Abstract:
Dynamical Decoupling is a popular strategy to protect quantum information against a variety of noise sources. The strength of the environmental interaction causing such noise determines the experimental viability of decoupling. In many cases, though, the relevant energy scales are not only determined by the Hamiltonian, but also by the environmental state, which tends to be thermal. In this mini course we aim to understand basic properties of this common set up and derive performance bounds of decoupling as properties of the environment.

Abstracts:

Product Formulas in Quantum Dynamics
Prof. Paolo FACCHI (University of Bari, Italy)

Product formulas provide a valuable method for decomposing the dynamics of a quantum system into simpler components. These formulas possess a rich mathematical structure, and their history is extensive, with significant contributions from mathematicians and physicists such as Lie, Trotter, Feynman, Kac, Chernoff, Kato, and Nelson. Recently, these formulas have proven invaluable in quantum technologies, ranging from digital quantum simulation to quantum control and dynamical decoupling. In these lectures, I will provide an introduction to the subject, featuring several examples and recent results.

Quantum Correlations
Prof. Saverio PASCAZIO (University of Bari, Italy)

Quantum correlations are fascinating and complex phenomena described by quantum physics that cannot be explained by classical physics. These correlations have intrigued scientists for decades, as they reveal the profoundly non-intuitive nature of the quantum world. The concept of quantum correlations was first explored by Albert Einstein, Boris Podolsky, and Nathan Rosen in their famous 1935 paper, which introduced the idea of quantum entanglement. This pioneering work was further developed by Erwin Schrödinger, who coined the term "entanglement" (Verschränkung in German) to describe these quantum correlations.

Thirty years after the initial explorations by Einstein, Podolsky, Rosen, and Schrödinger, John Bell made a significant breakthrough in understanding quantum correlations. In 1964, Bell formulated what is now known as Bell's Theorem, which provided a way to test the predictions of quantum mechanics against those of classical physics. Bell's work demonstrated that no local hidden variable theory could reproduce all the predictions of quantum mechanics, thereby solidifying the idea that quantum correlations are fundamentally different from classical correlations.

Our journey into the realm of quantum correlations begins with a historical timeline of key discoveries and milestones. We examine the initial controversies and debates that surrounded the nature of entanglement and quantum correlations, which were hotly debated topics among physicists. These early discussions laid the groundwork for the profound insights and experimental confirmations that followed.

Moving beyond the historical perspective, we delve into the physical and mathematical aspects of quantum correlations. We explore the intricate mathematical formalism that describes these correlations, such as the use of Hilbert spaces, operators, and entanglement measures.

Finally, we summarize the current status of what has been learned and is known about quantum correlations.

Abstract:
We study a system of two-level quantum emitters coupled to a closed waveguide, in different geometries, and consider states in which one or two excitations are shared between the field and the emitters.

We summarize the research activity carried out during the Prin research project "Taming complexity with quantum strategies: a hybrid integrated photonic approach". We focus on relaxation to bound states, entanglement generation, multimers, and correlated photon emission, including directional single-photon emission and directional two-photon bunching.

Abstracts:

Universal Bounds on Quantum Dynamics
Prof. Paolo FACCHI (University of Bari, Italy)

I will consider the unitary evolution of quantum systems with time-dependent Hamiltonians and derive universal non-perturbative bounds in terms of the Hamiltonians and their integral actions. I will show the effectiveness of the bounds in various interesting situations of quantum asymptotics such as adiabatic evolution, strong-coupling limit, rotating-wave approximation, Trotter product formulas and dynamical decoupling.

Quantum Evolutions and Complete Positivity
Prof. Saverio PASCAZIO (University of Bari, Italy)

The state of a closed quantum mechanical system evolves according to the Schrödinger equation. If a quantum system is open, i.e. it interacts with its environment, evolution is clearly more complicated, particularly when some features of the environment cannot be described with sufficient accuracy. Surprisingly, quantum evolutions can still be characterized and precise evolution equations can be written. One of the most striking features of quantum evolutions is the notion of complete positivity. In these lectures we will give many examples and try to understand where these ideas come from.

Controlling Open Quantum Systems
Prof. Daniel BURGARTH (University of Erlangen-Nürnberg, Germany)

Noise is the primary obstacle to quantum technologies. Quantum control offers some remedies here: by carefully applying pulses to a system coupled to an environment one can shape its interaction. We will introduce dynamical decoupling and discuss error mitigation strategies.

Abstract:
We derive and discuss one- and two-dimensional (classical) field-theoretical models by making use of Hadamard's method of descent. We focus on electromagnetism (Maxwell) and spin-1/2 fields (Dirac). Low-dimensional field models are conceived as a specialization of the higher dimensional ones, in which the fields are uniform along the additional spatial directions. We consider the free situation and then the interacting fields. The basic properties of these theories, as well as their relation with existing models for two-dimensional matter, are discussed. We focus on the relevance of these findings for the quantum simulation of (lattice) gauge theories.

Abstract:
We show that every finite-dimensional quantum system with Markovian (i.e. GKLS-generated) time evolution has an autonomous unitary dilation which can be dynamically decoupled. Since quantum stochastic calculus provides also an autonomous unitary dilation which cannot be dynamically decoupled, this highlights the role of dilations in the control of quantum noise. We construct our dilation via a time-dependent version of Stinespring in combination with Howland's clock Hamiltonian and certain point-localised states. We show this dilation is very flexible and can be applied to every analytic (not necessarily Markovian) family of quantum channels.

Reference:
D. Burgarth, P. Facchi, and R. Hillier, Ann. Henri Poincaré 24, 325 (2023).

Abstract:
Control theory takes an engineering perspective on physical systems. Rather than asking "How does a system move?" it asks "How can we move a system?". I will introduce linear control which shows how control enters classical systems. In particular, I will derive the Kalman rank criterion, which decides if control is universal. I will then move to how control enters quantum physics. Here, the goal is to control quantum computers, and I will give the quantum analogue of the Kalman criterion.

Abstract:
We derive and discuss one- and two-dimensional models for classical electromagnetism by making use of Hadamard's method of descent. Low-dimensional electromagnetism is conceived as a specialization of the higher dimensional one, in which the fields are uniform along the additional spatial directions. We then consider two-dimensional models for a charged spin-1/2 particle, both in the free case and in the presence of the electromagnetic field, by applying the same reduction technique. The basic properties of these theories, as well as their relation with existing models for two-dimensional matter, are discussed.

Abstract:
We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser theorem in classical mechanics.

References:
D. Burgarth, P. Facchi, H. Nakazato, S. Pascazio, and K. Yuasa,
"Kolmogorov-Arnold-Moser Stability for Conserved Quantities in Finite-Dimensional Quantum Systems," Phys. Rev. Lett. 126, 150401 (2021);
"Eternal Adiabaticity in Quantum Evolution," Phys. Rev. A 103, 032214 (2021).

Abstract:
The rotating wave approximation is a commonly used approximation in quantum physics. Often, it is taught in a very ad-hoc manner, without a proper justification. In this lecture I will give a gentle introduction to the subject and show how the quality of the approximation can be bounded.

Abstract:
It is well known that almost all quantum systems are controllable. That is, if we have a random time-independent Hamiltonian H, and the ability to switch on and off an additional Hamiltonian C, which might describe a laser or a magnetic field, then with probability one the resulting dynamics is universal for quantum computation. But is this practical? In this seminar we will explore the consequences of this generic controllability for practical quantum computation.

Abstract:
We give a brief introduction to graph theory. After defining walks, trails and paths, as well as eccentricity, radius and diameter, we define the adjacency and incidence matrices and briefly describe the general features of their spectra. We then introduce random networks, by following the definition by Erdős and Rényi, and discuss the associated percolation phase transition. We finally present a novel characterization of complex networks, based on the potential of an associated Schrödinger equation. The potential is designed so that the energy spectrum of the Schrödinger equation coincides with the graph spectrum of the normalized Laplacian. We show that this quantum-inspired technique can be successfully employed in the study of random networks, as an alternative indicator of the percolation phase transition.

Abstract:
We will give a brief tutorial to the operational description of a physical system in terms of its algebra of observables. The basic structures of C*-algebras and of states will be discussed, and the famous GNS construction of a Hilbert-space representation will be explained.

Abstract:
We will give a method to suppress noise, which is one of the major obstacles to obtain an optimal solution in quantum annealers. We generalize the conventionally used Hamiltonian, i.e., the transverse field Hamiltonian, by introducing an ancilla system, which leads to cancellation of the effect of noise on the system under consideration for some typical cases.

Abstract:
Dynamical decoupling is a common control technique to remove unwanted environmental interactions. The basic idea is to rotate the system actively such that environmental interactions average out. It is folk-knowledge that decoupling works only with well-behaved environments which induce non-Markovian decay, because such environments provide a time-scale on which they have a memory. The key object to understand such behaviour is the spectral density of the environment. To put such statements into a rigorous framework we need to introduce some notions from mathematical physics, in particular Trotter-Suzuki limits, allowing us to describe unbounded generators, which naturally occur as harmonic oscillators or in the continuum limit. We will show that the spectral density is not always a good object to understand the control properties of a noisy system. In particular there are systems with non-Markovian decay which cannot be dynamically decoupled.

講演要旨:
機械学習には多くの手法，アルゴリズムがあり，また，個々の方式の枝分かれ的発展も著しい．代表的な手法を2, 3ピックアップして，それらの数理的バックグラウンドを解説する．よく聞く「AIを活用して...に成功」といったニュースの背後で何がなされたかを想像できるようになろう．

講演要旨:
量子計算機や多くの量子技術の実現には，多自由度の量子系を自在に制御することが求められる．しかし，そのすべてを直接制御することは現実的ではないし，その必要もない．周辺環境や制御装置自体との相互作用を考えると，直接の制御対象とする自由度は少ない方がよい面もある．実際，既存の実験において半ば無意識的に外界からの制御プローブの数は制限している．では，制御プローブの制限を進めたとき，制御できることにどのような制限をうけるのだろうか？この問いを突き詰めていく過程で，直接アクセスの制限によって，間接的に制御される量子系に対応するヒルベルト空間に特徴ある構造が誘起されることがわかった．その構造は，原理的に制御不可能な部分空間や，プローブの空間次元の違いによる制御可能性の質的変化などを鮮やかに示していた．本講演では，こうした構造の完全な分類と，制御という観点で見たときにその構造の意味するところを紹介する．

参考文献:
Go Kato, Masaki Owari, and Koji Maruyama, arXiv:1803.11128 [quant-ph].

Abstract:
Several "weak" quantum computing models have been proposed recently, such as the Boson Sampling, the IQP model, and the DQC1 model. These models are severely restricted and do not seem to be universal, but have been shown to be impossible to classically efficiently simulate unless the polynomial-time hierarchy collapses. The collapse of the polynomial-time hierarchy is not believed to happen in computer science, and therefore these results suggest "quantum supremacy" of these sub-universal quantum computing models. In this talk, I explain basics of these results, and several protocols that can verify the correctness of these computing with almost classical technology.

Reference:
[1] Y. Takeuchi and T. Morimae, Phys. Rev. X 8, 021060 (2018).
[2] J. F. Fitzsimons, M. Hajdušek, and T. Morimae, Phys. Rev. Lett. 120, 040501 (2018).
[3] T. Morimae, K. Fujii, and J. F. Fitzsimons, Phys. Rev. Lett. 112, 130502 (2014).

Abstract:
Quantum many-body systems are ubiquitous in physics ranging from elementary particle physics to condensed matter physics. Recently, ultracold atomic gases, in the field of atomic physics, have been receiving attention, since they are ideal platforms to look at quantum many-body problems in detail. In this seminar, after reviewing basic aspects in this field, I will discuss quantum mesoscopic transport phenomena with ultracold atomic gases, among the themes of my recent studies.

Abstract:
Quantum simulators, machines that can replicate the dynamics of quantum systems, are being built as useful devices and are seen as a stepping stone to universal quantum computers. A key difference between the two is the ability to perform the logic gates that make up computational algorithms. We propose a method for learning how to construct these gates efficiently by using the simulator to perform optimal control on itself. This bypasses two major problems of purely classical approaches: the need to have an accurate model of the system, and a classical computer more powerful than the quantum one to carry out the optimisation. For numerical simulations of up to 9 qubits with Ising interactions, this in situ approach scales polynomially in the number of qubits for different topologies suggesting that this is a practical way of upgrading quantum simulators to computers.

Reference:
B. Dive, A. Pitchford, F. Mintert, and D. Burgarth, arXiv:1701.01723 [quant-ph].

Abstract:
Practical implementations of quantum technology are limited by unavoidable effects of decoherence and dissipation. Starting from a microscopic model of a structured environment (a crystal) sensed by two bosonic probes, here we show the failure of such conceptual relation, and identify the exact physical mechanism underlying this cross-over, displaying a sharp contrast between dephasing and dissipative baths. Depending on the frequency of the system and, crucially, on its orientation with respect to the crystal axes, collective dissipation becomes possible for very large distances between probes, opening new avenues to deal with decoherence in phononic baths. In addition another phenomenon occurring is the spontaneous synchronization between the two bosonic probes. We provide a scenario for the emergence of local coherent dynamics in an extended harmonic system. We address the mechanisms favoring synchronization, even between not directly coupled oscillators comparing the possible buildup of quantum correlations during synchronization.

Abstracts:

Wave Propagation: From d'Alembert to Dirac
Prof. Saverio PASCAZIO (University of Bari, Italy)

The mechanism of wave propagation has fascinated mathematicians and physicists for centuries. Waves characterize many physical phenomena and arise in many fields: familiar examples are sound in acoustics, light in electromagnetism, and ocean waves in fluid dynamics. In quantum mechanics, waves coexist with particles (giving rise to the so-called wave-particle dualism).

The one-dimensional wave equation was first understood and discovered by Jean le Rond d'Alembert in 1746. A few years later, Leonhard Euler wrote the three-dimensional wave equation. The main mathematical and physical ingredients at the origin of wave propagation were also studied by Daniel Bernoulli and Joseph-Louis Lagrange. They were all trying to understand vibrating strings and sound propagation in musical instruments.

James Clerk Maxwell first predicted the existence of electromagnetic waves, by careful examination of the equations of electromagnetism. The unification of light and electrical phenomena was one of the greatest achievements of the 19th century.

Mathematically, the wave equation is a second-order linear hyperbolic partial differential equation. As such, it models processes which evolve over time. It can be rigorously studied, and admits existence and unicity theorems. However, when a wave meets an obstacle, part of it is reflected off the surface of the material, while some is transmitted through the material, and the problem often becomes too complicated to admit analytic (simple) solutions. In such a case, a famous principle is often invoked, that enables one to guess the form of the solution.

Every point on the wave front of a propagating wave is a source of secondary wavelets, which spread forward at the same speed as the source wave. The wave front at later times is then given by the surface tangent to the secondary wavelets. This principle was proposed by Christiaan Huygens in 1678, to explain the laws of reflection and refraction. It was used again more than a century later, in 1816, by Augustin-Jean Fresnel, to interpret the diffraction effects that occur when visible light encounters slits, edges and screens.

The principle provides crucial insight into the nature of wave propagation and it is a milestone in the physics of undulatory phenomena. For this reason, its universal validity is usually taken for granted. However, yet one century later, Jacques Hadamard noticed that Huygens' principle is valid only when waves propagate in an odd number n > 1 of spatial dimensions. The principle is therefore neither valid for n = 1, nor for even n.

Both quantum mechanics and quantum field theory make use of wave equations in their formulation. It is therefore interesting to ask whether Huygens' principle holds for the seminal equations that are the backbone of these theories. The Schrödinger equation, being non-relativistic, does not admit a satisfactory formulation of this question. What about the Dirac equation?

We shall discuss the validity of Huygens' principle for the massless Dirac-Weyl equation. We shall find that the principle holds for odd space dimension n, while it is invalid for even n. We explicitly discuss the cases n = 1, 2 and 3.

Operator Semigroups and Product Formulae
Prof. Paolo FACCHI (University of Bari, Italy)

One-parameter semigroups of transformations describe the time evolution of autonomous deterministic systems. The concept of semigroups encompasses the structure of the solutions of initial-value problems for ordinary and partial differential equations. It ranges from the heat and the wave equation of classical physics, up to the Schrödinger and the master equation of quantum physics. The continuity property of these semigroups is the key to a deep and beautiful theory, that has become an indispensable tool in a great number of areas of modern mathematics and physics.

In these lectures I will first introduce the basic properties of strongly continuous semigroups of linear operators on Banach spaces. Then I will touch upon perturbation theory and Dyson expansion. Finally we will have a look at approximation theory and at limit product formulae, ranging from Trotter-Kato to the quantum Zeno effect.

Introduction to Completely Positive Maps
Prof. Daniel BURGARTH (Aberystwyth University, UK)

The natural generalization of a probability distribution to quantum mechanics is the density matrix, which is a positive matrix with unit trace. Stochastic maps are generalized as completely positive maps, which have a rich and beautiful structure, and map density matrices to density matrices. The purpose of these lectures is to introduce some of the interesting properties such maps have and outline some applications in quantum theory.

Abstract:
We investigate the possibility to suppress unbounded interactions between a finite dimensional system and an infinite dimensional environment through a fast sequence of unitary kicks on the finite dimensional system. This method, called dynamical decoupling, is known to work for bounded interactions, but physical environments such as bosonic heat baths are usually modeled with unbounded interactions, whence here we initiate a systematic study of dynamical decoupling for unbounded operator. We develop a sufficient decoupling criterion for arbitrary Hamiltonians and a necessary decoupling criterion for semibounded Hamiltonians. We give examples for unbounded Hamiltonians where decoupling works and the limiting evolution as well as the convergence speed can be explicitly computed. We show that decoupling does not always work for unbounded interactions and provide both physically and mathematically motivated examples.

Abstract:
Quantum Control is a highly successful approach to understanding quantum dynamics and tailoring it to one's needs. In this lecture, we will discuss the mathematical background to control, which includes results such as Trotter's Formula and the Quantum Recurrence Theorem with high relevance in other areas of quantum mechanics. We will then introduce the central concept of Reachability in control and characterize its properties in the context of Quantum Computation. Finally we discuss optimal control and its numerical implementation.

Abstract:
We consider a finite disordered 1D quantum lattice with an attached impurity. We study the transport of a single electron from the impurity to the lattice at zero temperature. The transport is dominated by Anderson localization and in general it has a random character due to the lattice disorder. However we show theoretically that, by adjusting the impurity energy, the electron can attain quite regular, periodic motions. In particular, by a precise tuning of the impurity energy, the electron can be set to oscillate between the impurity and lattice sites far from the impurity, at distances larger than the Anderson localization length, and with long periods of oscillation. This phenomenon may be observable in analogous optical systems.

講演要旨:
Why does the time flow only in one particular direction? This is a very big and fundamental question of physics, called the problem of "the arrow of time." There are a few types of the arrow of time. The cosmological arrow of time refers to the direction of time in our universe as we see it. The psychological arrow of time refers to the direction of time that we perceive.

The arrow of time that I am going to discuss in the present talk is the dynamical arrow of time, which refers to the following problem. The fundamental equations of motion, namely the Newton equation for classical mechanics and the Schrödinger equation for quantum mechanics, are symmetric if we flip the arrow of time. Nonetheless, we see the excited states always decay and the entropy always increases. How can this inconsistency arise? There have been many answers to it but most of them attribute the reason to the limitation of our ability of measuring physical systems. My collaborator Gonzalo Ordonez and I have reservations because it almost sounds like the dynamical arrow of time could be flipped or eliminated if we have more abilities!

We answer the question of the dynamical arrow of time in the realm of quantum mechanics in the following way. We succeeded in deriving a time-reversal symmetric decomposition of the time evolution operator which contains both decaying states and growing states symmetrically. If we watch the time evolution starting from an initial condition, we pick all the decaying states, whereas if we watch the time evolution ending in a terminal condition, we pick all the growing states. This shows that the arrow of time is generated by the difference in the way we watch the time evolution.

Abstract:
In this talk we will discuss the relation between the incompatibility of quantum measurements and quantum nonlocality. In a recent paper we show that any set of measurements that is not jointly measurable (i.e. incompatible) can be used for demonstrating EPR steering, a form of quantum nonlocality. This implies that EPR steering and (non)joint measurability can be viewed as equivalent. Moreover, we discuss the connection between Bell nonlocality and joint measurability, and give evidence that both notions are inequivalent. This suggests the existence of incompatible quantum measurements which are Bell local, similarly to certain entangled states which admit a local hidden variable model. Finally, we will discuss applications of these results to problems in joint measurability, and for EPR steering using randomly chosen measurements.

Reference:
[1] Marco Túlio Quintino, Tamás Vértesi, Daniel Cavalcanti, Remigiusz Augusiak, Maciej Demianowicz, Antonio Acín, and Nicolas Brunner, arXiv:1501.03332 [quant-ph].
[2] Marco Túlio Quintino, Tamás Vértesi, and Nicolas Brunner, Phys. Rev. Lett. 113, 160402 (2015).
[3] Joseph Bowles, Tamás Vértesi, Marco Túlio Quintino, and Nicolas Brunner, Phys. Rev. Lett. 112, 200402 (2014).

Abstract:
We show that the mere observation of a quantum system can turn its dynamics from a very simple one into a universal quantum computation. This effect, which occurs if the system is regularly observed at short time intervals, can be rephrased as a modern version of Plato's Cave allegory. More precisely, in the original version of the myth, the reality perceived within the Cave is described by the projected shadows of some more fundamental dynamics (the Ideals) which is intrinsically simpler (intelligible). We found that in the quantum world the "projected" reality perceived through sequences of measurements is even more complex than in a classical world. After discussing examples we go on to show that this effect is generally to be expected: almost any quantum dynamics will become universal once "observed" as outlined above. Conversely, we show that any complex quantum dynamics can be "purified" into a simpler one in larger dimensions.

Reference:
[1] D. Burgarth, P. Facchi, V. Giovannetti, H. Nakazato, S. Pascazio, and K. Yuasa, "Exponential Rise of Dynamical Complexity in Quantum Computing through Projections," Nature Communications 5, 5173 (2014).
[2] D. Orsucci, D. Burgarth, P. Facchi, H. Nakazato, S. Pascazio, K. Yuasa, and V. Giovannetti, "Hamiltonian Purification," arXiv:1411.0316 [quant-ph].

講演要旨:
エンタングルメントは量子力学的相関を特徴付ける概念であり，量子情報科学の発展とともに，その重要性が再認識されつつある．現在エンタングルメントは，量子情報科学のみならず，統計物理学，物性物理学，高エネルギー物理学，計算物理学など諸般の物理学における普遍的な概念として，その地位を確立している．エンタングルメントの統計物理学的側面に話を限ると，これまでは，1次元量子系におけるエンタングルメント特性が精力的に研究されてきた．2次元量子系の場合は計算の困難さから，エンタングルメント特性の探求が未開拓の領域であった．私は複数の 2次元可解量子系に注目し，そのエンタングルメント特性を検討した．一つは，2次元格子上に定義された VBS (Valence Bond Solid) 状態 [1,2]．もう一つは，ラダー上に定義された量子格子気体模型の基底状態である [3]．これらの状態について，エンタングルメントエントロピー，エンタングルメントスペクトル，ネステッド・エンタングルメントエントロピー [2,4] を用いてエンタングルメント特性を検討した．その結果，これらの 2次元可解量子系のエンタングルメント特性が，1次元量子系の物理特性と対応することを明らかにした．

本講演に関する研究は，桂法称氏 (東大院理)，川島直輝氏 (東大物性研)，Anatol N. Kirillov 氏 (RIMS)，Vladimir E. Korepin 氏 (C. N. Yang Institute for Theoretical Physics)，Jie Lou 氏 (Fudan University)，田村亮氏 (物材機構) との共同研究である．

参考文献:
[1] H. Katsura, N. Kawashima, A. N. Kirillov, V. E. Korepin, and S. Tanaka, J. Phys. A 43, 255303 (2010).
[2] J. Lou, S. Tanaka, H. Katsura, and N. Kawashima, Phys. Rev. B 84, 245128 (2011).
[3] S. Tanaka, R. Tamura, and H. Katsura, Phys. Rev. A 86, 032326 (2012).
[4] S. Tanaka, Interdisciplinary Information Sciences 19, 101 (2013).

講演要旨:
1988年に Aharonov らによって提案された弱測定は，状態をほぼ壊さずに測定する量子測定であり，重ね合わせ状態の測定や運動量と位置のような非可換な物理量の連続した測定を可能にする．しかし弱測定は多くの場合，確率が負や複素数となる結果をもたらす．弱測定がもたらすこのような奇妙な結果は，従来の延長線上では考えにくいものである．近年，我々は全く測定しない状態から完全に射影した状態まで測定の強さを制御できる光子の偏光測定の実現に成功し，この装置を使って二つの非可換物理量の連続した測定を行った．この実験は初段を完全に射影測定をせずに二段目に射影測定をすることによって実現することが可能である．我々は初段の測定装置が及ぼす不確定性を調べ，量子測定理論や弱測定理論を使わずに実験データから測定前の確率分布を推定したところ，予想された確率分布 (Kirkwood-Dirac 擬確率分布) と一致した．この結果は，直接得られない測定前の確率分布と測定後の確率分布は基本的に異なること，測定が及ぼす影響によって測定後は正の確率分布が得られること，負や複素数の確率分布が量子系の本来の統計的性質である可能性が高いこと，などを示している．このような解釈は弱測定を使わず，弱い領域から強い領域にまで行った測定結果からの帰結であり，これは量子状態は負や複素数の値を含む拡張された確率分布で記述できることを示唆する結果である．さらにこの確率分布は測定から直接得られないが，その意味では密度行列と同じであり，しかも数学的には完全に 1対1 で対応する．セミナーでは偏光を使った連続測定実験の結果を紹介し，上記の解釈ができる根拠を示す．

A weak measurement, which was proposed by Aharonov, et al. in 1988, is a quantum measurement leaving the coherence of the initial state intact and far from a normal projection measurement. It enables us to measure a superposition state with a tiny destruction or perform a sequential measurement of one observable followed by a non-commuting observable, such as position and momentum. However, the results by weak measurements sometimes provide negative and complex probabilities, which are quite different from normal statistics. This strange phenomenon is confusing in terms of our normal sense. Recently, we realized a variable strength measurement of photon polarization, which is capable of controlling a measurement strength from zero (no measurement) to fully projection (completely destructive measurement) and performed the sequential measurement of two non-commuting observables in photon polarization. Such sequential measurement becomes feasible by performing the non-fully projective measurement in the first stage with our apparatus and the fully projective measurement in the second stage. We investigated the role of measurement uncertainties caused by the first measurement and estimated inaccessible joint probabilities before the measurements without the use of quantum measurement theory or weak measurement theory. Consequently, the obtained joint probabilities were consistent with the predicted ones (Kirkwood-Dirac pseudo-probability distribution). These results show that the inaccessible joint probabilities before the measurement are essentially different from the ones after the measurement, the inaccessible joint probabilities are converted to the normal positive statistics due to the influence of the measurement, and there is a high possibility that negative and complex probability just show intrinsic statistical properties of quantum systems. These interpretations can be derived from the experimental results in not only weak regime, but also strong regime of the measurement strength without the use of weak measurement concept. In addition, these results also imply that quantum state can be described with extended probabilities including negative and complex numbers. These joint probabilities are mathematically equivalent to density matrix and also have the common characteristics of no directly-measurable quantities. In this seminar, I will introduce our experimental results and show the possible reasons for the above interpretation.

参考文献:
Y. Suzuki, M. Iinuma, and H. F. Hofmann, New J. Phys. 14, 103022 (2012).

Abstract:
Noise is a limiting factor for quantum technologies. Any strategy to overcome it is therefore important. I will give an introduction to dynamical decoupling as a method to suppress noise in finite dimensional quantum systems. Then I will discuss the important case of infinite dimensional systems, were some more powerful mathematical tools need to be applied, but the essential results remain the same. Finally I will suggest applications of dynamical decoupling for foundational problems in quantum mechanics.

Reference:
C. Arenz, R. Hillier, M. Fraas, and D. Burgarth, arXiv:1405.7644 [quant-ph].

講演要旨:
量子系をフィードバックで制御しようとしたとき，2つの選択肢があります．1つ目は，系を測定してその測定結果を用いてフィードバックする，というもので，2つ目は，測定を介さずに量子レベルでフィードバックする，というものです．前者は古典信号処理プロセスを含んでいるので古典フィードバック，他方，後者は量子フィードバックと呼べます．これらを比較し，性能の優劣を議論することは興味深い問題です．広く捉えると，この問題は古典情報処理系と量子情報処理系の優劣を論ずるものになるからです．この研究では，特に BAE (Back-Action Evasion) 実現，QND (Quantum Non-Demolished) 物理量の生成，DFS (Decoherence Free Subsystem) の生成，という 3つのゴールを，上の 2種類のフィードバック機構で達成できるか否かを議論します．これらは重力波検出，非古典状態生成，量子メモリ構築などを含む，量子情報処理問題全般で重要なゴールです．鍵となるアイデアは，「線形システム理論の利用」です．これは，入出力を有するシステムを解析・制御するための一般理論で，それ自体は非常に成熟したものですが，近年では量子系をはじめとする様々な物理システムの解析・制御でも非常に有効であることが分かっています．この理論に基づいて BAE，QND，DFS の概念の整理統合を行い，上記問題へアタックする足がかりとします．

参考文献:
arXiv:1406.6466.

講演要旨:
達成可能な熱機関効率の上限としてのカルノー効率の導出は，熱力学の最も重要な結果の一つであるが，その統計力学的な理解は未だ不完全な状態にある [1]．我々はこの達成証明を情報理論の数学的テクニック [2,3] を用いて与え，かつそれを熱浴と熱機関が共に有限粒子系である場合にも拡張する [4]．我々はまず，n 粒子によって構成される熱浴と接触する，L 粒子の熱機関の効率上界を与える．次に，この上界を 1/n について漸近展開し，その second order までが等号達成可能であることを示す．漸近展開の first order である定数項はカルノー効率であるので，我々の結果は粒子数無限大の極限でカルノー効率の統計力学的達成証明を corollary として含む．我々の結果は，有限サイズ効果によって熱機関の効率がいかに低下するかを与え，かつ，熱機関効率に影響を与える有限サイズ効果は，本質的に熱浴由来の物のみである事を与える．熱機関自体の有限サイズ性は，その表面積の有限性によって熱浴を有限にするような，間接的な物に限られる．

参考文献:
[1] 清水明，「熱力学の基礎」，東京大学出版会 (2007)．
[2] R. Bahadur and R. R. Rao, "On Deviations of the Sample Mean," Ann. Math. Stat. 31, 1015 (1960).
[3] C. Joutard, "A Strong Large Deviation Theorem," Math. Meth. Stat. 22, 155 (2013).
[4] H. Tajima and M. Hayashi, arXiv:1405.6457.

Abstract:
Quantum Control is a highly successful approach to understanding quantum dynamics and tailoring it to one's needs. In this lecture, we will discuss the mathematical background to control, which includes results such as Trotter's Formula and the Quantum Recurrence Theorem with high relevance in other areas of quantum mechanics. We will then introduce the central concept of Reachability in control and characterize its properties in the context of Quantum Computation.

Abstract:
The definition and characterization of the quantum features of a physical system is an important problem, both for its fundamental implications and its practical aspects, due to the advent of quantum information processing. There are tasks in computation and communication that can be efficiently performed only if quantum resources are available. Classical systems are not suitable for such applications. For these reasons, it is very important to give proper (and rigorous) definitions of classicality and quantumness. In this talk we propose a definition of classical states and quantumness witnesses by focusing on the algebra of observables of the system. A central role is assumed by the anticommutator of the observables, namely the so-called Jordan symmetric product. This approach turns out to be suitable for generalizations to infinite dimensional systems.

Reference:
P. Facchi, L. Ferro, G. Marmo, and S. Pascazio, J. Phys. A 47, 035301 (2014).

Abstract:
One major contemporary application of quantum mechanics is cryptography. In quantum key distribution (QKD), the goal is to distribute secret keys between two parties. Once a secret key is shared, it is used to encrypt a message on a public channel, where a third party has negligible probability of successful decryption. A drawback, however, is that the key and the message must have equal length. This yields a major challenge for QKD, i.e., the distribution of secret keys at bit rates comparable to standard communication technology. We consider a different approach based on the phenomenon of quantum data locking (QDL). As a cryptographic primitive, QDL allows the direct transmission of the encrypted message, while it is ill-suited for QKD. Due to a strong form of the uncertainty principle, a small secret key can be used to "lock" an exponentially longer message, if this message is suitably encoded into a quantum system. To find useful application, QDL should be proven to be robust under noise. The aim of our work is to put noisy QDL on a solid theoretical ground. We formally define the locking capacities of a quantum channel, and prove bounds and results concerning these quantities. What are the quantum channel with nonzero locking capacity? Is the channel required to preserve entanglement, or discord is sufficient? Can the locking capacity surpass the private capacity? Is QDL possible with coherent states? We address all these questions and, while several challenging questions remain open, we give complete or partial answers to a few them.

Reference:
arXiv:1307.5368.

Abstract:
The entanglement of the pure states of a large quantum system is investigated by applying the tools of statistical mechanics. The spectral distribution of the bipartite entanglement is computed by mapping the problem onto a random matrix model and by using a Coulomb gas method due to Dyson. Entanglement exhibits a rich structure, with three different regimes, corresponding to two phase transitions of the matrix model. The two critical points are associated with sudden changes in the shape of the entanglement spectra. We also briefly touch upon the characterization of multipartite entanglement and the search for maximally multipartite entangled states, highlighting the links with frustration.

Abstract:
We discuss the quantum-mechanical motion of a non-relativistic particle in a bounded domain of physical space, bringing out both the analogies and the differences with the corresponding classical motion. We will show that on physical as well as mathematical grounds the classical and quantum problems are not equivalent. In particular, we will consider the case of varying domains and/or boundary conditions, that can be viewed as a model of spacetime topology change. We will explicitly show how geometric terms naturally arise in the Hamiltonian and, in the case of rapidly changing boundary conditions, we will exhibit a dynamical composition law ruling the superposition of different topologies. Possible experimental implementations with superconducting quantum interference devices and microwave cavities are explored and analyzed.

References:
[1] M. Asorey, P. Facchi, G. Marmo, and S. Pascazio, J. Phys. A 46, 102001 (2013).
[2] S. Di Martino, F. Anzà, P. Facchi, A. Kossakowski, G. Marmo, A. Messina, B. Militello, and S. Pascazio, J. Phys. A 46, 365301 (2013).

Abstract:
The understanding of the microscopic mechanisms which determine the macroscopic laws of heat and particle transport is one of the main problems of statistical mechanics. On the other hand, thermoelectric phenomena, which involve the conversion between thermal and electrical energy, and provide a method for heating and cooling materials, are expected to play an increasingly important role in meeting the energy challenge of the future. Here we discuss a new approach to this problem, which is rooted in nonlinear dynamical systems. More precisely we will discuss idealized models of interacting particles in one and two dimensions.

Reference:
G. Benenti, G. Casati, and J. Wang, Phys. Rev. Lett. 110, 070604 (2013).

Abstract:
We study the dynamics of a spin that interacts with a surrounding spin environment. This system has recently gained renewed interest due to its experimental implementation in NV-centers in diamond. The experiments are amongst the most promising candidates for quantum computation, and it is therefore interesting to investigate their properties from a control perspective.

We show that it is possible to fully control the central spin by applying a classical field on it. The dimension of the dynamical Lie-algebra depending on the number of surrounding spins is compared with the time one needs to implement a target unitary transformation on the central spin. Furthermore we investigate the question how the control field influences the dynamics, and come to a surprising conclusion.

講演要旨:
複数の熱浴によって駆動される量子非平衡系の解析には，縮約行列の方法がよく用いられる．縮約行列の方法とは，注目系と相互作用する無限に広がる環境系を考え，そこから環境系をトレースアウトして得られる注目系の密度行列を調べる手法である．縮約密度行列の時間発展を記述する最も一般的な時間発展生成子に Lindblad 型が挙げられる．Lindblad 時間発展子は，完全正値かつ Markov 的な時間発展を記述する．近年，講演者らは，注目系に加えて環境系の有限の一部 (メゾ熱浴) を残した系を注目系として縮約行列の方法を適用することで，熱浴の性質も含めた系の振る舞いを調べる方法 (メゾ熱浴の方法) を提案した．この際，メゾ熱浴は Lindblad 型の演算子によって平衡状態近傍に保たれる．本講演では，第一に，メゾ熱浴の方法が Landauer 公式などの既知の結果を再現することを示す．また，メゾ熱浴の平衡状態からのずれが粒子流を記述する上で重要であることを説明する．第二に，メゾ熱浴の方法を用いて，XY 鎖，平均場 Hubbard モデルについて調べた結果を紹介する．XY 鎖については，ユニタリー時間発展により構成される定常状態との比較も行う．

講演要旨:
We study collective excitations in three- and four-band superconductors with inter-band frustration, which may lead to a violation of time-reversal symmetry. Using a low-energy spin-Hamiltonian, we find that a massless Leggett mode occurs in a wide parameter region of this four-band system. This massless mode is related to a degeneracy between a time-reversal-symmetry-breaking state and a time-reversal-symmetric state. Therefore, the mass of this collective mode characterizes well the time-reversal symmetry in frustrated multi-band superconductors.

講演要旨:
コンピュータの発達に伴う高度情報化により，ウェブ上での人々の書き込み，携帯電話の通話記録，PASMO などの乗車記録など詳細な行動履歴がデータとして記録されるようになり，最近ではビッグデータ解析という言葉も新聞やテレビなどでしばしば聞くようになった．本発表では，詳細なデータ解析から経験則を確立し，単純化した数理モデルによってこの経験則を再現することにより現象を理解し，応用を目指すという基礎科学，特に物理学の手順でビッグデータ解析を行った結果を紹介する．最初に，約3000万人のブロガーからランダムサンプリングされた30万人のブロガーが投稿した約5000万記事のデータを解析した結果を報告する．個人が自由意志で書き込んだブログも大量に集めると，例えば流行語がどのように広がったかなど，ブームの形成や収束を定量的に解析することが可能となる．また，ブロガーの行動をモデル化したブログ投稿モデルを構築しブロガーというミクロな構成要素がどのような行動をすると，ブームというマクロな現象が創発されるかを明らかにする．

参考文献:
高安美佐子，山田健太 (3,4章担当)，他 "ソーシャルメディアの経済物理学"，日本評論社，2012.

講演要旨:
近年，熱的アンサンブルをエルゴード仮説によらず，純粋状態が内包する統計的性質から直接導く興味深い試みが国内外のグループによって独立に示された．この理論によると熱平衡値はヒルベルト空間における一個の典型的な純粋状態における期待値として計算できる．本講演では，扱う物理量のクラスを多体相関を含むものまで拡張することで，この理論を非平衡系に適用する．特に，エントロピー生成の分布関数や輸送係数等を一個の純粋状態によって計算できることを示す．

講演要旨:
Entangled states can potentially be used to outperform the standard quantum limit by which every classical sensor is bounded. However, entangled states are very susceptible to decoherence, and so it is not clear whether one can really create a superior sensor to classical technology via a quantum strategy which is subject to the effect of realistic noise. This talk presents an investigation of how a quantum sensor composed of many spins is affected by independent dephasing. We adopt general noise models including non-Markovian effects, and in these noise models the performance of the sensor depends crucially on the exposure time of the sensor to the field. We have found that, by choosing an appropriate exposure time within the non-Markovian time region, an entangled sensor does actually beat the standard quantum limit. Since independent dephasing is one of the most typical sources of noise in many systems, our results suggest a practical and scalable approach to beating the standard quantum limit.

参考文献:
Phys. Rev. A 84, 012103 (2011).

Abstract:
The properties of bipartite entanglement of a quantum system in a pure state are encoded in its entanglement spectrum, given by the eigenvalues of the reduced density matrix of one of its two subsystems. We will show that for large systems the typical spectrum is given by the minimal-energy equilibrium distribution of a 2D Coulomb gas of charges. We will consider the case of an unbiased ensemble of pure states and the case of isoentangled manifolds with a given value of the local purity or von Neumann entropy. We will unveil the existence of first-order and continuous phase transitions, characterized by changes in the physiognomy of the entanglement spectrum.

References:
[1] Phys. Rev. Lett. 101, 050502 (2008).
[2] Phys. Rev. A 81, 052324 (2010).
[3] J. Phys. A 45, 015308 (2012).

Abstract:
A closed quantum system evolves according to the Schrödinger equation. When the quantum system is "open," namely in interaction with its own environment, the picture becomes richer and more complicated. The objectives of these lectures is to try and understand how an open quantum system behaves. This will shed light on the very structure of quantum mechanics and some of its most interesting aspects.

1. BASIC IDEAS
Evolution of quantum states: vectors and density matrices. The exponential decay law. The Fermi Golden Rule.

2. OPEN QUANTUM SYSTEMS
Role of the environment. Short, intermediate and long times. The quantum Zeno effect.

3. MASTER EQUATION
Density matrices and superoperators. The Master equation. The role of Markovianity. Memory effects.

4. FROM POSITIVITY TO COMPLETE POSITIVITY
The modern approach. The Gorini-Kossakowski-Sudarshan-Lindblad structure of the evolution.
The Kraus-Sudarshan representation. Positivity and complete positivity.
The role of entanglement. The foundations of quantum mechanics.

Abstract:
We discuss an implementation of quantum Zeno dynamics in a state-of-the-art cavity quantum electrodynamics experiment. We show that repeated measurements or unitary operations performed on the atoms probing the field confine its evolution to tailored subspaces. This confinement leads to nontrivial field dynamics and to the generation of interesting nonclassical states, including Schrödinger's cat states.

Joint work with: M. Brune, I. Dotsenko, S. Haroche, S. Gleyzes, S. Pascazio, B. Peaudecerf, J. M. Raimond, and C. Sayrin.

References:
[1] Phys. Rev. A 86, 032120 (2012).
[2] Phys. Rev. Lett. 105, 213601 (2010).

講演要旨:
量子情報処理を実装するにあたっては，実験機器のエラーや環境系からのノイズに対して耐性のある操作を用いることが望ましい．そのような操作の構成法として，複合パルス法と呼ばれる方法が知られている．複合パルス法では，目的の操作を複数の操作に小分けして実装する．このとき，小分けの仕方を工夫して，エラーの影響を全体として相殺するように設計しておくのである．本講演では，二種類のエラーに対して同時に耐性のあるユニタリ操作の設計や，多量子ビット系への拡張などについて，我々が得た具体的な構成法や操作の特徴などを述べる．

参考文献:
[1] Phys. Rev. A 84, 062311 (2011).
[2] arXiv:1204.2437.
[3] arXiv:1204.2750.

講演要旨:
本セミナーでは，まず，量子系を時間連続的に測定し，その結果をリアルタイムでフィードバックし制御を行うための一般的方法論を示す．次いで，このスキームの応用例として，
(i) 光の量子状態を固体系へ転写する「state transfer」の精度向上，
(ii) 原子時計への応用，
(iii) 測定結果に応じて測定を変化させる「適応測定」による状態生成
の概要を紹介する．

講演要旨:
We show a systematic construction for implementing general measurements on a single qubit, including both strong (or projection) and weak measurements. We mainly focus on linear optical qubits. The present approach is composed of simple and feasible elements, i.e., beam splitters, wave plates, and polarizing beam splitters. We show how the parameters characterizing the measurement operators are controlled by the linear optical elements. We also propose a method for the implementation of general measurements in solid-state qubits.

参考文献:
Phys. Rev. A 85, 043808 (2012).

Abstract:
Resonantly enhanced tunneling is a quantum effect in which the probability for the tunneling of a particle between two potential wells is increased when the quantized energies of the initial and final states of the process coincide.

A Bose-Einstein condensate in an accelerated optical lattice potential can be used to study the phenomenon of resonantly enhanced tunneling. In a tilted periodic potential, atoms can escape by tunneling to the continuum via higher-lying levels. The tunneling process of atoms out of the tilted lattice is resonantly enhanced when the energy difference between lattice wells matches the distance between the energy levels in the wells.

We study here the time evolution of a Bose-Einstein condensate with a narrow quasi-momentum distribution in a shallow (accelerated) optical lattice. The decay of the survival probability in the ground band has a step-like structure. In this regime we establish a connection between the wave function renormalization parameter Z, characterizing non-exponential decay, and the phenomenon of resonantly enhanced tunneling.

Reference:
Phys. Rev. A 85, 053602 (2012).

Abstract:
We study the efficiency of quantum state tomography of a qubit (quantum target) that is indirectly monitored by looking at the scattering state of a probe qubit in a Heisenberg-type interaction. We show that the accuracy efficiency that one can get by adopting entanglement-assisted strategies is significantly larger than that achievable when entanglement resources are not available.

Reference:
J. Phys. A 45, 105309 (2012).

講演要旨:
量子測定の誤差や擾乱を量子推定論の立場から定量化し，それらの間に成り立つ不確定性関係について報告する．ハイゼンベルグのガンマ線顕微鏡を現代的な観点から捉え直すと，量子測定の測定誤差は，測定結果から粒子の位置を推定した際の推定精度によって定量化される．また，擾乱は，測定後の状態から測定前の状態の運動量の推定精度によって定量化される．量子測定の測定結果は確率的に出力されるため，推定値の精度を議論するためには量子推定理論を用いなければならない．我々は，量子推定理論を用いることで，測定誤差や擾乱が Fisher 情報量によって表されることを示し，それらの間に成り立つ不確定性関係を求めた[1,2]．

参考文献:
[1] Phys. Rev. A 84, 042121 (2011).
[2] arXiv:1106.2526.

Abstract:
We discuss a method to witness the quantumness of a system. The method relies on the fact that the anticommutator of two classical states is always positive. We show that there is always a nonpositive anticommutator due to any two quantum states. We notice that interference depends on the trace of the anticommutator of two states and it is therefore more natural to detect quantumness by looking at anticommutators of states rather than their commutators.

Abstract:
The aim of quantum system identification is to estimate the ingredients inside a black box, in which some quantum-mechanical unitary process takes place, by just looking at its input-output behavior. Here we establish a basic and general framework for quantum system identification, that allows us to classify how much knowledge about the quantum system is attainable, in principle, from a given experimental setup. Unlike usual quantum metrology, our theory applies to cases where the set of observables are not topographically complete, and where knowledge about the system can only be retrieved indirectly. When the topology of the system is known, the framework enables us to establish a general criterion for the estimability of the coupling constants in its Hamiltonian.

Reference:
Phys. Rev. Lett. 108, 080502 (2012).

講演要旨:
本セミナーでは，古典確率論，量子力学を含む，操作主義的に最も一般的な確率論を説明し，特に情報理論と量子力学基礎への応用を紹介する．前者の応用は，量子情報理論を一般化する最も一般的な情報理論の構築である．古典物理学や量子力学といった具体的な物理モデルに依存しない情報理論を構築することで，情報処理（計算や通信）や，情報の本質に迫ることができると期待される．後者の応用は，量子力学の物理原理を追求する試みである．量子力学の理論体系は，それ自体物理的（あるいは操作的）意味を持たない数学（例えば，Hilbert空間と線形作用素論）によって表現される．これに対し，量子力学を（実験で確認され得る）原理ベースで説明することができれば，より直観的に量子力学を理解することができると期待される．

Abstract:
We describe a cavity QED experiment on quantum Zeno effect with Rydberg atoms and a microwave superconducting cavity. We propose an implementation of quantum Zeno dynamics leading to promising methods for tailoring nonclassical field states.

講演要旨:
Maxwellの悪魔のパラドックスは，物理学 (特に熱力学) と情報の概念が明確に結びついた，おそらくもっとも歴史ある例だろう．端的に言えば，このパラドックスは次のような主張をする．超人的だが (Newton力学の) 物理法則は破らない悪魔がいれば，熱力学第2法則を破ることができる，つまり第2種永久機関の実現が可能である．いかにも怪しい (間違っている) 話だが，どこに物理的問題があるかを明らかにするのに1世紀以上もかかった，知恵の輪のようなパズルである．このパズルの難しさは，"観測による情報取得" を物理の議論の中に定量的に組み入れる必要があったことにある．情報の物理を巡る議論は，"情報処理は物理的過程である" との認識を強め，微細加工・制御技術の発展と相まって，量子情報科学隆盛の礎のひとつとなった．本講演では，このパラドックスが紆余曲折を経ながらもいかに解決に至ったかを解説しつつ，カギとなったLandauerの消去原理，および第2法則の興味深い帰結のいくつかを紹介する．

参考文献:
K. Maruyama, F. Nori, and V. Vedral, "The Physics of Maxwell's Demon and Information," Rev. Mod. Phys. 81, 1-23 (2009).

講演要旨:
Dynamical error in quantum gate is caused by interactions between qubits and ancilla systems that control quantum states of the qubits to implement the target gate operation. The dynamical error is inevitable in the sense that it remains even if the environmental noise would be perfectly suppressed during the gate operation. In this talk, I will discuss the dynamical error in quantum gates under conservation laws, with current research achievements.

Abstract:
A new detection scheme is presented which allows one to saturate asymptotically the Holevo bound. Differently from previous proposals, it is based on performing a sequence of (projective) YES/NO measurements which in N steps determines which codeword was sent by the sender (N being the number of the codewords). Our analysis shows that as long as N is below the limit posed by the Holevo bound the error probability can be sent to zero asymptotically in the length of the codewords.

Abstract:
By applying general methods of statistical mechanics, we analyze the distribution of the Schmidt coefficients of the reduced density matrix of a quantum system in a pure state. By introducing a fictitious temperature and a partition function, the problem is translated in terms of the distribution of the eigenvalues of random matrices. We investigate the appearance of two phase transitions, one at a positive temperature, associated with very entangled states, and one at a negative temperature, signaling the appearance of a significant factorization in the many-body wave function.

講演要旨:
We theoretically present a method to realize a deterministic photon-photon root-SWAP gate using a three-level Lambda system interacting with single photons in reflection geometry. The Lambda system is used completely passively as a temporary memory for a photonic qubit; the initial state of the Lambda system may be arbitrary, and active control by auxiliary fields is unnecessary throughout the gate operations. These distinct merits make this entangling gate suitable for deterministic and scalable quantum computation.

講演要旨:
2008年の鉄系超伝導の発見以降, その超伝導状態の解明と超伝導デバイス応用は物理学・工学の両分野における重要なトピックスとなっている. 我々は代表的な超伝導デバイスであるジョセフソン接合について, その理論構築を実施し, 従来までのジョセフソン効果の理論を本質的に革新する成果を得た. 鍵となるのは,多バンド超伝導体である鉄系超伝導特有の集団励起モード, Leggettモードの存在である. 我々はLeggettモードを利用により鉄系超伝導の基礎物性が明らかになることを示す. さらに新規デバイス応用の上で重要な役割を果たすことを示す.

参考文献:
[1] Y. Ota, M. Machida, T. Koyama, H. Aoki, arXiv:1008.3212.
[2] Y. Ota, M. Machida, T. Koyama, arXiv:1006.2184.
[3] Y. Ota et al., Phys. Rev. B 81, 214511 (2010).
[4] Y. Ota, M. Machida, T. Koyama, H. Matsumoto, Phys. Rev. B 81, 014502 (2010).
[5] Y. Ota, M. Machida, T. Koyama, J. Phys. Soc. Jpn. 78, 103701 (2009).
[6] Y. Ota, M. Machida, T. Koyama, H. Matsumoto, Phys. Rev. Lett. 102, 237003 (2009).

講演要旨:
The state of a single particle after it passes through a beam splitter (BS) is expressed as a superposition of two states, each of which corresponds to the particle traveling along a path, e.g., |left>+|right>.However, if we represent the same state in the number basis it is written as |01>+|10>, which looks like one of the maximally entangled Bell states.Then, does the single particle state after a BS exhibit the quantum non-locality?Can the Bell inequalities be violated?Particularly for single massive particles, the Bell test seems tricky because we cannot rotate a measurement basis to the form |0>+|1> due to the particle-number superselection rule. Here, we show a scheme to convert the mode entanglement of a massive particle to a Bell-testable form.The conversion can be done using a BEC as a resource, but we also consider the case where no resource is provided initially.

講演要旨:
多体系の量子状態は，一般にその部分系の状態の直積状態であるときは‘もつれて’(エンタングルして) おらず，一方 Bell 状態のような非直積状態は‘もつれて’いるとされる．ところが，例えばフェルミ粒子の多体系状態は，粒子の統計性から反対称状態であって直積状態ではないが，必ずしも量子もつれ状態とはならない．このような同種粒子系の量子もつれについては，近年，Ghirardi らにより物理量の完全性の実在論的立場から議論されているが，量子相関を基礎とした操作論的な立場から，より一般的な理解と整理が可能になる．本講演では，量子相関から見た同種粒子系の量子もつれとは何か，またその有無の判定規準について述べるとともに，量子もつれは測定の設定に依存する相対的概念であることを簡単な例を用いて説明する．

参考文献:
G.-C. Ghirardi, L. Marinatto, and T. Weber, J. Stat. Phys. 108, 49 (2002).
T. Ichikawa, T. Sasaki, and I. Tsutsui, J. Math. Phys. 51, 062202 (2010).

Abstract:
Quantum correlations in composite systems lie at the heart of quantum mechanics and play a fundamental role in protocols of quantum information and computation. Environmental effects on the dynamics of quantum systems are ubiquitous and typically lead to loss of correlations in open quantum systems.

In this talk I review the dynamics of different features of quantum correlations, ranging from entanglement to nonlocality, in several configurations of composite systems made by qubits embedded in structured environments [1-6]. The roles of environmental spectral properties, interactions and initial preparation of the systems are pointed out.

References:
[1] L. Mazzola, B. Bellomo, R. Lo Franco, and G. Compagno, Phys. Rev. A 81, 052116 (2010).
[2] B. Bellomo, G. Compagno, A. D'Arrigo, G. Falci, R. Lo Franco, and E. Paladino, Phys. Rev. A (in press) [quant-ph/1001.4875].
[3] B. Bellomo, R. Lo Franco, S. Maniscalco, and G. Compagno, Phys. Rev. A 78, 060302(R) (2008).
[4] B. Bellomo, R. Lo Franco, and G. Compagno, Phys. Rev. A 78, 062309 (2008).
[5] B. Bellomo, R. Lo Franco, and G. Compagno, Phys. Rev. A 77, 032342 (2008).
[6] B. Bellomo, R. Lo Franco, and G. Compagno, Phys. Rev. Lett. 99, 160502 (2007).

Abstract:
Recent research on superconducting qubits has been accompanied by innovative ideas and technological advances in measurement techniques. For example, some experiments have demonstrated measurements of qubit states where the back-action was limited by the Heisenberg uncertainty principle. I will briefly mention some of the measurement ideas employed in recent experiments. I will then present some of our recent results on the information about the state of a qubit gained by a weakly coupled detector. In particular, we analyze the case where the measurement is concurrent with coherent dynamics or decoherence. In this case, extracting meaningful measurement results from the observed signal can become a complicated task. However, with proper analysis several pieces of information are accessible. Our approach complements the stochastic master equation approach, which describes the evolution of the qubit's state but does not keep track of the acquired measurement information.

Abstract:
If very frequent measurements are performed on a quantum system, in order to ascertain whether it is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place.This phenomenon stems from very general features of the Schroedinger equation, that yield quadratic behavior of the survival probability at short times.

However, the quantum Zeno effect does not necessarily freeze everything.On the contrary, for frequent projections onto a multi-dimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement.This continuing time evolution within the projected subspace is named "quantum Zeno dynamics" and has interesting features.

Some significant examples will be proposed and their practical relevance discussed.We focus on the control of decoherence, irreversibility and the risks deriving from the inverse Zeno effect.

講演要旨:
A measurement scheme of atomic qubits pinned at given positions is studied by analyzing the interference pattern obtained when they emit photons spontaneously. In the case of two qubits, a well-known relation is revisited, in which the interference visibility is equal to the concurrence of the state in the infinite spatial separation limit of the qubits. By taking into account the super-radiant and sub-radiant effects, it is shown that a state tomography is possible when the qubit spatial separation is comparable to the wavelength of the atomic transition. In the case of three qubits, the relations between various entanglement measures and the interference visibility are studied, where the visibility is defined from the two-qubit case. A qualitative correspondence among these entanglement relations is discussed. In particular, it is shown that the interference visibility is directly related to the maximal bipartite negativity.

参考文献:
J. Suzuki, C. Miniatura, and K. Nemoto, arXiv:1002.4716.

Abstract:
Recent experiments on solid state qubits have focused on the implementation of high-fidelity operations on two qubits. It may seem that scaling these experiments to say 30 qubits requires a further big step in controlling the qubits. We show that this is not the case: the control implemented today is enough to indirectly control the remaining qubits, which can remain passive. The control we have on the two qubits is mediated by the system Hamiltonian in a scalable way.

講演要旨:
量子情報の分野のおいて，ベネットらのよって提案された量子テレポーテーション (QT) は重要な概念となっている．QT では空間的に離れた 2者 A と B が持っている量子系が，スピン 1重項状態のように量子的にもつれた状態にある場合を考える．QT の驚くべきことは，A がある量子系の任意の量子状態を B に A と B の局所的操作と古典通信 (Local Operations and Classical Communication, LOCC) だけ送ることができる点である．この性質ために将来の量子情報ネットワークにおいても QT は重要な働きをすると期待されている．一方従来の QT は量子情報は送信できるが，転送される励起状態のエネルギーそのものは送ることができない．しかし，もしエネルギーも LOCC で送信できれば，任意の励起状態をエネルギーを受信地で調達することなくテレポートできることになり大変有用である．ではこのようなことは可能なのだろうか．答えは驚くべきことに YES である．スピン鎖系や量子場の基底状態がもつ量子もつれを使用して，因果律や局所的エネルギー保存則を壊すことなく LOCC のみでエネルギーを転送できるプロトコルを今回提案する．スピン鎖系を例にとると，離れた位置にいる A と B が基底状態 (これは何もない状態に対応する) にあるスピン鎖の端点にそれぞれいるとする． それぞれのスピン鎖に対して，A が基底状態にエネルギーを注入しながら自分のスピンに局所的量子測定を行ってその測定結果を得る．A は B にその古典的測定結果を古典通信路で相手に知らせ，B はその結果に依存した局所的量子操作を自分のスピンに作用させる．この過程において B はスピン鎖系からエネルギーを集団平均の意味で取り出すことができるプロトコルになっている．この新しいプロトコルでは励起した物理的媒体を直接送ることなく，古典情報のみを伝達することで局所的基底状態のスピン鎖からエネルギーを取り出せるために，この輸送過程では熱の発生を抑制することができる．このような特性からナノマシン等のパワー源への応用も期待される．また発表では量子場の真空の量子揺らぎを用いたエネルギー転送プロトコルも議論する予定である．このプロトコルでは量子揺らぎの局所的 POVM 測定と測定結果に依存した局所的スクィージング操作を用いることで，離れた位置にあるエネルギーを有効的な意味で転送して使用することができる．

Abstract:
We analytically and numerically study the effects of pulsed control on the decoherence of a qubit coupled to a quantum spin bath. When theenvironment is critical, decoherence is faster and we show that thecontrol is relatively more effective. Two coupling models areinvestigated, namely a qubit coupled to a bath via a single link and aspin star model, yielding results that are similar and consistent.

Abstract:
Einstein, Podolsky and Rosen's (EPR) seminal paper on quantumcorrelations of entangled states, and Bell's subsequent discovery thatquantum theory is incompatible with any locally realistic hiddenvariable theory, have spurred substantial discussion and manyexperiments on the nature of quantum nonlocality and entanglement. Mostexperimental tests follow Bohm and Aharonov's reformulation of EPR'sgedanken experiment. Hence, the experiments involve two space-likeseparated entangled particles, typically two polarization entangled photons.

In 1990, Tan, Walls, and Collet proposed a test of nonlocality involvingonly one particle, a single photon. Their proposal initiated a debate ifit is indeed possible for a single particle to have measurable (andcorrelated) properties at two space-like separated locations? The twomain argument against this possibility were: 1) that detection of theparticle at one location would immediately nullify any possibility ofrecording any property associated with the particle (except for itsabsence) at any other location, and 2) that detection of the particlewould typically involve the simultaneous detection of particles fromauxiliary states (e.g., a local oscillator), compromising the notion ofsingle particle nonlocality.

We have performed an experiment where the properties of a single photon,split between two spatial modes, are measured. We measure the relativephase between the superposition between zero and one photon at twolocations, by co-propagating a local oscillator in the same spatialmode, but with an orthogonal polarization. The obtained correlationviolates classical physics and clearly support the quantum mechanicalpredictions.

Abstract:
Homogenization protocols model the quantum mechanical evolution of asystem into a fixed state independently from the initial configurationof the system by repeatedly coupling it with a collection of identicallyprepared controller ancillas. Here we analyze these protocols within the formalism of "relaxing" channels providing an easy to check sufficient condition for homogenization. In this context we describe full homogenization schemes which allow usto derive the relaxation to a fixed state in a network of connectedqudits by only interacting with few of them. Furthermore, we studyconfigurations which allow us to introduce entanglement among theelements of a network.

Abstract:
We calculate the energy spectrum of the electrons emitted from a superconductor under an applied field, including two-particle processes. Among these two-particle processes we examine the Andreev processes, in which a whole Cooper pair tunnels out of the superconductor. The Andreev processes are the only ones that can happen with subgap energies. In this range we find an explicit formula for the spectrum of the two-particle processes.

Abstract:
The characterization and quantification of entanglement is an open andchallenging problem. It is possible to give a good definition ofbipartite entanglement in terms of the von Neumann entropy and theentanglement of formation. The problem of defining multipartiteentanglement is more difficult and no unique definition exists. I introduce the notion of maximally multipartite entangled states (MMES) of n qubits as a generalization of the bipartite case. Theirbipartite entanglement does not depend on the bipartition and ismaximal for all possible bipartitions. Some examples of MMES for n = 3,4,6 and 7 are investigated. These states are the solutions of anoptimization problem, that can be recast in terms of statistical mechanics.

Abstract:
We consider a system of n qubits and characterize its multipartiteentanglement in terms of the distribution function of the bipartitepurity over all balanced bipartitions. We search for those maximallymultipartite entangled states whose average purity is minimum forall bipartitions and recast this optimization problem into a problemof statistical mechanics.

Abstract:
I will introduce a random matrix model for the statistical properties ofthe partial purity of a pure state in a bipartite quantum system at afinite fictitious temperature. This enables one to write the generatingfunction for the cumulants, for both balanced and unbalancedbipartitions. It also unveils an unexpected feature of the system,namely the existence of two phase transitions, characterized bydifferent spectra of the partial density matrices. One of the criticalphases is described by the statistical mechanics of 2-D gravity, theother is a second-order phase transition.

Abstract:
Superconducting devices have attracted significant interest both becausethese artificial macro-atoms allow us to investigate fundamentalproperties of quantum theory on a mesoscopic/macroscopic scale and morerecently in view of possible applications in quantum communication andinformation processing. So far, experimental research in this field hasmostly focused on the behaviour of a single isolated qubit while in thelast few years, significant achievements on superconducting two-qubitsystems were reported, i.e., the generation of entangled states insystems of coupled flux and phase qubits, as well as the observation ofquantum coherent oscillations and conditional gate operations using twocoupled superconducting charge qubits. Generation of multi-qubitentanglement will be the next significant and very challenging steptowards quantum information processing based on these scalablesolid-state systems. Within this framework the performances oftheoretical coupling schemes for the controlled generation andmanipulation of entangled states and of nonclassical superposition will be discussed.

Abstract:
Decoherence theory has a fundamental interest since it provides explanations for the emergence of classicality in a world governed by the laws of quantum mechanics. It is also relevant for actual implementation of any quantum computation and communication protocol. Here we address the question if a many-body environment can be substituted, without changing theeffects on system's dynamics, by a closed deterministic system with a small number of degrees of freedom, yet chaotic. We give a positive answer and in particular we show that the complexity of the environment arises from the chaotic dynamics rather than from the many-body nature.

Abstract:
A pure quantum state of N qubits has maximal bipartite entanglementif, for a given bipartition, the reduced density matrix of N/2 qubitsis totally mixed. The notion of maximally multipartite entangled states generalizes thisnotion. These states have a bipartite entanglement that does not dependon the bipartition and is maximal for all possible bipartitions. They are solutions of a minimization problem. Examples for small N areinvestigated, both analytically and numerically.