PHYSICAL REVIEW A 84, 052303 (2011)

Entanglement generation by interaction with semiclassical radiation

Amir Leshem and Omri GatRacah Institute of Physics, Hebrew University of Jerusalem, Jerusalem IL-91904, Israel

(Received 5 July 2011; published 4 November 2011)

We address a fundamental issue in quantum mechanics and quantum information theory—the generation of anentangled pair of qubits that interact solely through a third, semiclassical degree of freedom—in the frameworkof cavity quantum electrodynamics. We show that finite, although not maximal, entanglement is obtainable inthe classical limit, at the price of a diverging effective interaction time. The optimal atomic entanglement derivesfrom a trade-off between the atomic entanglement in a sub-wave packet and the purity of the atomic state.Decoherence by photon loss sets an upper limit on the degree of excitation of the cavity mode, beyond which theachievable entanglement decreases as the inverse mean photon number to the sixth power.

DOI: 10.1103/PhysRevA.84.052303 PACS number(s): 03.67.Bg, 03.65.Sq, 42.50.Pq

I. INTRODUCTION

Field-atom entanglement is one of the hallmarks of stronglyinteracting cavity quantum electrodynamics (CQED) systems.This fundamental process is often used as a building block inthe formation of entangled states of two or more atoms thatserve as a resource for quantum information processing. Anotable example is the generation of a two-atom Bell state bymeans of consecutive interaction with a resonant microwavecavity mode prepared in the vacuum state [1]. In effect, thecavity mode stores the entanglement with one atom, which isthen transferred to the second atom. The question addressedin this work is the following: Can entanglement be generatedwhen the initial field state becomes semiclassical? On the onehand, quantum fluctuations persist also in the semiclassicallimit while, on the other hand, the correspondence principlestates that, in this limit, classical physics should be recovered,where the notion of entanglement is not even defined.

The semiclassical states of the field mode are wavepackets that are localized in phase space, where the canonicalcoordinates are the field quadratures, such as coherent statesand squeezed coherent states, and have a large mean photonnumber. When a two-level atom interacts with a coherent-statewave packet, its polarization undergoes Rabi oscillationswhose amplitude exhibits collapse-revival dynamics [2] asa consequence of the splitting of the initial wave packetinto two mutually orthogonal sub-wave packets and of theirperiodic collisions [3,4]. Each sub-wave packet is a product ofa squeezed field state and an atomic state. After splitting, theatom-field system is highly entangled, but slow atomic stateevolution turns the field-atom state again into a product state[3,4], and the field state is a superposition of two well-separatedwave packets, sometimes called a Schrodinger cat state. Thissemiclassical dynamics can be described as a flow in a doublephase space [5].

Here we study a field mode interacting consecutively withtwo two-level atoms. The interaction of the field wave packetwith the first atom causes it to split, as explained, and tobecome entangled with the atom. We show in Sec. II that,when the interaction with the second atom begins, each ofthese sub-wave packets splits again, and the atom-field-atomsystem becomes a superposition of four wave packets; seeFig. 1. Atom-atom entanglement can only be obtained if twoor more of the sub-wave packets overlap, and this occurs when

the normalized interaction times of the field with the two atomsare equal. The system state is then, in general, a superpositionof three wave packets, and the atom-atom state is an entangledmixed state.

We show in Sec. III that it is possible to generateentanglement that reaches a finite, less-than-maximal limit,when the photon number tends to infinity, although the timerequired to generate the entangled state diverges in this limit.The optimal interaction time is determined as a tradeoffbetween the mixing of three pure atomic states and the degreeof entanglement of one of them, as shown in Figs. 2 and 3.We consider the effects of photon loss in Sec. IV and showthat the entanglement generation is degraded by dephasing ofthe sub-wave packets, and that a coherent superposition ofwave packets cannot be maintained when the mean photonnumber is larger than the inverse decoherence rate to the 2

3power. Beyond this point, entanglement generation proceedsonly through small quantum fluctuations and it decreases asthe inverse of the mean photon number to the sixth power; seeFig. 4.

The analysis demonstrates that a semiclassical preparationcan serve as a mediator for appreciable entanglement ofatoms, if the decoherence is weak enough. For this purposeit is necessary that the field becomes entangled with theatoms and evolves into a macroscopic superposition state. Incontrast, when decoherence is too strong to allow wave-packetsplitting, atom-atom entanglement is generated only throughsmall quantum fluctuations and is therefore weak.

The resonant interaction of a field mode in excited stateswith two atoms has been investigated theoretically in [6–8]for simultaneous interaction and [9,10] for consecutive in-teraction, revealing wave-packet splitting, slow atomic stateevolution, and collapse-revival dynamics of entanglementsimilar to those observed in one-atom–field systems. Theoff-resonance interaction of a field wave packet with two atomswas studied experimentally and theoretically in [11,12] as ameans of measurement of the cavity decay rate and its effecton the dephasing of Schrodinger cat states. The steady stateentanglement of two atoms in a highly excited leaky cavitywas studied in [13]. Our focus is the process of entanglementof two two-level systems by interaction with an intermediarythat tends to the semiclassical limit, for which CQED providesa prominent realization.

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AMIR LESHEM AND OMRI GAT PHYSICAL REVIEW A 84, 052303 (2011)

6 4 2 0 2 4 6

6

4

2

0

2

4

6

q

p

5 0 5

5

0

5

q

p

FIG. 1. (Color online) Contour plots of the Wigner function of the field state Tratoms |2〉〈2|, for unequal effective interaction times γ1 = π

2 ,γ2 = 2π

7 (left) and equal interaction times γ1 = γ2 = π

2 (right), numerically calculated for α = 4. Bright and dark shades signify positive andnegative values, respectively. In the left panel the arrows signify the direction of phase-space rotation of the four sub-wave packets. The Wignerfunction of the initial coherent-state wave packet is circularly symmetric—the deformation of the four sub-wave packets in the left panel, andthe two flank sub-wave packets in the right panel is a consequence of squeezing.

Field-mediated atom-atom entanglement is often carriedout in the highly detuned Raman regime [14,15] where the fieldis only virtually excited to avoid field-induced decoherence.In a sense this is the opposite limit to the one studiedhere, and indeed we show that the atom-atom entanglementis very sensitive to dephasing of the field wave packet.Furthermore, although it is often argued that in the Ramanregime entanglement is independent of the field state, theRaman regime entails an arbitrarily large detuning when thefield state photon number tends to infinity.

II. ATOM-FIELD-ATOM INTERACTION

We study the dynamics of two two-level atoms interactingconsecutively on resonance with a single electromagneticmode. Modelling the interaction by the Jaynes-Cummings

Hamiltonian in the rotating wave approximation, the Hamil-tonian is H (t) = H1, 0 < t < t1, and H2, t1 < t < t1 + t2,where

Hk = hω(a†a + σ†k σk) + h�k(a†σk + aσ

†k ), (1)

where ω is the frequency of the electromagnetic mode withenergy states |n〉, n = 0,1, . . . created by a†, and coherentstates |α〉, hω is the level spacing of the two-level atoms withenergy states |g〉k,|e〉k (atom k) raised by σ

†k , and h�k are the

field-atom interaction energies. The system is prepared in aproduct state |α〉 ⊗ |g〉1 ⊗ |g〉2.

When the field interacts with the first atom, the energy statesare polariton states |n〉± ≡ 1√

2(|n − 1〉 ⊗ |e〉1 ± |n〉 ⊗ |g〉1)

with energies h(nω ± √n�1) and the absolute ground state

|0〉 ⊗ |g〉1. The field-atom state evolves into a superposition of

0.0 0.2 0.4 0.6 0.8 1.0

0.1

0.2

0.3

0.4

0.5

Ec

0.2 0.4 0.6 0.8 1.0

0.4

0.6

0.8

1P

FIG. 2. (Color online) Entanglement entropy Ec of the center wave-packet two-atom state |c〉 (left) and the purity P of the full two-atomstate ρa (right) as a function of phase-space rotation angle γ , shown for α = 4 (thin red), α = 8 (medium green), and asymptotically for|α| → ∞ (thick blue). The initial rise and later drop in the degree of entanglement of the full two-atom state can be understood as a tradeoffbetween the increase in P and the decrease of Ec. P (α → ∞) is discontinuous at γ = 0,π and these points are excluded from the graph.

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ENTANGLEMENT GENERATION BY INTERACTION WITH . . . PHYSICAL REVIEW A 84, 052303 (2011)

0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

Ef

0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

Entanglement

FIG. 3. (Color online) (left) Entanglement of formation Ef of the two-atom state ρa as a function of the phase-space rotation angle γ ,shown for α = 4 (thin red), α = 8 (medium green), and asymptotically for |α| → ∞ (thick blue). (right) Entanglement of formation (upperblue) and negativity (lower violet) as a function of γ for |α| → ∞. The two entanglement monotones attain their maxima (shown by verticaldashed lines) at close but different values of γ , showing that they are nonequivalent.

two products of a squeezed coherent state and an atomic state[2–4]. Expressing the Hamiltonian in terms of the polaritonnumber operator n and the projections P± on the ± subspaces[5], H1 = H+P+ + H−P− where H± = h(nω ± √

n�1), theinitial state is 1√

2(|α〉+ − |α〉−), where |α〉± = ±√

2P±(|α〉 ⊗|g〉1) and the field-first atom state at time t1 is

|1〉 = |1+〉 − |1−〉, |1±〉 = 1√2e− i

hH±t1 |α〉±. (2)

For large |α| the dynamics generated by H± is wellapproximated by classical dynamics in the polariton phasespaces—phase spaces corresponding to the ± sub-Hilbertspaces, where the canonical coordinates q,p are the fieldquadratures. For convenience, we join |0〉± states to the± subspaces (respectively) with occupations that remainexponentially small throughout. The initial states in both phasespaces are then coherent-state Gaussian wave packets, andthey evolve according to the rules of semiclassical phase-space dynamics [16]. In this approximation the wave packetevolution is determined completely by classical data generatedby the classical Hamiltonians

H(cl)± = 1

2(q2 + p2 − h)ω +

√h

2(q2 + p2 − h)�1, (3)

which are the Weyl phase-space representations of the quan-tum Hamiltonians H±, respectively. H

(cl)± generate nonlinear

oscillations; that is, rotation in phase space with an amplitude-dependent frequency.

The wave-packet propagation consists of three parts [16]:A phase-space translation from the initial point (q,p), α =

1√2(q + ip) to the final point (q±,p±), α± = 1√

2(q± + ip±)

determined by the classical trajectory of the center of the wavepacket, squeezing with squeeze parameters ξ± determinedby the phase-space deformation generated by the nonlinearoscillations, and an overall phase factor e−iφ± determined bythe classical action of the classical orbits. The evolved wavepackets are squeezed coherent states

|1±〉 = e−iφ±|α±,ξ±〉, (4)

where

|α,ξ 〉± = eαb†−α∗b exp{

12

[ξ ∗b2 − ξ (b†)2

]}|0〉±, (5)

with b being the polariton annihilation operator defined withthe usual properties b|n〉± = √

n|n − 1〉±, [b,b†] = 1. Thevalues of the wave-packet parameters in a frame rotating

0.0 0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

E f

0 0.04 0.08 0.12

4 10 5

8 10 5

12 10 5

E f

FIG. 4. (Color online) Entanglement of formation Ef of the two-atom state as a function of phase-space angle γ for, from top to bottom,

y ≡ λ|α|32 ( 1

�1+ 1

�2) = 0,0.15,0.4,0.7,1 (left) and y = 0,4,5,6,7,8 (right), where λ is the cavity loss rate and �1 and �2 are the atom-field

interactions. The dashed line in the right panel traces the maxima of Ef over γ for given values of y.

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AMIR LESHEM AND OMRI GAT PHYSICAL REVIEW A 84, 052303 (2011)

with angular speed ω are φ± = φ(±γ1), α± = α(±γ1), andξ± = ξ (±γ1), where

φ(γ ) = γ

2|α|2 − 1

2arctan

(1

4γ

), (6)

α(γ ) = αe− i2 γ , (7)

ξ (γ ) = arcsinh

(1

4γ

)exp

{i

[γ + arccot

(1

4γ

)]}. (8)

Here, γ1 = �1t1|α| is twice the phase-space angle of rotation

clockwise and counterclockwise of the + and − wave packet,respectively, and �1

2|α| is the classical (angular) frequency ofnonlinear oscillations.

The atom-field states can be expressed in terms of photonstates with the help of the identity |n〉± = 1√

2(a 1√

a†a|n〉 ⊗

|e〉1 ± |n〉 ⊗ |g〉1) as

|1±〉 = e−iφ(±γ1)|α(±γ1),ξ (±γ1)〉⊗ (ei(γ0∓ 1

2 γ1)|e〉1 ± |g〉1), (9)

where γ0 = arg α. Here we used the fact that the squeezed-statewave packets are localized in phase space so that they areapproximate eigenstates of a and a† with eigenvalues α(±γ1)and α(±γ1)∗, respectively.

The atom-field state |1〉 is therefore a superposition of twowave packets that are well separated in phase space exceptwhen γ1 is close to an integer multiple of 2π . The splittingof the wave packet generates field-atom entanglement that isalmost maximal when γ1 � |α|−1 and decreases to zero forγ1 = π [3].

The interaction with the second atom proceeds analogously.The final field–two-atom state at time t1 + t2 is

|2〉 = e− ihH2t2 (|1+〉 − |1−〉) ⊗ |g〉2, (10)

where each of the two wave packets |1〉± ⊗ |g〉2 splitsagain into two sub-wave packets, which continue to undergophase-space rotation and squeezing governed by Hamiltonianssimilar to H± constructed from polaritons of the second atom.|2〉 is therefore a superposition of four localized wave packetslabeled by two sign choices:

|2〉 = 12 (|2++〉 − |2+−〉 − |2−+〉 + |2−−〉), (11)

|2rs〉 = 12e−iφ(�rs )|α(�rs),ξ (�rs)〉 ⊗ (ei(γ0− 1

2 rγ1)|e〉1 + r|g〉1)

⊗ (ei(γ0− 12 �rs )|e〉2 + s|g〉2), (12)

where �rs = (rγ1 + sγ2) and γ2 = �2t2|α| .

III. ATOM-ATOM ENTANGLEMENT

The field–two-atom state |2〉 is a superposition of four sub-wave packets each having a well-defined atomic product state.Superpositions of product states can give rise to entanglementbut, for most values of γ1 and γ2, the four sub-wave packetsare separate in phase space (see left panel of Fig. 1) andlabel the different atomic components, and then the two-atomdensity matrix ρa = Trfield |2〉〈2| is an incoherent mixture offour product states in the limit |α| � 1, which is by definitionseparable. The atomic states superpose coherently when thecorresponding sub-wave packets overlap in phase space. We

therefore let γ1 = γ2 = γ , so that the field factors in the |2+−〉and |2−+〉 terms are identical and equal to the initial coherentstate |α〉 (right panel of Fig. 1), while being separate from thefield factors of |2++〉 and |2−−〉, unless γ is an integer multipleof 2π . If γ is not an integer multiple of π , the field factorsof |2++〉 and |2−−〉 are also phase-space separate, so that theatomic state is a mixture

ρa = 14 (|l〉〈l| + 2|c〉〈c| + |r〉〈r|) (13)

of the states

|l〉 = 12 (|g〉1 − eiγ0e

i2 γ |e〉1) ⊗ (|g〉2 − eiγ0eiγ |e〉2),

|c〉 = 1√2

[e2iγ0 cos

(12γ

)|e〉1 ⊗ |e〉2

+ ieiγ0 sin(

12γ

)|e〉1 ⊗ |g〉2 − |g〉1 ⊗ |g〉2],

|r〉 = 12 (|g〉1 + eiγ0e− i

2 γ |e〉1) ⊗ (|g〉2 + eiγ0e−iγ |e〉2).

The state |c〉 is entangled for all γ not an integer multipleof π , with monotonically decreasing entanglement entropyas a function of γ between 0 and π in the limit |α| → ∞(see Fig. 2), while the states |l〉 and |r〉 are separable. Thefull two-atom state is entangled if ρa cannot be expressedas a mixture of separable states. The degree of entanglementof the two-atom state can be tested with the entanglement offormation [17], which is displayed in Fig. 3. The entanglementof formation is an entanglement measure and, in particular,vanishes if and only if the state is separable, so that Fig. 3shows that the two atoms are entangled for most values of γ forfinite α, and for all γ except integer multiples of π in the limit|α| → ∞. The entanglement of formation of the two-atomstate obtains its maximum of ≈0.21 at a kink singularity atγ = π

2 . Unfortunately, there exist nonequivalent entanglementmonotones for mixed states [18]. As an example, the negativitythat equals minus the sum of the negative eigenvalues of thepartial transpose of ρa , obtains its maximum at γ ≈ 0.511π

(see Fig. 3). This result indicates that, although there is noprecise meaning for optimal interaction time, values of γ

near 12π are best for entanglement generation in this system.

Furthermore, changing the initial condition of the secondatom to |e〉 leaves the entanglement of formation unchanged,and slightly shifts the maximum of the negativity to γ ≈0.513π .

An important aspect of the two-atom entanglement-generation process is that, although the atom-field entangle-ment and therefore the entanglement entropy of the centerwave packet |c〉 reach maximal value after a short effectiveinteraction time γ of O(|α|−1), the degree of entanglement ofthe full two-atom state is very low for such small γ . This isa result of the incoherent mixing of |c〉 with the flank wavepackets |l〉,|r〉 whose overlap with |c〉 is low for small γ ,which is seen as low purity of ρa for these γ , as shown inFig. 2. When γ increases, the entanglement entropy of |c〉decreases until it becomes separable for γ = π but, at the sametime, the overlap of |c〉 with |l〉 and |r〉 and, consequently, thepurity of ρa increase. As a result, the degree of entanglementof ρa increases initially, reaches a maximum near γ = 1

2π ,and decreases to 0 for γ = π , as shown in Fig. 3. AlthoughEq. (13) is not valid for γ an integer multiple of π , thepreceding argument is valid for these values of γ showing

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ENTANGLEMENT GENERATION BY INTERACTION WITH . . . PHYSICAL REVIEW A 84, 052303 (2011)

that the entanglement of formation is a continuous functionof γ .

IV. EFFECTS OF DECOHERENCE

The preceding results apparently contradict the corre-spondence principle by displaying entanglement, a purelyquantum phenomenon, at the classical limit. However, a givenvalue of entanglement is reached for a fixed γ = �1t1

|α| = �2t2|α| ,

and therefore the required interaction time diverges when|α| → ∞. This is a demonstration of the singular nature ofthe semiclassical limit: for a fixed interaction time classicalphysics is recovered as |α| → ∞, but the classical and thelong-time limits do not commute. A similar phenomenon wasobserved in [19], where the classical limit and the limit of largesqueezing do not commute.

Physically, longer interaction times imply a stronger effectof environment coupling, and it is this effect that guaranteesthe emergence of classical physics for large |α|, both becauseof stronger dephasing and because of longer interaction times.We demonstrate this statement using the standard Markovianmodel of cavity loss, with Lindblad generator a and rate λ, sothat the master equation for the full system density matrix ρ

is [20]

∂tρ = − i

h[H,ρ] − 1

2λ(a†aρ − 2aρa† + ρa†a). (14)

A superposition of distinct coherent states c(|α〉 + |β〉) ex-periences, as a result of cavity loss and in addition to theoverall decay with rate λ, a much faster dephasing withrate λ|α − β|2 that affects the coefficients of the coherenceterms in the density matrix proportional to |α〉〈β| and itsconjugate [20].

In order to analyze this process in conjunction with thewave-packet dynamics, we assume that λ|α| �1,�2 so thatwe can neglect the energy decay of the state. The full-systemdensity matrix after the interaction with the first atom isthen

ρ1 = |1+〉〈1+| + x1|1+〉〈1−| + x∗1 |1−〉〈1+| + |1−〉〈1−|, (15)

where xk = exp {− λ|α|3�k

[γ + i(e−iγ − 1)]}. A similar dephas-ing affects the coefficient of the twelve coherence terms duringthe interaction with the second atom.

The significance of dephasing for the generation of two-atom entanglement is the reduction of the coherence betweenthe two wave packets |2+−〉 and |2−+〉 that generates theentangled atomic state |c〉 when they collide, so that the term|2+−〉〈2−+| and its conjugate in the full-state density matrixare multiplied by x1x2 and (x1x2)∗, respectively. It follows thatthe term 1

2 |c〉〈c| in ρa is then replaced by

ρc = 14 [|cA〉〈cA| + x1x2|cA〉〈cB |+ (x1x2)∗|cB〉〈cA| + |cB〉〈cB |], (16)

with

|cA,B〉 = 12 (|g〉1 ∓ e

i2 γ0e± i

2 γ |e〉1) ⊗ (|g〉2 ± ei2 γ0 |e〉2),

where upper (lower) sign corresponds to A (B), while theterms proportional to |l〉〈l| and |r〉〈r| are negligibly affectedby decoherence. For |x1x2| < 1, ρc is the density matrix of

a mixed state. Since |cA〉 and |cB〉 are product states, thedegree of entanglement of ρc, and therefore ρa , is a decreasingfunction of the decoherence rate λ.

It follows from Eq. (16) that the degree of entanglementdepends on the decoherence rate only through the combinationy = λ|α|3

2 ( 1�1

+ 1�2

). Figure 4 shows the entanglement offormation as a function of y and γ . Evidently, the maximumachievable entanglement decreases with increasing y, andthis maximum is achieved earlier. Detailed analysis showsthat, when y is large, the maximum of the entanglement isachieved at γ = 1

yand its value is inversely proportional to

y4 log y. This phenomenon has the simple interpretation that ashorter effective interaction time not only allows less time forentanglement generation, but also less time for decoherence,and that this tradeoff leads to a shorter optimal interactiontime for stronger decoherence; for large y the entanglementis completely destroyed when γ = 3

2y. In this limit the

optimal interaction time is shorter than the time neededfor the initial wave packet to split so that entanglement isgenerated only by weak quantum fluctuations, and this allowsa power law rather than an exponential decay in the degree ofentanglement—the entanglement of formation for the optimalinteraction time is 1

9 that of the entanglement achieved in anideal system for the same interaction time; see Fig. 4 (rightpanel).

V. CONCLUSIONS

Our first conclusion is that two qubits can be entan-gled solely by interaction mediated by a macroscopic fieldpreparation, and the degree of entanglement tends to apositive value in the classical limit. The entanglement processproceeds through splitting of the field wave packet into fourcomponents, each carrying a different atomic product state,and a subsequent merging of two of the sub-wave packetsto form an entangled superposition of two atomic productstates. Thus, although the initial field state is semiclassicalwith a large photon number and well-defined quadratures,the field necessarily evolves into highly nonclassical statesduring the entanglement-generation process. Nevertheless,the semiclassical wave-packet-dynamics approximation forthe propagation of the wave packet stays valid during thefull entanglement generation process, although in a multi-ple phase space—one phase-space copy for each sub-wavepacket.

The splitting of the wave packet is a slow process where amacroscopic field state evolves by interaction with two qubits.Naturally, it is not an efficient method of creating entangledpairs, because the interaction time required for entanglementdiverges in the classical limit. The semiclassical entanglementis obtained in a double limit where the photon number andinteraction time tend to infinity together; when the interactiontime is fixed the degree of entanglement tends to zero inthe limit of large photon number, in agreement with thecorrespondence principle.

The final two-atom state is a coherent superposition oftwo product states associated with the center wave packet,incoherently mixed with two additional products associatedwith the flank wave packets. Thus, the two-atom entanglement

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AMIR LESHEM AND OMRI GAT PHYSICAL REVIEW A 84, 052303 (2011)

is determined by the overlap of the evolving atomic states. Inparticular, wave-packet splitting and atom-field entanglementare necessary but not sufficient for the generation of atom-atomentanglement.

The two atoms are not maximally entangled by interac-tion with the semiclassical field mode. Although for shortinteraction periods the center wave packet has an almost-Bellatomic state, the atom-atom entanglement is degraded almostcompletely by mixture with the flank wave packets and, while

for long interaction times the field and atoms decouple, theatomic state at this stage is separable. Unlike the vacuum field,therefore, the semiclassical field does not function as an idealquantum gate, which we conjecture is an example of a generalprinciple.

We thank D Aharonov, I Arad, H Eisenberg, and N Katz,for helpful discussions. This work was supported by the ISFgrant 1002/07.

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