle. We also assume that

even for the N-qubit state j1i, the rate g of electrontunnelling through the QPC is rather low, so that wecan distinguish single tunnelling events (technically,this would require an additional single-electron tran-sistor). If we perform the measurement during timet and see no tunnelling through the QPC, thensimilarly to the case of Section 3.1, the correspondingnull-result Kraus operator Mn shrinks the j1i axis ofthe Hilbert space by the factor exp(7gt/2), whileleaving all perpendicular axes unchanged. For thematrix elements this means h1jMnj1i ¼ exp ð�gt=2Þ;hc?j jMnjc

?j 0 i ¼ djj 0; hc

?j jMnj1i ¼ h1jMn þ c

?j i ¼ 0,

where we introduced a set of 2N 7 1 states jc?j ispanning the subspace orthogonal to j1i.

The general strategy to implement the operator ~L isthe following. We first note that in the basis jii thatdiagonalises ~L, this diagonal matrix can be representedas a product of 2N diagonal matrices, where each termin the product has all diagonal entries as 1, exceptthe ith entry: diag {1, 1, . . . , ~Lii, . . . ,1}. Each of thesematrices may be interpreted as a separate Krausoperator that can be sequentially implemented. Thus,the explicit physical procedure consists of 2N steps,each of which has three substeps. First, we apply aunitary transformation U1 which transforms the firstbasis vector ji ¼ 1i into the state j1i. Then theevolution of all qubits is stopped, and the detector isturned on for a time t1. This time is chosen so thatthe null-result Kraus operator L(1) has the desiredmatrix element h1jL(1)j1i ¼ L11; this condition yieldst1¼ –2g–1 ln ~Lii. The measurement is then followed bythe reverse unitary, U

yi , to take the state j1i back to

state ji ¼ 1i. This three-substep procedure is thenrepeated for i ¼ 2,3, . . . ,2N, sequentially transformingthe state jii to j1i with unitary Ui, and performingmeasurement with the detector for a time ti ¼ –2g–1 ln~Lii, followed by the reverse unitary, U

yi . This sequence

of steps decomposes the uncollapsing operator ~L as

~L ¼ Uy2NLð2

NÞU2N . . .Uy2Lð2ÞU2U

y1Lð1ÞU1: ð59Þ

The uncollapsing procedure is successful only if therewere no tunnelling events in the QPC. By construction,the success probability PS for this procedure maximisesthe general bound (28).

The success probability PS for the uncollapsing

process rm ! rin with rin ¼ ~L~r ~Ly=Trð ~Ly ~L~rÞ and~r ¼ UymrmUm, can be calculated as

PS¼Trð ~Ly ~L~rÞ¼Xi

~L2

ii~rii¼Xi

~rii expð�gtiÞ; ð60Þ

where ~rij are the matrix elements of ~r in the basis jii,which diagonalises ~L. We may also find this resultfrom another perspective by realising that the success

Figure 3. Schematic set-up for uncollapsing of N entangledqubits. The tunnel junction detector (QPC) is in a stronglynonlinear regime, so that an electron can tunnel through itwith rate g only when all qubits are in state j1i.



probability is simply the product of the null-resultprobabilities p

ðiÞS of all 2

N measurements,

PS ¼Y2Ni

PðiÞS ; p

ðiÞS ¼

Pij¼1~rjj exp ð�gtjÞ þ

P2N

j¼iþ1 ~rjjPi�1j¼1 ~rjj exp ð�gtjÞ þ

P2Nj¼iþ1 ~rjj

;

ð61Þwhere the expression for p

ðiÞS comes from comparing

the traces of unnormalised density matrices after eachof 2N steps of the procedure. It is instructive to showexplicitly that this expression for p

ðiÞS is equal to the

expected expression

pðiÞS ¼ 1� ½1� exp ð�gtiÞ�~r

ðiÞii ; ð62Þ

in which ~rðiÞ is the normalised density matrix before theith step of the procedure (after i 7 1 null-result steps).This can be done if we prove the relation

Yki¼1½1� ~rðiÞii ð1� expð�gtiÞÞ� ¼ 1�

Xki¼1½1� expð�gtiÞ�~rii;

ð63Þ

(notice that the right-hand side of this equation isequal to the numerator in Equation (61) withsubstitution k ! i). Equation (63) can be proven byinduction using the relation ~rðiÞii ¼ ~rii=

Qi�1j¼1 ½1�

½1� exp ð�gtjÞ�~rðjÞjj �, which can be easily derived recur-sively, ~rðjÞii ! ~r

ðjþ1Þii , starting from ~r

ð1Þii ¼ ~rii. In this way

we show consistency between the null-result probabil-ities given by Equations (61) and (62), permitting thecalculation of PS in two independent ways.

Let us mention again that the uncollapsing proce-dure considered in this section reaches the upperbound (28) for the success probability PS, that can beseen both by construction and explicitly.

8. Recent developments in wavefunction uncollapse

Before concluding, we wish to give a summary ofsome interesting recent developments in this area ofresearch. We will briefly discuss two theory proposalsand one experiment.

8.1. Spin qubit

The examples given above mainly concern quantumdot charge qubits. It is a natural question if a similarkind of partial collapse/uncollapse can be carried overto spin qubits. An analysis of this situation was carriedout by Trauzettel, Burkard, and one of the authors[41]. There it was shown how an uncollapse measure-ment can be realised using a scheme similar to therecent experiments by Koppens et al. [42]. The essentialidea of the spin-qubit experiments [42–45] is tomanipulate and measure the spin of a single electron

through the charge degree of freedom. This techniquecircumvents the otherwise difficult problem of control-ling the weakly interacting spin. While we refer thereader to [41–45] for the details, we will give asimplified thumb-nail sketch of the physics here.

The qubit is encoded with two electron spins,where each electron is confined in a separate quantumdot. In contrast to our charge qubit discussion, thesedots are open, with electrons able to enter and leave.Electrical bias is applied across this double quantumdot leading to charge transport. Electrons can tunnelsequentially, but spin blockade [46] restricts transportto situations where the two electrons form a spinsinglet (0,2)S on the right dot while the spin triplet(0,2)T is outside the transport energy window due tothe large single quantum dot exchange energy (here(n, m) refers to n electrons on the left dot and melectrons on the right dot). This blockade physicsprovides an interesting initialisation procedure of thequantum register – when the single-electron currentstops flowing, we are confident that the two-electronstate is in a (1,1)T state, because in the absence ofspin flip processes, the tunnelling transition to the (0,2) state is forbidden. From this configuration, it ispossible to manipulate the system by applyingelectron spin resonance pulses [42], transitioning thestate to have overlap with the singlet state. Thus, theelectron on the left dot may tunnel (with rate �) tothe right dot and exit the system, giving rise to asmall electrical current at the drain when this processis repeated many times. Of course, this will happenwith some probability controlled by the overlap of thestate with the singlet.

Drawing on our experience with the phase qubit(see Section 3), it is clear how to devise a weakmeasurement experiment and an uncollapsing experi-ment: the allowed transition can be permitted fora time of one’s choosing and then forbidden bydetuning the energy levels with a voltage pulse to oneof the quantum dot’s gates. In this way one canweakly probe the two-electron state, and in the null-result case (no single electron tunnelling) partiallycollapse it to the triplet subspace. In order to proposethe uncollapse part of the experiment, it is easiest toconsider the case when the surrounding nuclear spins[42] quickly admix the singlet state with a triplet state,permitting the two-qubit state to encode one effectivequbit: parallel or anti-parallel spins. The weakmeasurement technique described above will thenpartially collapse the state toward the parallel stateunder a null-measurement (no single electron tunnel-ling). If now a p-pulse is applied to one of the spinswith electron spin resonance, followed by a secondnull-measurement, this was shown to uncollapse thestate of the effective qubit [41].



8.2. Optical polarisation qubit

Another interesting development is the experimentalimplementation of wavefunction uncollapse for opticalqubits using the polarisation degree of freedom ofsingle photons by Kim et al. [22] The weak measure-ment was implemented by passing the photon througha glass plate oriented at the Brews

contemporary physics uncollapsing the wavefunction by ...first published on: 12 february 2010 to...

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