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Workshop Programme

for period 15 - 19 November 2004

Focus Week: Quantum Statistics - Quantum Measurements, Estimation and Related Topics

15 - 19 November 2004


Monday 15 November
10:30-11:15 D'Ariano, M (Pavia)
  On convex structures of states, POVM's and channels, and their mutual relations Sem 1

After briefly reviewing the structure of the convex sets of POVM's and channels in finite dimensions, we will consider maps between different types of convex sets, corresponding to different kinds of quantum information processing, e. g. quantum calibration, programmable channels and POVM's, universal POVM's, pre-processing and post-processing of POVM's. In particular, we will focus attention on programmability of POVM's and pre-processing, introducing the problem of "clean POVM's", and concluding with a list of open problems.

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11:15-12:00 Chiribella, G (Pavia)
  Extremel covariant POVM's Sem 1

The random choice between two different apparatuses measuring the same physical parameter can be viewed as a convex combination in the space of quantum measurements. In particular, the set of positive operator valued measures (POVM) pertaining to a given parameter is a convex set. The aim of this work is to characterize the extreme points of the set of POVM's which are covariant with respect to a finite dimensional representation of a Lie group. Necessary and sufficient conditions are given, also relating extremality with uniqueness and stability of measurements arising in concrete optimization problems.

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14:00-14:45 Artiles, LM (EURANDOM)
  Efficient minimax rates for Wigner function estimation from quantum homodyne tomography data Sem 1

It was just recently that the problem of quantum tomography has been called to the attention of the statistical community throughout the proof of consistency results for Projection Pattern Function and Sieve Maximum Likelihood estimators of the density matrix and the Wigner function of the quantum state of light. Proving consistency means that the estimator converges to the desired state in appropriate norm. A step further in this study is to prove how fast (rate of convergence) an estimator converges as the number of samples increases. We show pointwise rates of convergence of the minimax error for the classical Kernel estimator of the Wigner function from quantum homodyne tomography data, assuming the Wigner function is in certain class. As a whole, we also give an overview of previous works and motivate future directions.

14:45-15:30 Owari, M (Tokyo)
  Local copying of orthogonal maximally entangled states and its relationship to local discrimination Sem 1

In the quantum system, perfect copying is impossible without prior knowledge. But, perfect copying is possible, if it is known that unknown states to be copied is contained by the set of orthogonal states, which is called the copied set. However, if our operation is limited to local operations and classical communications, this problem is not trivial. Recently, F. Anselmi, A. Chefles and M.B. Plenio constructed theory of local copying when the copied set consists of maximally entangled states. They also classified the copied set when it consists of two orthogonal states (quant-ph/0407168). In this paper, we completely classify the copied set of local copying of the maximally entangled states in the prime dimensional system. That is, we prove that, in the prime dimensional system, the set of locally copiable maximally entangled states is equivalent to the set of Simultaneously Schmidt decomposable canonical form Bell states. As a result, we conclude that local copying of maximally entangled states is much more difficult than local discrimination at least in prime dimensional systems.

Tuesday 16 November
10:30-11:15 Bagan, E (Barcelona)
  Optimal qubit mixed state estimation Sem 1

Given a number N of identically prepared qubit mixed states, I analyse the optimal estimation protocol, based on collective measurements. I discuss two scenarios: completely general states (3D), and states known to lie on the equatorial plane of the Bloch sphere (2D). I will derive the optimal POVMs for finite number of copies and obtain the analytical expressions of the fidelities in the large N regime. In 3D, I will show that the optimal POVM's are independent of the prior distributions, provided they are isotropic. In 2D this is only true asymptotically. However, for the latter, the fidelity is F=1-1/(2N), independently of the prior.

11:15-12:00 Munoz-Tapia, R (Barcelona)
  Estimation of qubit mixed states with local measurements Sem 1

I discuss the problem of estimating an unknown qubit mixed state from a large sample of N identically prepared states with local measurements. I use as a prior the Bures distribution and focus, for simplicity, on states belonging to the equatorial plane of the Bloch sphere. I will show that the average fidelity for a large sequence of fixed measurements tends to 1-1/N^{3/4}. This is a very suprising result that contrast with both, estimation of pure states and estimation of mixed states with collective measurements, where the fidelity tends to 1 as 1/N.

14:00-14:45 Ghosh, S (York)
  State estimation on a circle Sem 1

Gisin and Popescu [PRL, 83, 432 (1999)] showed that more information about the direction of the Bloch vector of a pure qubit |\psi(\theta, \phi)> = cos(\theta/2)|0> + exp[i\phi] sin(\theta/2)|1> can be obtained from anti-parallel states |{\Psi}_{11}(\theta, \phi)> = |\psi(\theta, \phi)> X |\psi(\pi-\theta, \pi+\phi)>, compared to parallel states |{\Psi}_{20}(\theta, \phi)> = |\psi(\theta, \phi)> X |\psi(\theta, \phi)>, where (\theta, \phi) is distributed uniformly over [0, \pi] x [0, 2\pi]. As a cause behind this difference, they pointed out the difference between the dimensions of the subspaces spanned by parallel and anti-parallel states separately. When \theta = \pi/2, there is no difference in the amount of information as in that case (and only in that case) exact spin-flip is possible. For any fixed \theta, the dimension of the space spanned by N no. of same and M no. of its orthogonal qubits is (N+M+1). We found that whenever we fix \theta, anti-parallel states always give more information if \theta is different from 0 or \pi/2 or \pi, in the case when we estimate the direction of the Bloch vector of the qubit. We generalized this to the case of N no. of same and M no. of its orthogonal qubits. Here the measurement basis for optimal estimation strategy always turns out to be a quantum Fourier transform. But in the case of estimating the direction of the Bloch vector of the qubit |\psi(\theta, \phi)> = cos(\theta/2)|0> + exp[i\phi]sin(\theta/2)|1>, we found that both the sets S_P(\theta) = {|{\Psi}_{20}(\theta, \phi)> : \phi \in [0, 2\pi]} U {|{\Psi}_{20}(\pi-\theta, \pi+\phi)> : \phi \in [0, 2\pi]} and S_A(\theta) = {|{\Psi}_{11}(\theta, \phi)> : \phi \in [0, 2\pi]} U {|{\Psi}_{11}(\pi-\theta, \pi+\phi)> : \phi \in [0, 2\pi] give the same information.

14:45-15:30 Hayashi, M (ERATO)
  Asymptotic theory of quantum estimation Sem 1

We discuss the asymptotic bound the accuracy of the estimation when we use the quantum correlation in the measuring apparatus. It is also proved that this bound can be achieved in any model in the quantum two-level system and in the general case. Moreover, we show that in several specific model this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus. That is, in such a model, the quantum correlation can improve the accuracy of the estimation in an asymptotic setting.

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16:15-17:00 Sacchi, M (Pavia)
  Quantum measurements that maximise the likelihood and optimal reference frame transmission Sem 1

We determine the covariant measurements that maximize the likelihood according to the symmetries of quantum states, and apply our results to provide optimal schemes for reference frames transmission.

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Wednesday 17 November
10:30-11:15 Jupp, P (St Andrews)
  Some aspects of quantum statistical inference Sem 1

In classical parametric statistical inference, an important question is `What parts of the data are informative about the parameters of interest?'. Key concepts here are those of sufficient statistic, ancillary statistic and cut. Some analogous concepts for quantum instruments will be outlined.

11:15-12:00 Petz, D (Alfred Renyi Institute of Math, Budapest)
  Sufficiency in quantum statistical inference Sem 1

Coarse-grining is a basic concept in mathematical statistics and its analogue may be defined in the formalism of quantum mechanics. In the lecture sufficiency of a coarse-graining is discussed with respect to a family of states. Charaterizations of sufficiency are given. The equality case for the strong subadditivity of the von Neumann entropy and the Koashi-Imoto theorem are among the applications. The lecture is partially based on joint work with Anna Jencova.

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14:00-14:45 Keyl, M (Pavia)
  Quantum state estimation and large deviations Sem 1

In this talk we propose a method to estimate the density matrix \rho of a d-level quantum system by measurements on the N-fold system. The scheme is based on covariant observables and representation theory of unitary groups and it extends previous results concerning the estimation of the spectrum of \rho. We show that it is consistent (i.e. the original input state is recovered with certainty for N \to infinity) and analyze its large deviation behavior. In addition we calculate explicitly the corresponding rate function which describes the exponential decrease of error probabilities in the limit of infinitely many input systems. For pure input states, or if \rho is mixed but only information about its spectrum is required, we then show that the proposed scheme is optimal in the sense that it provides the fastest possible decay of error probabilities. In the general case, however, the optimality question remains open.

14:45-15:30 Guta, M (Utrecht)
  On the relation between information and disturbance in quantum measurements Sem 1

The state of a quantum system is perturbed due to the interaction with the environment. In quantum control one tries to undo the perturbation by using measurement results to perform correction operations. The correction is perfect if the measurement results do not contain any information about the state of the system which is considered unknown. If some information is obtained then inevitably the system cannot be brought back in the initial state. We show however that the perturbation is bounded by a power of the information. We then look at the difference between in the Fisher information before and after the measurement and show that it is bounded by the perturbation. An application of these ideas is the use of squeezed states for controlling a system shown recently by L. Bouten.

Thursday 18 November
10:30-11:15 Fujiwara, A (Osaka)
  Differential geometry of quantum channel estimation Sem 1

Since almost every quantum protocol assumes a priori knowledge of the behavoir of the quantum channel under consideration, there is no doubt that identifying the channel is of fundamental importance in quantum information theory. In this talk, I will review some recent developments in quantum channel estimation theory, putting emphasis on an active interplay between noncommutative statistics and information geometry.

11:15-12:00 De Martini, F (Roma)
  Optimal realisation of non-unitary maps for quantum information Sem 1

It is well known that several state transformations, allowed in the framework of classical information theory, cannot be realized exactly in the quantum realm. Precisely, these correspond to the non “Completely Positive” (Non-CP) Maps wich play a substantial role in fundamental issues as the superluminal signaling in Einstein-Podolsky-Rosen correlations or within any universal “spin flip” process. We outline the behaviour of the most important Non-CP Maps of general relevance in quantum information and the strategy adopted to implement them “optimally” and “universally”. All that will be substantiated by corresponding experiments based on two different methods of modern quantum optics: the photon stimulated emission process in a quantum injected optical parametric amplifier or the Ou-Mandel quantum-state photon symmetrization procedure.

14:00-14:45 Tsuda, Y (Chuo)
  Quantum estimation for non-differentiable models Sem 1

State estimation is a classical problem in quantum information. In optimization of estimation scheme, to find a lower bound to the error of the estimator is a very important step. So far, all the proposed tractable lower bounds use derivative of density matrix. However, sometimes, we are interested in quantities with singularity, e.g. concurrence etc. In the paper, lower bounds to a Mean Square Error (MSE) of an estimator are derived for a quantum estimation problem without smoothness assumptions. Our main idea is to replace the derivative by difference, as is done in classical estimation theory. We applied the inequalities to several examples, and derived optimal estimator for some of them. (quant-ph/0207150)

14:45-15:30 Hradil, Z (Palacky)
  Maximum likelihood methods in quantum mechanics Sem 1

The principle of Maximum Likelihood (MaxLik) is not a rule that requires justification. It does not need to be proved and nowadays it is widely used in many applications. What makes this technique so attractive and powerful is it efficiency and versatility. MaxLik estimation may be advanatageously applied to the inverse problems in quantum mechanics. Variables, which cannot be directly measured may always be estimated obeying the rules of quantum theory. Such a tight relationship will be demonstrated on the estimation of the phase shift and number-phase uncertainty relations. This motivates the application of the MaxLik for the quantum state reconstruction, the so called quantum tomography. Extremal equation for the MaxLik estimate of quantum state will be found and interpreted as the closure relation for quantum state measurement. Though the operator equation is nonlinear, it may be solved by iterations. The accuracy may be evaluated by means of Fisher information matrix. MaxLik estimation may be easily modified in order to treat the insufficient data. In this sense the MaxLik reconstruction represents an advantageous alternative to the linear reconstruction techniques based for example on the Radon transformation, which are prone to artifacts of various origin. MaxLik estimation will be demonstrated on several examples including the operational phase concepts, reconstruction of spin and entangled spin states, reconstruction of higher dimensional states of photons with angular momentum, reconstruction of photon statistics counted by inefficient detectors or absorption and phase X-ray tomography.

16:15-17:00 Ballester, M (Utrecht)
  Estimation of SU(d) using entanglement the d $>$ 2 case Sem 1

In recent papers (Refs [1-4]) it is shown that if N copies of an SU(2) gate are available, one can estimate it with a square error that goes to 0 as 1/N^2 (instead of 1/N as one normally expects in statistics). This is achieved by using an Nfold entangled state as input. In my talk I will try to show that this is also possible for SU(d) with d > 2.

[1] M. Hayashi, (2004), quant-ph/0407053. [2] E. Bagan, M. Baig, and R. Munoz-Tapia, Phys. Rev. A 69, 050303 (2004), quant-ph/0303019. [3] E. Bagan, M. Baig, and R. Munoz-Tapia, Phys. Rev. A 70, 030301 (2004), quant-ph/0405082. [4] G. Chiribella, G. D.Ariano, P. Perinotti, and M. Sacchi, (2004), quant-ph/0405095.

Friday 19 November
10:30-11:15 Larsson, J-A (Linkoping)
  The Bell inequality and the coincidence time loophole Sem 1

This paper analyzes effects of time dependence in the Bell inequality. A generalized inequality is derived for the case when coincidence and non-coincidence (and hence whether or not a pair contributes to the actual data) is controlled by timing that depends on the detector settings. Needless to say, this inequality is violated by quantum mechanics and could be violated by experimental data provided that the loss of measurement pairs through failure of coincidence is small enough, but the quantitative bound is more restrictive in this case than in the previously analyzed "efficiency loophole".

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11:15-12:00 Loubenets, E (Moscow State Institute)
  Class of quantum states satisfying the original Bell inequality Sem 1

We introduce the analytic property of a quantum state (separable or nonseparable) to satisfy the perfect correlation form of the original Bell inequality under any quantum measurements of Alice and Bob.

14:00-14:45 Buscemi, F (Pavia)
  Repeatable measurements without eigenstates Sem 1

We show that, contrarily to the widespread belief, in quantum mechanics repeatable measurements are not necessarily described by orthogonal projectors-the customary paradigm of observable. Nonorthogonal repeatability, however, occurs only for infinite dimensions. We also show that when a non orthogonal repeatable measurement is performed, the measured system retains some "memory" of the number of times that the measurement has been performed.

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14:45-15:30 Ozawa, M (Tohoku)
  Universal uncertainty principle Sem 1

In 1927 Heisenberg [1] proposed a reciprocal relation for measurement noise and disturbance, or equivalently for joint measurement noises, by the famous gamma ray microscope thought experiment and claimed that the relation is a straightforward mathematical consequence of the commutation relation. However, his proposed proof did not really consider the measurement noise or disturbance, and was immediately reformulated by Kennard [2] as the famous inequality for the standard deviations of position and momentum. Kennard's relation was soon generalized by Robertson [3] to arbitrary pairs of observables. In spite of the above development, text books of quantum mechanics have treated Robertson's relation as a mathematical expression of Heisenberg's original relation for noise and disturbance.

In this talk, we discuss a new relation (the universal uncertainty principle) found and rigorously proven recently by the present speaker [8] for the well-defined root-mean-square noise and disturbance in all the physically possible quantum measurements. In the previous work [4], it has been shown that jointly unbiased measurements satisfy the reciprocal joint measurement noise relation analogous to Robsertson's. However, it should be pointed out that there is no jointly unbiased spin measurements for two different components, and that no projective measurements of finite level systems satisfy the reciprocal noise disturbance relation. Counter examples of the original Heisenberg relation also include the contractive state position measurement and the indirect position measurement for the EPR pair [6,8]. The universal uncertainty relation reveals the correct bound for the noise in non-disturbing measurements. Using this noise bound, we construct a proof of a quantitative generalization of the Wigner-Araki-Yanase theorem, setting a lower bound for the accuracy of measurement under conservation laws [5,9]. Then, this relation is used to give a lower bound for the accuracy of quantum gate operations realized by interactions between qubits and the controller (or the ancilla) obeying given conservation laws [7,9]. The relations to the no-cloning theorem and to security arguments for quantum cryptography will be also discussed briefly.

References. 1. W. Heisenberg, Z. Phys. 43, 172 (1927). 2. E. H. Kennard, Z. Phys. 44, 326 (1927). 3. H. P. Robertson, Phys. Rev. 34, 163 (1929). 4. M. Ozawa, in Lecture Notes in Physics 378 (Springer, Berlin), 3 (1991); S. Ishikawa, Rep. Math. Phys. 29, 257 (1991). 5. M. Ozawa, Phys. Rev. Lett. 88, 050402 (2002). 6. M. Ozawa, Phys. Lett. A 299, 1 (2002). 7. M. Ozawa, Phys. Rev. Lett. 89, 057902 (2002); 91, 089802 (2003). 8. M. Ozawa, Phys. Rev. A 67, 042105 (2003); Phys. Lett. A, 318, 21 (2003); Phys. Lett. A 320, 367 (2004); Ann. Phys. 31, 350 (2004). 9. M. Ozawa, Intern. J. Quant. Inf. (IJQI) 1, 569 (2003).


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