Recenti sviluppi della Meccanica Quantistica:

dalla tomografia quantistica alla caratterizazione dei rivelatori quantistici

3 Maggio 2005, IEN Galileo Ferraris Anno Mondiale della Fisica

L’eredità di EinsteinAlcuni recenti progressi

Giacomo Mauro D’Ariano Università degli Studi di Pavia

“Measuring” the quantum state

• It contains the whole information in-principle available on the system.

• Quantum complementarity forbids to obtain all the possible information from the same measurement: if we are measuring the wave aspects of the system, we are missing out on all its particle aspects.

• Thus, to recover the state, we need to perform a set of incompatible measures on an ensemble of equally prepared systems.

The quantum state

Pre-history: “measuring” the state

Wolfgang Pauli [1958]: The mathematical problem, as to whether for given functions W(q) and W(p) the wave function ψ, if such function exists is always uniquely determined, has still not been investigated in all its generality.

Eugene Paul Wigner [1983]: There is no way to determine what the wave function (or state vector) of a system is— if arbitrarily given, there is no way to “measure” its wave function. Clearly, such a measurement would have to result in a function of several variables, not in a relatively small set of numbers ... In order to verify the [quantum] theory in its generality, at least a succession of two measurements are needed. There is in general no way to determine the original state of the system, but having produced a definite state by a first measurement, the probabilities of the outcomes of a second measurement are then given by the theory.

Pre-history: “measuring” the state

Bernard d’Espagnat [1976]: The question of determining which operators correspond to observables and which do not is a very difficult one. At the present time, no satisfactory answer appears to be known. Neverthless, it is interesting to investigate the relationship of this question to another, similar one: “What are the systems whose density matrices are measurable?” Should we, for instance, say that if a given type of systems corresponding to a given Hilbert space has a measurable density matrix, then all the Hermitean operators defined on that space are measurable? And is the reverse proposition true? What do we mean when we say that the density matrix corresponding to a given type of system is measurable? Let an ensemble E of a sufficiently large number of this type be given. Let us first separate it into subensembles Eλ, the elements of which are chosen at random in E. If from the results of appropriate measurements on the Eλ, we can derive the value of every element of the matrix ρ that describes E in some fixed representation, we say that ρ is measurable.

Pre-history: “measuring” the state

• O. Alter, and Y. Yamamoto, Inhibition of the Measurement of the Wave Function of a Single Quantum System in Repeated Weak Quantum Nondemolition Measurements, Phys. Rev. Lett. 74 4106 (1995).

• Y. Aharonov, J. Anandan, L. Vaidman, Meaning of the Wave Function, Phys. Rev. A 47 4616 (1993); See also the Comment: W. G. Unruh, Reality and Measurement of the Wave Function, Phys. Rev. A 50 882 (1994).

• M. Ueda and M. Kitagawa, Phys. Rev. Lett. Reversibility in Quantum Measurement Processes, 68 3424 (1992).

• A. Imamoglu, Logical Reversibility in Quantum-Nondemolition Measurements, Phys. Rev. A 47 R4577 (1993).

• A. Royer, Reversible Quantum Measurements on a Spin 1/2 and Measuring the State of a Single System, Phys. Rev. Lett. 73 913 (1994); Errata, Phys. Rev. Lett. 74 1040 (1995).

• G. M. D'Ariano and H. P. Yuen, On the Impossibility of Measuring the Wave Function of a Single Quantum System, Phys. Rev. Lett. 76 2832 (1996) [NO CLONING]

“Measuring” the quantum state of a syngle system

• No cloning theorem <---> It is impossible to determine the state of a single quantum system.

• To “measure the state” we need an ensemble of equally prepared identical quantum systems.

“Measuring” the quantum state

• How to measure concretely the matrix elements of the quantum state?

• In order to determine the density matrix, one needs to measure a “complete” set of observables, the quorum, [Fano, d’Espagnat, Royer,...]

• The problem remained at the level of mere speculation for many years...

• It entered the realm of experiments only in 1994, after the experiments by Raymer’s group in the domain of Quantum Optics.

“Measuring” the quantum state

• For particles, it is difficult to devise concretely measurable translational observables—other than position, momentum and energy.

• Quantum optics: unique opportunity of measuring all possible linear combinations of position Q and momentum P of a harmonic oscillator, here a mode of the electromagnetic field.

• Such a measurement is achieved by means of a balanced homodyne detector, which measures the quadrature of the field at any desired phase with respect to the local oscillator (LO).

“Measuring” the quantum state

Homodyne Detector

Homodyne Tomography

K. Vo gel, H . Ris ken, Phy s. Re v. A 40 2847 (1989 )

Homodyne Tomography

• The inverse Radon transform is nonanalytical

• One needs a cutoff, which gives an uncontrollable bias in the matrix elements

Problems with the Radon transform

Exact methodIt is possible to bypass the Radon transform and obtain the density matrix elements by simply averaging suitable functions on homodyne outcomes

G. M. D'Ariano, C. Macchiavello and M. G. A. Paris, Phys. Rev. A 50 4298 (1994)

Exact methodMeasurement statistical errors on the density matrix

elements can make them useless for the estimation of ensemble averages

In the same way different representations of the state can be “experimentally” inequivalent

Exact methodRobust to noise, such as gaussian noise from loss or nonunit quantum efficiency

Bound for quantum efficiency for estimation of the density matrix in the Fock basis

Exact method

G. M. D'Ariano, U. Leonhardt and H. Paul, Phys. Rev. A 52 R1801 (1995)

QuickTime™ and aGIF decompressor

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QuickTime™ and aGIF decompressor

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Exact method

G. D'Ariano, P. Kumar, M. Sacchi, Phys. Rev. A 61, 13806 (2000)

Multimode field: full joint multimode density matrix via random scan over LO

modes

Tomography of a twin beam

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Marginal distributions for the signal and idler beams

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Tomography of a twin beam

Tomography of a twin beam

Exact method: adaptive techniquesThe estimators are not unique, and can be

“adapted” to data to minimize the rms error

G. M. D'Aria no an d M

. G . A. P aris, Phys. Rev. A 60 5 18 ( 1999)

Max-likelihood techniquesMaximize the likelihood function of data

Positivity constraint via Cholesky decomposition

Statistically optimally efficient!Drawbacks: exponential

complexity with the number of modes; estimation of parameters

of the density operator only

K. Banaszek, G. M. D'Ariano, M. G. A. Paris, M. Sacchi, Phys. Rev. A 61, 010304 (2000) (rapid communication)

Angular momentum tomography G. M. D'Aria no, L. M

acc one, a nd M. Paini , J. Opt . B 577 (20 03)

General quantum tomography

General approach: theory of operator frames (frames on Banach spaces): the operator form of wavelet theory

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General quantum tomography

G. M. D'Ariano, Scuola "E. Fermi" on Experimental Quantum Computation and Information,  F. De Martini and C. Monroe eds. (IOS Press, Amsterdam 2002) pag. 385.

4. Any quantum system5. Any observable6. Many modes, or many quantum systems7. Unbiasing noise ...

The method is very powerful:

Pauli Tomography

Tomography of quantum operation of a device

Tomography of quantum operation of a device

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Phys. Rev. L ett. 8 6 419 5 (20 01)

Tomography of quantum operation of a device

Twin beam

Fork scheme

Tomography of quantum operation of a device

F. De Martini, M. D'Ariano, A. Mazzei, and M. Ricci, Phys. Rev. A 87 062307 (2003)

Tomography of quantum operation of a device

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. D 'Aria no, A. Mazz ei, an d

M. Ric ci, Ph ys. Re v. A 87 06230 7 (20 03)

• Is it possible to make a tomography of a quantum operation using entangled mixed states, or even separable states?

• Answer: yes! as long as the state is faithful.

Tomography of quantum operation of a deviceFaithful states

Tomography of quantum operation of a device

M. D'A riano and P . Lo P resti, Phys. Rev. L ett. 91 04790 2-1 ( 2003)

Faithful states

Tomography of quantum operation of a device

M. D'A riano and P . Lo P resti, Phys. Rev. L ett. 91 04790 2-1 ( 2003)

Tomography of quantum operation of a device

M. D'A riano and P . Lo P resti, Phys. Rev. L ett. 91 04790 2-1 ( 2003)

We can perform a complete quantum calibration of a

measuring apparatus experimentally, without knowing its functioning!

Quantum Calibration

How we describe a measuring apparatus?

Quantum Calibration

pre calibration

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4)Quantum Calibration

Quantum calibration of a photocounter

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Quantum calibration of a photocounter

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Quantum tomography for imaging

G. M. D’Ariano and L. Maccone (submitted to

Discrete Tomography and Applications, N.Y. City,

June 13-15(2005)

• Quantum tomography is a method to measure experimentally the quantum state, or any ensemble average.

• There is a setup for any quantum system.

• Robust to noise.

• Statistically efficient.

• Can be used for fully calibrating devices and measuring apparatuses. Robust to input state.

• Maybe useful also for ACT.

Conclusions

www.qubit.it

THE END