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数理解析研究所講究録

OPERATIONAL STRUCTURE OF ENTANGLED QUANTUM SYSTEMS (Information and mathematics of non-additivity and non-extensivity : contacts with nonlinearity and non-commutativity)

Ichiro Fujimoto, 英男 宮田

Published Feb 1, 2008• 0 citations
OpenAlex

Abstract

We investigate the quantum interaction between two quantum sys- tems in terms of $C^{*}$ -algebras.We introduce the $co$ rrelation CP-map between the bipartite systems and study the operational and statistical structure of the quantum interaction.We then generalize the informational quantities into this quantum setting, and introduce a new entropy of CP-maps, which vanishes at extreme CP-maps in CP-convexity.Introduction.We prepare two quantum states described by density operators $\rho$ and $\sigma$ on Hilbert spaces $H_{1}$ and $K_{1}$ respectively, and so initially we have the compound state $\omega_{0}=\rho\otimes\sigma$ on the tensor product Hilbert space $H_{1}\otimes K_{1}$ .Suppose that there exists an interaction between the two subsystems for some time interval, and that the systems are in equilibrium with the exterior for this time period, then the compound state is changed to a state $\omega$ described by $\omega=U(\rho\otimes\sigma)U^{*}$ where $U$ is a unitary operator representing the quantum evolution of the systems.After the interaction, the states of the subsystems on $H_{2}$ and $K_{2}$ are described by $Tr_{K_{2}}w$ and $R_{H_{2}}\omega$ respectively.Then the channels $\varphi^{*}$ from $T(H_{1})$ to $T(H_{2})$ , and $\phi^{*}$ from $T(K_{1})$ to $T(K_{2})$ , are defined by $\varphi^{*}(\rho)=h_{K_{2}}\omega=r_{b_{K_{2}}U(\rho}\otimes\sigma)U^{*}$ and $\phi^{*}(\sigma)=b_{H_{2}}\omega=h_{H_{2}}U(\rho\otimes\sigma)U^{*}$ ,The detailed version of this paper will be submitted for publication elsewhere.Typaeet by $A_{A}\#?ffl$

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数理解析研究所講究録•Dec 1, 2012•0 citations
We investigate the quantum interaction between two quantum sys- tems in terms of C∗C^{*}C∗ -algebras.We introduce the cococo rrelation CP-map between the bipartite systems and study the operational and statistical structure of the quantum interaction.We then generalize the informational quantities into this quantum setting, and introduce a new entropy of CP-maps, which vanishes at extreme CP-maps in CP-convexity.Introduction.We prepare two quantum states described by density operators ρ\rhoρ and σ\sigmaσ on Hilbert spaces H1H_{1}H1​ and K1K_{1}K1​ respectively, and so initially we have the compound state ω0=ρ⊗σ\omega_{0}=\rho\otimes\sigmaω0​=ρ⊗σ on the tensor product Hilbert space H1⊗K1H_{1}\otimes K_{1}H1​⊗K1​ .Suppose that there exists an interaction between the two subsystems for some time interval, and that the systems are in equilibrium with the exterior for this time period, then the compound state is changed to a state ω\omegaω described by ω=U(ρ⊗σ)U∗\omega=U(\rho\otimes\sigma)U^{*}ω=U(ρ⊗σ)U∗ where UUU is a unitary operator representing the quantum evolution of the systems.After the interaction, the states of the subsystems on H2H_{2}H2​ and K2K_{2}K2​ are described by TrK2wTr_{K_{2}}wTrK2​​w and RH2ωR_{H_{2}}\omegaRH2​​ω respectively.Then the channels φ∗\varphi^{*}φ∗ from T(H1)T(H_{1})T(H1​) to T(H2)T(H_{2})T(H2​) , and ϕ∗\phi^{*}ϕ∗ from T(K1)T(K_{1})T(K1​) to T(K2)T(K_{2})T(K2​) , are defined by φ∗(ρ)=hK2ω=rbK2U(ρ⊗σ)U∗\varphi^{*}(\rho)=h_{K_{2}}\omega=r_{b_{K_{2}}U(\rho}\otimes\sigma)U^{*}φ∗(ρ)=hK2​​ω=rbK2​​U(ρ​⊗σ)U∗ and ϕ∗(σ)=bH2ω=hH2U(ρ⊗σ)U∗\phi^{*}(\sigma)=b_{H_{2}}\omega=h_{H_{2}}U(\rho\otimes\sigma)U^{*}ϕ∗(σ)=bH2​​ω=hH2​​U(ρ⊗σ)U∗ ,The detailed version of this paper will be submitted for publication elsewhere.Typaeet by AA#?fflA_{A}\#?fflAA​#?ffl
One of the most challenging open problems in quantum information theory is to clarify and quantify how entanglement behaves when part of an entangled state is sent through a quantum channel. Of central importance in the description of a quantum channel or completely positive map (CP-map) is the dual state associated to it. The present paper is a collection of well-known, less known and new results on quantum channels, presented in a unified way. We will show how this dual state induces nice characterizations of the extremal maps of the convex set of CP-maps, and how normal forms for states defined on a Hilbert space with a tensor product structure lead to interesting parameterizations of quantum channels.
G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed
Quantum channels, also called quantum operations, are linear, trace preserving and completely positive transformations in the space of quantum states. Such operations describe discrete time evolution of an open quantum system interacting with an environment. The thesis contains an analysis of properties of quantum channels and different entropies used to quantify the decoherence introduced into the system by a given operation. Part I of the thesis provides a general introduction to the subject. In Part II, the action of a quantum channel is treated as a process of preparation of a quantum ensemble. The Holevo information associated with this ensemble is shown to be bounded by the entropy exchanged during the preparation process between the initial state and the environment. A relation between the Holevo information and the entropy of an auxiliary matrix consisting of square root fidelities between the elements of the ensemble is proved in some special cases. Weaker bounds on the Holevo information are also established. The entropy of a channel, also called the map entropy, is defined as the entropy of the state corresponding to the channel by the Jamiolkowski isomorphism. In Part III of the thesis, the additivity of the entropy of a channel is proved. The minimal output entropy, which is difficult to compute, is estimated by an entropy of a channel which is much easier to obtain. A class of quantum channels is specified, for which additivity of channel capacity is conjectured. The last part of the thesis contains characterization of Davies channels, which correspond to an interaction of a state with a thermal reservoir in the week coupling limit, under the condition of quantum detailed balance and independence of rotational and dissipative evolutions. The Davies channels are characterized for one-qubit and one-qutrit systems.
The interactions and correlations among the constituents of manybody systems are manifested in characteristic physical properties such as ferromagnetism, superconductivity, etc. A list of some that have been studied in the last century is given in the Box (see below). A parallel development in quantum information was initially slow but over the past two decades progress has been very rapid. Fundamentally this is another aspect of quantum correlations in composite systems arising from the twin features of the superposition principle and the tensor product structure of state space. These features are not utilized in the same manner in quantum many-body physics. In the Box, a corresponding parallel list with properties of many-body systems is given because there has been an interplay between the two research efforts and their understanding. The basic quantum mechanical principles apply to both cases except different aspects are utilized because the goals are different in each. Both areas of investigation are based on a probabilistic foundation with a variational underpinning founded on an “entropy” maximization, which may be called Quantum Statistical Mechanics [1]. The specific form of the entropy as a functional of the density matrix will be made explicit presently. See Table I for definitions of density matrix and associated quantities. The equilibrium properties of the many-body systems are then given by a maximization of von Neumann entropy subject to certain constraints such as the average value of the Hamiltonian of the system “energy”. This leads to the familiar exponential probabilities of the Boltzmann-Gibbs (BG) form. Any nonequilibrium properties are studied by a quantum time evolution equation. Non-equilibrium properties such as the anomalous relaxation in time are often analyzed with the quantum version of the Tsallis entropy with an entropic index, q, which leads to powerlaw probabilities in contrast to the BG type [2]. (See the Box in the Introduction of Tsallis and Boon). In quantum information, the maximum entropy scheme is not of use because its origins are elsewhere as will be made clear presently. The evolution is replaced by processing of information
In scope of CPCPCP -convexity theory for C∗C^{*}C∗ -algebras (quantization of convexity, measure, entropy for completely positive maps), we investigate the operational stmcture of quantum interactions of entangled systems, and propose new information quantities which naturally generalize the classical information theory.