GSIS Workshop on
Quantum information theory

Dates: 5-7 November, 2008
Place: Lecture Room (Room 206), 2F Graduate School of Information Sciences, Tohoku University
Organizer: a MEXT Grant-in-Aid for Young Scientists (A) and Deepening and Expansion of Statistical Mechanical Informatics (DEX-SMI), Grant-in-Aid for Scientific Research on Priority Areas, MEXT & Graduate School of Information Sciences, Tohoku University
Contact: Masahito Hayashi

## Program:

5th, November, 2008

10:00- 10:50 Matthias Christandl (Ludwig-Maximilians-University, Germany)
Post-selection technique for quantum channels with applications to quantum cryptography

10:50-11:10 Tea Break

11:10- 11:40 Tokishiro Karasawa (NII)
Practicable verifications of performance for mode-qubit systems
11:40- 12:10 Toyohiro Tsurumaru (Mitsubishi Electric)
Security proof for QKD systems with threshold detectors

12:10-14:00 Lunch Time

14:00- 14:30 Shun Watanabe (Tokyo Institute of Technology)
Tomography increases key rates of quantum-key-distribution protocols
14:30- 15:20 Nilanjana Datta (Cambridge University, UK)
Relative entropies and entanglement monotones

15:20-15:40 Tea Break

15:40- 16:30 Igor Bjelakovic (TU-Berlin, Germany)
Approximate quantum error correction under channel uncertainty

16:30-16:50 Tea Break

16:50- 17:40 Masahito Hayashi (Tohoku University)
Universal protocols in quantum information

6th, November, 2008

10:00- 10:50 Arleta Szkola (Max Planck Institute for Mathematics, Germany)
Operational meaning of selected entropy-like quantities for finite-dimensional quantum systems

10:50-11:10 Tea Break

11:10- 12:00 Andreas Winter (Bristol University, UK)
Distinguishability of quantum states under restricted families of measurements

12:00-13:40 Lunch Time

13:40- 14:10 Milan Mosonyi (Tohoku University)
Generalized relative entropies and the capacity of classical-quantum channels
14:10- 14:40 Gen Kimura (AIST)
On Generic Probability Models --- classicality and optimal state discrimination

14:40-15:00 Tea Break

15:00- 15:50 Jonas Kahn (Centre National de la Recherche Scientifique, France)
Quantum local asymptotic normality and asymptotically optimal estimation procedure for d- dimensional states
15:50- 16:20 Hiroshi Imai (NII)
Fourier Analytic Method in Phase Estimation Problem

16:20-16:40 Tea Break

Uncertainty relation between arbitrary POVMs
17:10- 17:40 Go Kato (NTT)
Quantum cloning of qubits with orthogonal states as hints

Banquet

7th, November, 2008

10:00- 10:30 Peter Turner (Univ. Tokyo)
Continuous variable two designs
10:30- 11:00 Jun Suzuki (NII)
Symmetric construction of reference-frame-free qudits

11:00-11:20 Tea Break

11:20- 12:10 Aram Harrow (Bristol University, UK)
Pseudo-random quantum states and operations

12:10-14:00 Lunch Time

14:00- 14:30 Ryo Namiki (Kyoto Univ.)
Verification of quantum-domain process using two non-orthogonal states
14:30- 15:00 Koji Azuma (Osaka Univ.)
Quantum catalysis of information and its implications

15:00-15:20 Tea Break

15:20- 16:10 Kae Nemoto (NII)
Scalable architecture and Qubus computation

16:10-16:30 Tea Break

16:30- 17:00 Satoshi Ishizaka (NEC)
Retrieving quantum operation from quantum state
17:00- 17:30 Paul Slater (University of Califolnia, USA)
Two-Qubit Separability Functions and Probabilities

## Organizing Committee:

• Masahito Hayashi (Chair), Tohoku University
• Hiroshi Imai, NII
• Gen Kimura, AIST

## Abstracts:

Matthias Christandl (Ludwig-Maximilians-University, Germany)
Title: Post-selection technique for quantum channels with applications to quantum cryptography
Abstract: We propose a general method for studying properties of quantum channels acting on an n-partite system, whose action is invariant under permutations of the subsystems. Our main result is that, in order to prove that a certain property holds for any arbitrary input, it is sufficient to consider the special case where the input is a particular de Finetti-type state, i.e., a state which consists of n identical and independent copies of an (unknown) state on a single subsystem. A similar statement holds for more general channels which are covariant with respect to the action of an arbitrary finite or locally compact group. Our technique can be applied to the analysis of information-theoretic problems. For example, in quantum cryptography, we get a simple proof for the fact that security of a discrete-variable quantum key distribution protocol against collective attacks implies security of the protocol against the most general attacks. The resulting security bounds are tighter than previously known bounds obtained by proofs relying on the exponential de Finetti theorem [Renner, Nature Physics 3,645(2007)]. This is joint work with Robert Koenig and Renato Renner.
Related manuscript: http://arxiv.org/abs/0809.3019v1

Tokishiro Karasawa (NII)
Title: Practicable verifications of performance for mode-qubit systems
Abstract: We consider how to experimentally characterize a mode-qubit system used in quantum repeaters and quantum key distributions with the optical continuous variable. In this talk, we will discuss on a state tomography, performance of a quantum gate, entanglement verifications, and their practical issues for this system. In particular, we will focus on the state tomography using a Wigner function on this system, which can be experimentally given by local measurements with the homodyne detection.

Toyohiro Tsurumaru (Mitsubishi Electric)
Title: curity proof for QKD systems with threshold detectors
Abstract: In this presentation, we rigorously prove the intuition that in security proofs for BB84 one may regard an incoming signal to Bob as a qubit state. From this result, it follows that all security proofs for BB84 based on a virtual qubit entanglement distillation protocol, which was originally proposed by Lo and Chau [H.-K. Lo and H. F. Chau, Science 283, 2050 (1999)], and Shor and Preskill [P. W. Shor and J. Preskill, Phys. Rev. Lett. 85, 441 (2000)], are all valid even if Bob's actual apparatus cannot distill a qubit state explicitly. As a consequence, especially, the well-known result that a higher bit error rate of 20% can be tolerated for BB84 protocol by using two-way classical communications is still valid even when Bob uses threshold detectors. Using the same technique, we also prove the security of Bennett-Brassard-Mermin 1992 (BBM92) protocol where Alice and Bob both use threshold detectors.

Shun Watanabe (Tokyo Institute of Technology)
Title: Tomography increases key rates of quantum-key-distribution protocols
Abstract: We construct a practically implementable classical processing for the BB84 protocol and the six-state protocol that fully utilizes the accurate channel estimation method, which is also known as the quantum tomography. Our proposed processing yields at least as high key rate as the standard processing by Shor and Preskill. We show two examples of quantum channels over which the key rate of our proposed processing is strictly higher than the standard processing. In the second example, the BB84 protocol with our proposed processing yields a positive key rate even though the so-called error rate is higher than the 25% limit.
Related manuscript: http://jp.arxiv.org/abs/0802.2419

Nilanjana Datta (Cambridge University, UK)
Title: Relative entropies and entanglement monotones
Abstract: We introduce two new relative entropy quantities, called the min- and max-relative entropies. The well-known min- and max- entropies, introduced by Renner, are obtained from these. This leads us to define a new entanglement monotone, which we refer to as the max-relative entropy of entanglement. Its properties are investigated and its operational significance as the one-shot perfect catalytic entanglement dilution rate is discussed. We also generalize the min- and max-relative entropies to obtain smooth min- and max- relative entropies. These act as parent quantities for the smooth Renyi entropies which are known to correspond to one-shot rates of various protocols of Quantum Information Theory. Further, these allow us to define smooth min- and max- relative entropies of entanglement, which we can relate to the regularised relative entropy of entanglement in the asymptotic limit.
Related manuscripts:
arXiv:0807.2536 : Max- Relative Entropy of Entanglement, alias Log Robustness,
arXiv:0803.2770 : Min- and Max- Relative Entropies and a New Entanglement Measure and, to some extent, the following paper written jointly with Renato Renner.
arXiv:0801.0282 : Smooth Renyi Entropies and the Quantum Information Spectrum

Igor Bjelakovic (TU-Berlin, Germany)
Title: Approximate quantum error correction under channel uncertainty
Abstract: Compound channels, either classical, classical-quantum (cq) or purely quantum, are among the simplest non-trivial models for communication via unknown channel. In this scenario the transmitter and receiver do not know the channel they are communicating over. They are merely provided with prior information that the actual channel belongs to a given set of memoryless channels.
In this talk we will present
1. capacity results on compound cq-channels,
2. coding theorem for finite compound quantum channels based on a channel estimation technique introduced by Datta and Dorlas, and
3. work in progress on general compound quantum channels.
This is joint work with Holger Boche and Janis Noetzel
Related manuscripts: http://arxiv.org/abs/0710.3027, http://arxiv.org/abs/0808.1007,

Masahito Hayashi (Tohoku University)
Title: Universal protocols in quantum information
Abstract: We have constructed universal codes for quantum lossless source coding and classical-quantum channel coding. In this construction, we essentially employ group representation theory. In order to treat quantum lossless source coding, universal approximation of multi-copy states is discussed in terms of the quantum relative entropy.
Related manuscripts: http://arxiv.org/abs/0806.1091, http://arxiv.org/abs/0805.4092

Arleta Szkola (Max Planck Institute for Mathematics, Germany)
Title: Operational meaning of selected entropy-like quantities for finite-dimensional quantum systems
Abstract: We review the quantum extensions of entropy-like quantities like von Neumann entropy, relative entropy, Chernoff distance or Hoeffding bound and discuss their operational meaning in the context of quantum hypothesis testing and/or quantum data compression.

Andreas Winter (Bristol University, UK)
Title: Distinguishability of quantum states under restricted families of measurements
Abstract: We investigate distinguishability of pairs of states on a quantum system under restricted classes of measurements. Any such restriction leads to a norm on trace class operators, in generalisation of the trace norm (which is recovered if all POVMs are allowed, thanks to Helstrom's classic result). We analyse properties of these norms for various classes of POVMs, in particular the constants of domination w.r.t. the trace norm: single POVMs (especially 2-, 4-, and $\infty$-designs) and POVMs obeying a locality restriction. The latter investigation is strongly related to the subject of "data hiding" [Terhal et al.], and we show that the performance of the originally proposed data hiding states is essentially optimal with regard to guessing probability.
This works is a joint work with William Matthews and Stephanie Wehner.
Related manuscript: http://arxiv.org/abs/0810.2327

Gen Kimura (AIST)
Title: On Generic Probability Models --- classicality and optimal state discrimination
Abstract: We investigate generic probability models in which probability plays a central role, including both classical and quantum theory. Two topics on the theory will be presented. First, we investigate a hidden variable problem in generic probability models, and show that a general theory is essentially classical if and only it is embeddable in a large classical theory. Second, a state discrimination problem in generic probability models is investigated. A family of ensembles is introduced which provides a geometrical method to find an optimal measurement for state discrimination. We illustrated the method in 2-level quantum systems, and reproduced the Helstrom bound for binary state discrimination and symmetric quantum states. Finally, the existence of the family is shown both in quantum and classical theory.
This work is done in collaboration with Takayuki Miyadera and Hideki Imai

Milan Mosonyi (Tohoku University)
Title: Generalized relative entropies and the capacity of classical-quantum channels (joint work with N.Datta)
Abstract: By the Holevo-Schumacher-Westmoreland theorem, the asymptotic information transmission capacity of a classical-quantum channel is equal to its Holevo capacity. By a result of Ohya, Petz and Watanabe, this in turn coincides with the divergence radius of the range of the channel, giving the capacity a geometric interpretation. In real-world applications, however, one has only finite resources, and hence it is important to have a good understanding of finite (i.e., non-asymptotic) performance measures, like the single shot capacity of a channel. Here we define generalizations of the Holevo capacity and the divergence radius usingHoeffding distances and the max-relative entropy, respectively, and show that bounds on the single shot capacity of a channel can be obtained in terms of these quantities. These results contribute to a better understanding of the operational significance of generalized relative entropies, and give also an additional insight into the deep relation between hypothesis testing and channel coding problems.
Related manuscript: http://arxiv.org/abs/0810.3478.

Jonas Kahn (Centre National de la Recherche Scientifique, France)
Title: Quantum local asymptotic normality and asymptotically optimal estimation procedure for d- dimensional states
Abstract: In classical statistics, the theory of local asymptotic normality allows to treat experiments with n independent identically distributed samples as if we were sampling only once a normal law with unknown parameter its mean. Similarly we show asymptotic strong equivalence (that is using embeddings) between experiments where we are given n copies of a qudit and one copy of a (multi- dimensional) gaussian state. We can then transpose results of one experiment to the other. In particular we get from the optimal estimation procedure for gaussian state an asymptotically optimal estimation procedure for n copies of a qudit.

Hiroshi Imai (NII)
Title: Fourier Analytic Method in Phase Estimation Problem
Abstract: For a unified analysis on the phase estimation, we focus on the limiting distribution. It is shown that the limiting distribution can be analyzed by treating the 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.
This work is a joint work with Masahito Hayashi.
Related manuscript: http://arxiv.org/abs/0810.5602

Title: Uncertainty relation between arbitrary POVMs
Abstract: Heisenberg's uncertainty principle is often considered as one of the most important features of quantum theory. In every text book on quantum theory one can find its explanation proposed by Heisenberg himself and its derivation" by Robertson. However, as recently claimed by several researchers, the above explanation and the derivation have a certain gap between them. On the one hand Heisenberg is concerned with a simultaneous measurement of position and momentum, on the other hand the Robertson's formulation treats two distinct (parallel) measurements. In this talk I will show our recent results on both formulations. For the former Heisenberg's original formulation, a limitation on simultaneous measurement of two arbitrary positive operator valued measures will be discussed. As a byproduct a necessary condition for two positive operator valued measures to be simultaneously measurable is obtained. For the later Robertson's formulation, I will show a generalization of Landau-Pollak type uncertainty relation. If possible, I would like to show some applications of our results to security proof of quantum key distribution.
Related manuscript: http://arxiv.org/abs/0809.1714

Go Kato (NTT)
Title: Quantum cloning of qubits with orthogonal states as hints
Abstract: A universal cloning machine is derived. The machine produces $c$ clones of an unknown qubit from $s$ identical replicas of qubit and $k$ identical replicas of its orthogonal qubit. For the standard cloning machine, i.e., k=0, the universal not machine, s=0, and some other cases, the optimum machine is well known. The universal cloning machine derived in this paper gives clones whose fidelity is the same value as that for the optimum machine. With some other numerical calculations, we extrapolate that the cloning machine we derived is the optimum machine.

Peter Turner (Univ. Tokyo)
Title: Continuous variable two designs
Abstract: I will discuss our ongoing attempts to construct Symmetric Informationally Complete Positive Operator Valued Measures, or (minimal) two designs, out of Gaussian states. This poses difficulties both in principle, such as how to introduce a measure on the noncompact group of symplectic transformations, as well as in practice, such as how to truncate the space of states in an experimentally useful way. Joint work with Robin Blume-Kohout.

Jun Suzuki (NII)
Title: Symmetric construction of reference-frame-free qudits
Abstract: By exploiting a symmetric scheme for coupling N spin-1/2 constituents (the physical qubits) to states with total angular momentum N/2 - 1, we construct rotationally invariant logical qudits ofdimension d = N - 1. One can encode all qudit states, and realize all qudit measurements, by this construction. The rotational invariance of all relevant objects enables one to transmit quantum information without having aligned reference frames between the parties that exchange the qudits. We illustrate the method by explicit constructions of reference-frame-free qubits and qutrits.
Related manuscript: http://arxiv.org/abs/0802.1609

Aram Harrow (Bristol University, UK)
Title: Pseudo-random quantum states and operations
Abstract: The idea of pseudo-randomness is to use little or no randomness to simulate a random object such as a random number, permutation, graph, quantum state, etc... The simulation should then have some superficial resemblance to a truly random object; for example, the first few moments of a random variable should be nearly the same. This concept has been enormously useful in classical computer science. In my talk, I'll review some quantum analogues of pseudo-randomness: unitary k-designs, quantum expanders (and their new cousin, quantum tensor product expanders), extractors. I'll talk about relations between them, efficient constructions, and possible applications.
Some of the material is joint work with Matt Hastings and Richard Low.
Related manuscript: http://arxiv.org/abs/0804.0011

Ryo Namiki (Kyoto Univ.)
Title: Verification of quantum-domain process using two non-orthogonal states
Abstract: If a quantum channel or process cannot be described by any measure-and-prepare scheme, we may say the channel is in quantum domain (QD) since it can transmit quantum correlations. The concept of QD clarifies the role of quantum channel in quantum information theory based on the local-operation-and-classical-communication (LOCC) paradigm: The quantum channel is only useful if it cannot be simulated by LOCC.
We construct a simple scheme to verify that a given physical process or channel is in QD by using two non-orthogonal states. We also consider the application for the experiments such as the transmission or storage of quantum optical coherent states, single-photon polarization states, and squeezed vacuum states.
Related manuscript: http://arXiv.org/abs/0807.0046 (Phys. Rev. A 78, 032333 (2008) )

Koji Azuma (Osaka Univ.)
Title: Quantum catalysis of information and its implications
Abstract: About 80 years ago, Heisenberg implied a groundbreaking notion that measurement on a quantum system inevitably disturbs its state, in contrast to classical systems. This suggests that our accessibility to the information contained in a quantum system is severely limited if we are to keep its state unchanged, namely if we are to use it as a $B!F(Bcatalyst$B!G(B of information. In fact, the no-cloning theorem and subsequent no-go theorems as well as the no-deleting theorem have corroborated the idea that we can never access quantum information without causing disturbance. Here, however, we deny this presumption by exhibiting a novel process, $B!F(Bquantum catalysis of information.$B!G(B In this process, a system (catalyst) helps to make a clone or to delete information without changing its state. Nevertheless, it requires interaction strong enough to enable transmission of a qubit in an arbitrary state, or to inevitably consume one ebit of entanglement if it is implemented by local operation and classical communication. This urges us to interpret that the information exchanged by the catalyst is not classical but quantum. Our result suggests that the boundary between the classical and the quantum world will not be determined on the basis of the fragility of physical states as one would expect from the current no-go theorems. This talk is based on a paper [quant-ph/0804.2426] entitled Quantum catalysis of information'' by KA, Masato Koashi, and Nobuyuki Imoto.
Related manuscript: http://arxiv.org/abs/0804.2426

Kae Nemoto (NII)
Title: Scalable architecture and Qubus computation
Abstract: Quantum computation requires two types of operations: single-qubit manipulation and two-qubit operation. However, it is very difficult to realize these two types of operations, keeping quantum coherence at the same time. Qubus computation (quantum computation via communication) was recently introduced to address these fundamental issues in implementation of scalable quantum information processing. We explain the concept of qubus computation and show some new developments and applications. We discuss the advantages and disadvantages of the qubus computation towards scalable quantum information processing.

Satoshi Ishizaka (NEC)
Title: Retrieving quantum operation from quantum state
Abstract: It is well-known that completely positive (CP) maps are associated with quantum states via the Choi-Jamiolkowski isomorphism. The associated state is easily obtained from a CP map if we apply the CP map to the half of a maximally entangled state. In this presentation, we consider the converse process: how to retrieve a CP map from the associated state. We propose and discuss a teleportation scheme to achieve the converse process in an asymptotical way.

Paul Slater (University of Califolnia, USA)
Title: Two-Qubit Separability Functions and Probabilities
Abstarct: We describe our efforts over the past several years to determine the probability that a generic two-qubit state is separable, in terms of various metrics of widespread interest, in particular, the Hilbert- Schmidt and Bures metrics. A useful concept, in this regard, is that of a separability function. In the two-qubit context, this can be a three-dimensional function of either the eigenvalues or the diagonal entries of the corresponding 4 x 4 density matrix. We have investigated the possibility that these separability functions can be exactly re-expressed as one-dimensional or univariate functions. This has, then, led us--making use of the random matrix theoretical concept of "Dyson indices"--to conjecture that the Hilbert-Schmidt separability probability for the 15-dimensional convex set of (complex) two-qubit states is 8/33, and for the 9-dimensional convex set of real two-qubit states, 8/17. The Bures case, substantially different in nature apparently, remains under active investigation.
Related manuscripts: http://arxiv.org/abs/0806.3294, http://arxiv.org/abs/0805.0267, http://arxiv.org/abs/0802.0197, http://arxiv.org/abs/0704.3723, http://arxiv.org/abs/quant-ph/0609006.