Recenti sviluppi della Meccanica Quantistica:
dalla tomografia quantistica
alla caratterizazione dei rivelatori quantistici
Giacomo Mauro D’Ariano
Università degli Studi di Pavia
3 Maggio 2005, IEN Galileo Ferraris
Anno Mondiale della Fisica
L’eredità di Einstein
Alcuni recenti progressi
“Measuring” the quantum state
The quantum state
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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.
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.
Pre-history: “measuring” the state
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.
“Measuring” the quantum state of a syngle system
•
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).
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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
•
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,...]
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The problem remained at the level of mere speculation for
many years...
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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
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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).
Homodyne Detector
Homodyne Tomography
Homodyne Tomography
Problems with the Radon transform
•
The inverse Radon transform is nonanalytical
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One needs a cutoff, which gives an
uncontrollable bias in the matrix elements
Exact method
It 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 method
Measurement 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 method
Robust 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 a
GIF decompressor
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QuickTime™ and a
GIF decompressor
are needed to see this picture.
Exact method
Multimode field: full joint
multimode density matrix via
random scan over LO modes
G. D'Ariano, P. Kumar, M. Sacchi,
Phys. Rev. A 61, 13806 (2000)
M. Vasilyev, S.-K. Choi, P. Kumar, and G. M.
D'Ariano, Phys. Rev. Lett. 84 2354 (2000)
Tomography of a twin beam
M. Vasilyev, S.-K. Choi, P. Kumar, and G. M.
D'Ariano, Phys. Rev. Lett. 84 2354 (2000)
Tomography of a twin beam
Marginal distributions for the signal and idler beams
Tomography of a twin beam
Exact method: adaptive techniques
The estimators are not unique, and can be
“adapted” to data to minimize the rms error
Max-likelihood techniques
Maximize 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
General quantum tomography
General approach: theory of operator frames (frames on Banach
spaces): the operator form of wavelet theory
General quantum tomography
The method is very
powerful:
1. Any quantum system
2. Any observable
3. Many modes, or many quantum systems
4. Unbiasing noise ...
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.
Pauli Tomography
Tomography of quantum operation of a device
Tomography of quantum operation of a device
Tomography of quantum operation of a device
Fork scheme
Twin beam
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
Tomography of quantum operation of a device
Faithful states
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Is it possible to make a tomography of a
quantum operation using entangled mixed
states, or even separable states?
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Answer: yes! as long as the state is
faithful.
Tomography of quantum operation of a device
Faithful states
Tomography of quantum operation of a device
Tomography of quantum operation of a device
Quantum Calibration
We can perform a complete
quantum calibration of a measuring
apparatus experimentally, without
knowing its functioning!
Quantum Calibration
How we describe a measuring apparatus?
G. M. D'Ariano, P. Lo Presti, and L. Maccone,
Phys. Rev. Lett. 93 250407 (2004)
Quantum Calibration
pre calibration
G. M. D'Ariano, P. Lo Presti, and L. Maccone,
Phys. Rev. Lett. 93 250407 (2004)
Quantum calibration of a photocounter
G. M. D'Ariano, P. Lo Presti, and L. Maccone,
Phys. Rev. Lett. 93 250407 (2004)
Quantum calibration of a photocounter
Quantum tomography for imaging
G. M. D’Ariano and L. Maccone
(submitted to Discrete
Tomography and Applications,
N.Y. City, June 13-15(2005)
Conclusions
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Quantum tomography is a method to measure
experimentally the quantum state, or any ensemble average.
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There is a setup for any quantum system.
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Robust to noise.
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Statistically efficient.
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Can be used for fully calibrating devices and measuring
apparatuses. Robust to input state.
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Maybe useful also for ACT.
www.qubit.it
THE END