DEX-SMI Workshop on
Quantum Statistical Inference

Dates: 2-4 March, 2009
Place: 12F, National Institute of Informatics (NII), Tokyo, Japan
Organizer: Deepening and Expansion of Statistical Mechanical Informatics (DEX-SMI), Grant-in-Aid for Scientific Research on Priority Areas, MEXT.

## Organizing Committee:

• Masahito Hayashi (Chair), Tohoku University
• Kae Nemoto, NII
• Hiroshi Imai, NII
• Keiji Matsumoto, NII

Contact: Masahito Hayashi

## Program:

2th, March, 2009

13:20-13:30 Opening: Masahito Hayashi

13:30-14:15 Masahito Hayashi (Tohoku Univ.)
Group theoretical study of LOCC-detection of maximally entangled state using hypothesis testing Presentation file
14:25-15:10 Masato Koashi (Osaka Univ.) Fumitaka Takenaga, Takashi Yamamoto, and Nobuyuki Imoto
Quantum nonlocality without entanglement in a pair of qubits

15:10-15:30 Tea break

15:30-16:15 Miguel Navascues (Imperial College London), Masaki Owari, Martin B. Plenio
Merging state estimation and entanglement theory
16:25-16:55 Yutaka Shikano (Tokyo Institute of Technology), Seth Lloyd (MIT), Akio Hosoya (Tokyo Institute of Technology)
Optimal Quantum Covariant Clock

16:55-16:15 Tea break

17:15-18:00 Giacomo Mauro D'Ariano (Pavia Univ.)
Probabilistic theories: what is special about Quantum Mechanics? Presentation file

3rd, March, 2009

10:00- 10:45 Masaki Owari (Imperial College London), M. B. Plenio, E. S. Polzik, A. Serafini, M. M. Wolf
Squeezing the limit: Quantum benchmarks for the teleportation and storage of squeezed states Presentation file

10:45-11:05 Tea break

11:05- 11:50 John Calsamiglia (Universitat Autonoma Barcelona)
Phase-Covariant Quantum Benchmarks for Quantum Storage and Teleportation Presentation file

11:50-13:30 Lunch Time

13:30- 14:15 Masahiro Takeoka (NICT) K. Tsujino, M. Sasaki, C. Wittmann, K. N. Cassemiro, G. Leuchs, and U. L. Andersen
Experimental implementation of near-optimal quantum measurements of optical coherent state Presentation file
14:25-15:10 Emilio Bagan (Universitat Autonoma Barcelona)
Phase eatimation with thermal Gaussian states Presentation file

15:10-15:30 Tea break

15:30-16:15 Dominic Berry (Macquarie University), B. L. Higgins, H. M. Wiseman, S. D. Bartlett, M. W. Mitchell, and G. J. Pryde
Phase Measurements at the Theoretical Limit Presentation file

16:15-16:35 Tea break

16:35-17:15 Hiroshi Imai (NII), Masahito Hayashi (Tohoku Univ.)
Fourier Analytic Approach to Phase Estimation Presentation file
17:25-17:50 Yu Matsumoto (Tohoku Univ.), Masahito Hayashi (Tohoku Univ.)
Identification of rotation axis in Bloch Sphere

4th, March, 2009

Local asymptotic normality in quantum statistics Presentation file

10:45-11:05 Tea break

11:05-11:50 Keiji Matsumoto (NII)
Monotone metrics on quantum channels

11:50-13:30 Lunch Time

13:30-14:00 Yu Watanabe (Tokyo Institute of Technology), Takahiro Sagawa (Univ. of Tokyo) and Masahito Ueda (Univ. of Tokyo)
Visualization of Quantum Operations Presentation file
14:10-14:55 Akihisa Hayashi (Fukui Univ.)
Unitary process discrimination with error margin Presentation file

14:55-15:15 Tea break

15:15-16:00 Paolo Perinotti (Pavia Univ.)
Estimation and discrimination of quantum networks Presentation file
16:00-16:45 Giulio Chiribella (Pavia Univ.)
Optimal quantum learning and multiround reference frame alignment Presentation file

16:45-16:50 Closing remarks

## Abstracts:

Masahito Hayashi (Tohoku Univ.)
Title: Group theoretical study of LOCC-detection of maximally entangled state using hypothesis testing Presentation file
Abstract: In the asymptotic setting, the optimal test for hypotheses testing of the maximally entangled state is derived under several locality conditions for measurements. The optimal test is obtained in several cases with the asymptotic framework as well as the finite-sample framework. In addition, the experimental scheme for the optimal test is presented. Related manuscript: http://arxiv.org/abs/0810.3380

Masato Koashi (Osaka Univ.) Fumitaka Takenaga, Takashi Yamamoto, and Nobuyuki Imoto
Title: Quantum nonlocality without entanglement in a pair of qubits
Abstract: We consider unambiguous discrimination of two separable bipartite states, one being pure and the other being a rank-2 mixed state. There is a gap between the optimal success probability under global measurements and the one achieved by generalized measurements with separable measurement operators. We show that even the latter success probability cannot be achieved via local operations and classical communication, leaving a nonzero gap in between.
Related manuscript: http://arxiv.org/abs/0709.3196

Miguel Navascues (Imperial College London) Masaki Owari, Martin B. Plenio
Title: Merging state estimation and entanglement theory
Abstract: In this talk, I will try to convince the audience that there exists a connection between state estimation and entanglement theory from which both fields can mutually benefit. On one hand, I will use recent results on probabilistic state estimation in order to analyze the speed of convergence of the Doherty-Parrilo-Spedalieri (DPS) criterion for entanglement detection. More concretely, I will show how to derive non trivial upper bounds for the distance between the states returned by the computer and the set of separable states, in terms of trace distance, operator norm distance and entanglement robustness. On the other hand, I will show how the DPS criterion can be applied to provide two converging numerical sequences of upper and lower bounds for the optimal fidelity in any pure state estimation problem.

Yutaka Shikano (Tokyo Institute of Technology), Seth Lloyd (MIT), Akio Hosoya (Tokyo Institute of Technology)
Title: Optimal Quantum Covariant Clock
Abstract: We cannot measure time directly but we can measure a clock and then infer the time from the position of the hand we read. This feature of measuring time is more acute in quantum mechanics because there is no self-adjoint time operator if the Hamiltonian is bounded below. The time in the Schroedinger equation is only a parameter but not a physical observable. Any physical object can be a clock but its quality may vary. It is reasonable to consider a clock is good if it has a high precision and its accuracy is maintained for a long time. We are going to explore the best quantum clock of the highest possible precision given a finite energy range, which keeps accuracy in an arbitrarily long time passage.

Giacomo Mauro D'Ariano (Pavia Univ.)
Title: Probabilistic theories: what is special about Quantum Mechanics? Presentation file
Abstract: Quantum Mechanics (QM) is a very special probabilistic theory, yet we don't know which operational principles make it so. Here I will analyze the possibility of deriving QM as the mathematical representation of a "fair operational framework", i.e. a set of rules which allows the experimenter to make predictions on future "events" on the basis of suitable "tests", e.g. without interferences from uncontrollable sources. Two postulates need to be satisfied by any fair operational framework: NSF: "no-signaling from the future" (for the possibility of making predictions on the basis of past tests) FAITH: "existence of faithful states" (for the possibility of calibrating all tests and of preparing any state). I will show that all theories satisfying NSF admit a C*-algebra representation of events as linear transformations of effects. Based on a very general notion of dynamical independence, it is easy to see that all such probabilistic theories are "no-signaling without interaction" (shortly "no-signaling")--another requirement for a fair operational framework. Postulate FAITH then implies the "local observability principle", along with the tensor-product structure for the linear spaces of states and effects. What is special about QM is that also "effects make a C*-algebra. However, whereas the sum of effects can be operationally defined, the notion of effect abhors any kind of composition. Based on another very natural postulate--AE: atomicity of evolution--along with a purely mathematical postulate--CJ: Choi-Jamiolkowski isomorphism--it is possible to identify effects with atomic events, through which we can then define composition.
Related manuscript: http://arxiv.org/abs/0807.4383([v3] Tue, 31 Mar 2009) appear in "Philosophy of Quantum Information and Entanglement" Eds A. Bokulich and G. Jaeger (Cambridge University Press, Cambridge UK)

Masaki Owari (Imperial College London), M. B. Plenio, E. S. Polzik, A. Serafini, M. M. Wolf
Title: Squeezing the limit: Quantum benchmarks for the teleportation and storage of squeezed states Presentation file
Abstract: We derive fidelity benchmarks for the quantum storage and teleportation of squeezed states of continuous variable systems, for input ensembles where the degree of squeezing $s$ is fixed, no information about its orientation in phase space is given, and the distribution of phase space displacements is a Gaussian. In the limit where the latter becomes flat, we prove analytically that the maximal classical achievable fidelity (which is 1/2 without squeezing, for $s=1$) is given by $\sqrt{s}/(1+s)$, vanishing when the degree of squeezing diverges. For mixed states, as well as for general distributions of displacements, we reduce the determination of the benchmarks to the solution of a finite-dimensional semidefinite program, which yields accurate, certifiable bounds thanks to a rigorous analysis of the truncation error. This approach may be easily adapted to more general ensembles of input states.
Related manuscript: New Journal of Physics, vol. 10, 113014, (2008) http://arxiv.org/abs/0808.2260

John Calsamiglia (Universitat Autonoma Barcelona)
Title: Phase-Covariant Quantum Benchmarks for Quantum Storage and Teleportation Presentation file
Abstract: We give a quantum benchmark for teleportation and quantum storage experiments suited for pure and mixed test states. The benchmark is based on the average fidelity over a family of phase- covariant states and certifies that an experiment can not be emulated by a classical setup, i.e., by a measure-and-prepare scheme. We give an analytical solution for qubits, which shows important differences with standard state estimation approach, and compute the value of the benchmark for coherent and squeezed states, both pure and mixed.
Related manuscript: http://arxiv.org/abs/0807.5126

Masahiro Takeoka (NICT) K. Tsujino, M. Sasaki, C. Wittmann, K. N. Cassemiro, G. Leuchs, and U. L. Andersen
Title: Experimental implementation of near-optimal quantum measurements of optical coherent state Presentation file
Abstract: The optimal discrimination of optical coherent states is a fundamental task in quantum information theory as well as an important primitive in optical communication. We propose and experimentally demonstrate a new and simple near-optimal quantum measruement strategy capable of discriminating two coherent states with smaller average error probability than the homodyne detection. Our proof-of-principle experiment clearly shows the lower error probability than that of the homodyne limit after compensating quantum efficiency (QE) of the photon detector. We discuss the necessary QE to truly outperform the homodyne limit without compensating experimental imperfections and show that it is feasible with the current technology.
As a related topic, we discuss alternative figures of merit to evaluate the classical communication performance, that is reliability function and its cut-off rate. Interestingly, a variety of quantum measurements can attain the maximum (single-shot) cut-off rate while some of them are not optimal from the viewpoints of average error probability or mutual information.
Related manuscripts:
http://arxiv.org/abs/0809.4953 [Phys. Rev. Lett. 101, 210501 (2008)]
http://arxiv.org/abs/0706.1038 [Phys. Rev. A 78, 022320 (2008)]

Emilio Bagan (Universitat Autonoma Barcelona)
Title: Phase eatimation with thermal Gaussian states Presentation file
Abstract: We consider two classes of Gaussian states: displaced thermal states and squeezed thermal states; to perform phase estimation. We find that while for displaced thermal states an increase in temperature gives rise to a reduction in the precision, for squeezed thermal states it goes the other way around. In the multiple-copy scenario we analyze the most paradigmatic measurement strategies (including homodyne and heterodyne measurements) and discuss their performance as compared to the optimal schemes. We apply our results to investigate the influence of losses in an optical metrology experiment.
Related manuscript: http://arxiv.org/abs/0811.3408

Dominic Berry (Macquarie University), B. L. Higgins, H. M. Wiseman, S. D. Bartlett, M. W. Mitchell, and G. J. Pryde
Title: Phase Measurements at the Theoretical Limit Presentation file
Abstract: The interferometric measurement of optical phase is a vital tool for precision measurement. The performance of phase measurements is usually quantified by how the uncertainty scales in terms of the number of resources, N. Standard measurement techniques result in an uncertainty scaling as O(1/sqrt(N)) _ the standard quantum limit (SQL). In contrast, the fundamental limit on phase measurements is the Heisenberg limit of O(1/N). The SQL results from the independent use of resources, so methods of beating the SQL usually involve special nonclassical resource states, such as squeezed states or NOON_ states. Similar results can also be obtained using multiple passes of single photons.
Problems with using nonclassical states or multiple passes are that they may only work over a small range of phases, require initial knowledge of the phase or give ambiguous phase information. We have developed a range of techniques to eliminate these problems, providing unambiguous phase measurements with accuracy close to the theoretical limit. We have experimentally demonstrated a scheme that uses an adaptive technique and multiple passes of single photons to implement the quantum phase estimation algorithm (QPEA), as well as a generalised form of this algorithm.
The QPEA only achieves the SQL due to fat tails in the distribution. Simply combining the QPEA with standard measurements can improve the scaling, although not achieve the theoretical limit. In contrast, our generalised scheme uses multiple measurements to eliminate the fat tails, and achieve the theoretical limit with an overhead factor of only about 1.6. We have now further improved this scheme to achieve the theoretical limit without requiring adaptation, albeit with a slightly increased overhead factor of about 2. Reducing or removing adaptive measurements should make practical implementation of precision phase estimation far easier.

Hiroshi Imai (NII), Masahito Hayashi (Tohoku Univ.)
Title: Fourier Analytic Approach to Phase Estimation Presentation file
Abstract: For a unified analysis on the phase estimation, we focus on the limiting distribution. It is shown that the limiting distribution can be given by the absolute square of the Fourier transform of $L^2$ function whose support belongs to $[-1,1]$. Using this relation, we study the relation between the variance of the limiting distribution and its tail probability. As our result, we prove that the protocol minimizing the asymptotic variance does not minimize the tail probability. Depending on the width of interval, we derive the estimation protocol minimizing the tail probability out of a given interval. Such an optimal protocol is given by a prolate spheroidal wave function which often appears in wavelet or time-limited Fourier analysis. Also, the minimum confidence interval is derived with the framework of interval estimation that assures a given confidence coefficient.
http://arxiv.org/abs/0810.5602

Yu Matsumoto (Tohoku Univ.), Masahito Hayashi (Tohoku Univ.)
Title: Identification of rotation axis in Bloch Sphere
Abstract:

Title: Local asymptotic normality in quantum statistics Presentation file
Abstract: Quantum statistics deals with the processing of statistical information carried by quantum systems. One of the central problems in quantum statistics is that of estimating an unknown quantum state by performing general measurements on a (large) number of identically prepared quantum systems.
We will show how to optimally estimate a completely unknown quantum state by developing the quantum analogue of the classical statistical concept of "local asymptotic normality".
The latter means that the statistical model described by the collective state of n identically prepared quantum systems converges in a statistical sense to a model consisting of a classical Gaussian model and a quantum Gaussian state of d(d-1)/2 harmonic oscillators. Both Gaussian models have fixed variance and unknown mean which can be easily estimated by means of standard (heterodyne/homodyne) measurements. This optimal measurement can be pulled back to an optimal strategy for estimating the quantum state.
http://arxiv.org/abs/0804.3876

Keiji Matsumoto (NII)
Title: Monotone metrics on quantum channels
Abstract:

Yu Watanabe (Tokyo Institute of Technology), Takahiro Sagawa (Univ. of Tokyo) and Masahito Ueda (Univ. of Tokyo)
Title: Visualization of Quantum Operations Presentation file
Abstract: Recently many technologies of controlling quantum states have developed. In particular, methods of transferring arbitrary quantum states from one system to another have been proposed and some of them have been experimentally implemented. These technologies are important for making quantum computers and quantum networks. However there is no general measure of accuracy of transferring quantum states which can be applied for any experimental results.
By considering some examples, we relate the accuracy of transferring quantum states to that of estimating input states from output states. The accuracy of estimation is measured by Fisher information. We think that we can characterize any quantum operations (which is a mathematical formulation of quantum state transformations) by using Fisher information. In this talk, we introduce Fisher information for estimating input states from output states of any quantum operations and propose a method of visualizing quantum operation by quantifying the information.

Akihisa Hayashi (Fukui Univ.)
Title: Unitary process discrimination with error margin Presentation file
Abstract: Unitary process discrimination with general error margin m is considered. This interpolates two standard discrimination problems, minium-error discrimination for m=1 and unambiguous discrimination for m=0. We will report two solvable cases: discrimination between two unitary processes and discrimination among unitary processes generated by a group projective representation.

Paolo Perinotti (Pavia Univ.)
Title: Estimation and discrimination of quantum networks Presentation file
Abstract: We review the theory of quantum combs for description of quantum networks, and the theory of testers for measurements of network parameters. After an overview of the problems that can be tackled by means of the new theoretical framework, we show three important results recently obtained through its application.
The first problem is optimal discrimination of two transformations with a finite number of available uses. The measuring party has a fixed number of identical black boxes that perform one unknown transformation and must determine what transformation the single black box performs, out of two. In this case the theory simplifies the problem of taking into account all possible dispositions of black boxes, (e. g. in parallel or in a sequence), and provides the ultimate solution in the case of unitary transformations.
The second problem is optimal covariant estimation of unitary channels having a finite number of available uses. This problem is similar to the first one, but here the possible transformations are the whole unitary representation of a group. In this case the theory allows to prove under very general hypotheses that the parallel disposition is always optimal.
The third problem is optimal tomography of channels and quantum operations. One wants an experimental scheme that allows to reconstruct the unknown and completely free input/output transformation performed by a black box, using it and measuring it an arbitrary number of times, independently. Also in this case the optimal setup can be derived exploiting the general theory, providing a solution that exhibits some non trivial aspects that will be discussed.
Related manuscripts:
[1] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Phys. Rev. Lett. 101, 060401 (2008). http://arxiv.org/abs/0712.1325
[2] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Phys. Rev. Lett. 101, 180501 (2008).
[3] A. Bisio, G. Chiribella, G. M. D\$B!G(BAriano, S. Facchini, and P. Perinotti, Phys. Rev. Lett. 102, 010404 (2009).

Giulio Chiribella (Pavia Univ.)
Title: Optimal quantum learning and multiround reference frame alignment Presentation file
Abstract: Quantum combs and testers entail a new paradigm of quantum information processing where the input of transformations and measurements are quantum channels rather then states. In this talk, I will present two remarkable applications of this paradigm. The first application is the optimal automated learning of an unknown qubit unitary from a finite training set of N examples. The examples are first exploited in an optimal storing network, whose output state is then sent to a retrieving machine that optimally reproduces the unknown unitary M times. The second application is the optimal alignment of reference frames with multiple rounds of quantum and classical communication. In this case, quantum combs and testers provide a simple proof that the precision of reference frame alignment only depends on the total number of exchanged qubits, and that a single round of forward quantum communication is sufficient.
Related manuscripts:
[1] G. Chiribella, G. M. D'Ariano, P. Perinotti, Quantum Circuits Architecture, Phys. Rev. Lett. 101, 060401 (2008), http://arxiv.org/abs/0712.1325
[2] G. Chiribella, G. M. D'Ariano, P. Perinotti, Optimal covariant quantum networks, http://arxiv.org/abs/0812.3922
[3] A. Bisio, G. Chiribella, G. M. D'Ariano, S. Facchini, and P. Perinotti, Optimal quantum learning of an unknown qubit unitary, in preparation