Ohya, M., Petz, D.: Quantum Entropy and its Use. Berlin: Springer 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

..... n . 2.1) In the following we simplify notations by defining #(n) and : # (#(n) for n (respectively for n N) Obviously, because of the translation invariance any # (A # ) is uniquely defined by the family n#N # . The von Neumann entropy of # (cf. [17]) is defined by: tr #D# log D# , where tr # denotes the trace on . For tr #(n) we will write tr n . It is well known that for every # (A # ) the limit s(#) lim 6 I. Bjelakovic et al. exists. We call it the mean (von Neumann) entropy. Let l N and consider the subgroup G l ....

M. Ohya, D. Petz, Quantum Entropy and its Use, Springer, Berlin 1993

.... quantum probability density, contrary to the Heisenberg uncertainty principle (which is based on the standard deviation of the densities) or any of its generalizations based on moments other than the standard deviation which yield non useful information or no information at all in certain cases [3, 4, 5]. The analytical evaluation of the information entropies of quantum systems is a formidable task, even for one dimensional single particle systems whose Hamiltonian operator H has a discrete eigenvalue spectrum E 0 E 1 E 2 . 1) The corresponding eigenfunctions (time independent ....

.... the so called entropic uncertainty relation of this system [8] is which is a consequence of a well known inequality in Fourier analysis, first conjectured by Hirschman [6] and then proved by Beckner [7] and BialynickiBirula and Mycielski [8] This inequality strongly generalizes and improves [5, 8] the Heisenberg Kennard Robertson uncertainty principle #x#p # . Up to now the calculation of these information functionals has been undertaken only for the harmonic oscillator and the Coulomb potentials [9, 10] as well as for the infinite well potential [11, 12, 13] and the powertype ....

M. Ohya and D. Petz, Quantum Entropy and its Use (Springer Verlag, Berlin, 1993)

....mechanics, quantum dynamical systems, quantum Markovian processes, van Hove limit, linear response theory, etc. Even though we have tried to provide the reader with the most relevant references to these earlier works, we do not claim completeness in this respect, and refer the reader to [BR1, BR2, OP] for an extensive list of references. Acknowledgment. We are grateful to Jan Derezinski and David Ruelle for useful discussions on the subject of this review. A part of this work has been done during the visit of the first author to Johns Hopkins University. V.J. is grateful to Steve Zelditch for ....

....consists of multiples of the identity. The effectiveness of the algebraic formalism of quantum statistical mechanics is largely due to Tomita Takesaki modular theory of von Neumann algebras. We assume that the reader is familiar with the basic results of this theory as discussed, for example, in [BR1, BR2, Ha, OP]. For notational purposes we recall some well known facts. The state is called modular if is a separating vector for M i.e. if extends to a faithful normal state on M . Any KMS state at inverse temperature 2 R is modular. Assume that is a modular state on O and denote by = ....

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Ohya, M., Petz, D.: Quantum Entropy and its Use. Springer-Verlag, Berlin (1993).

....for the rigorous treatment of entropy in the context of classical information theory; an excellent survey of Wehrl [141] is devoted to the concept of entropy in statistical physics. For an abstract treatment of entropy in the context of operator algebras, we recommend the book by Ohya and Petz [93]. In statistical physics, the entropy of a system that can exist in N possible con gurations is given, up to a multiplicative constant, by Boltzmann s formula S : ln N: 64 65 This assumes, however, that all N con gurations of the system are equiprobable. When this is not the case, i.e. ....

M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer-Verlag, Berlin, 1993).

....the first author was partly supported by NSERC. A part of this work has been done during the visit of the second author to the McGill University which was supported by NSERC. We assume that the reader is familiar with basic results of Tomita Takesaki modular theory as discussed, for example, in [BR, DJP, Don, OP]. Let M (O) 00 be the enveloping von Neumann algebra. Since is ( 1) KMS state, the vector is separating for M , and we denote by P , J , the corresponding natural cone, 4 modular conjugation and modular operator. We recall that = e L , where L is the unique ....

Ohya, M., Petz, D.: Quantum Entropy and its Use. Springer-Verlag, Berlin (1993).

....consists of multiples of the identity. The effectiveness of the algebraic formalism of quantum statistical mechanics is largely due to Tomita Takesaki modular theory of von Neumann algebras. We assume that the reader is familiar with the basic results of this theory as discussed, for example, in [BR1, BR2, Ha, OP]. For notational purposes we recall some well known facts. The state is called modular if is a separating vector for M i.e. if extends to a faithful normal state on M . Any KMS state at inverse temperature 2 R is modular. Assume that is a modular state on O and denote by = ....

....of entropy production in spin systems. 4.2 Definition and Basic Properties Let (O; be a quantum dynamical system. For any state 2 N we denote by Ent( j ) the relative entropy of with respect to . The basic properties of the relative entropy are discussed in the monograph [OP]. We recall that Ent( j ) 0 and that for finite quantum systems Ent( j ) Tr ( log log ) 22) Note also that our notation for the relative entropy differs from the one originally introduced by Araki in [Ar3, Ar4] by a sign and the order of its two arguments. We shall need two ....

Ohya, M., Petz, D.: Quantum Entropy and its Use. Springer-Verlag, Berlin (1993).

....was partly supported by NSERC. Part of this work has been performed during a visit of the second author to University of Ottawa which was also supported by NSERC. 2Proofs We assume that the reader is familiar with the basic results of Tomita Takesaki modular theory as discussed, for example, in [BR1, BR2, H, OP]. We begin by setting the notation and recalling some well known facts. ## # ## # # # # # denotes the GNS representation of the algebra # associated to #.By (A1) the vector # # is cyclic and separating for # # # # ### ## . Moreover, A1) implies that # # is injective. We respectively ....

Ohya, M., Petz, D.: Quantum Entropy and its Use, Springer-Verlag, Berlin (1993).

....definitions and the results we will need. For positive linear functionals ; 2 O , let Ent( j ) be the relative entropy of Araki (we use the ordering and the sign convention of Bratelli Robinson [BR2, Don] For definition and properties of Araki s relative entropy we refer the reader to [Ar1, Ar2, BR2, Don, OP]. We make the following assumption. E(1) There exists a C dynamics such that is a ( 1) KMS state. 4 The choice of reference temperature = 1 is made for mathematical convenience. If (E1) holds, then for any 6= 0 there is a C dynamics ; such that is ( KMS ....

Ohya M., Petz D.: Quantum Entropy and its Use, Springer-Verlag, Berlin (1993).

....perturbations, which is restricted to perturbations bounded from below. One of its versions has been developed by Sakai [Sa2] another version (applicable to generalized positive operators which may not have a dense domain) is due to Donald [Don] his method is also discussed in monograph [OP]) The Sakai Donald theory does not cover perturbations which are unbounded from both sides, and in particular is not applicable to Pauli Fierz systems. The W algebraic form (1.1) of the Golden Thompson inequality was rst proven by Araki [Ar2] A di erent proof, based on an application of ....

....which goes beyond what we could nd in the literature. To make our paper more accessible, we have included in Section 4 the proof of the Uhlmann s monotonicity theorem [Uh] and Donald s proof of the Golden Thompson inequality [Don] A somewhat di erent presentation of this topic can be found in [OP]. Section 5 contains the perturbation theory of KMS states. The subject naturally splits into three levels. The most restrictive level concerns analytic perturbations. In this case the proofs are essentially algebraic and relatively simple. The next level concerns bounded Q. This is the case ....

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Ohya, M., Petz, D.: Quantum Entropy and its Use, Springer-Verlag, Berlin (1993). 45

....de nitions and the results we will need. For positive linear functionals ## # ## # , let ##### # ## be the relative entropy of Araki (we use the ordering and the sign convention of Bratelli Robinson [BR2, Don] For de nition and properties of Arakis relative entropy we refer the reader to [Ar1, Ar2, BR2, Don, OP]. We make the following assumption. E(1) There exists a # # dynamics # # such that # is a ## # # ### KMS state. 4 The choice of reference temperature # # ## is made for mathematical convenience. If (E1) holds, then for any # ###there is a # # dynamics # ### such that # is ## ### ....

Ohya M., Petz D.: Quantum Entropy and its Use, Springer-Verlag, Berlin (1993).

....was partly supported by NSERC. Part of this work has been performed during a visit of the second author to University of Ottawa which was also supported by NSERC. 2 Proofs We assume that the reader is familiar with the basic results of Tomita Takesaki modular theory as discussed, for example, in [BR1, BR2, H, OP]. We begin by setting the notation and recalling some well known facts. H ; denotes the GNS representation of the algebra O associated to . By (A1) the vector is cyclic and separating for M (O) 00 . Moreover, A1) implies that is injective. We respectively ....

Ohya, M., Petz, D.: Quantum Entropy and its Use, Springer-Verlag, Berlin (1993).

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M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, Berlin-Heidelberg, 1993.

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M. Ohya, D. Petz, Quantum Entropy and its Use, Springer, 1993.

....are involved. The in nite system is modeled by a C algebra and their states are normalized linear functionals instead of statistical operators. The rigorous statistical mechanics of quantum spin systems was one of the successes of the operator algebraic approach. 11] and Sec. 15 of [7] are suggested further readings about details of quantum spin systems. One of the key points in this approach is the de nition of entropy density of a state of the in nite system which goes back to the subadditivity of the von Neumann entropy. Let H 1 and H 2 be possibly nite dimensional ....

....the subadditivity property now: S(D 12 ) S(D 1 ) S(D 2 ) 19) This is a particular case of the strong subadditivity S(D 123 ) S(D 12 ) S(D 23 ) S(D 2 ) 20) for a system consisting of three subsystems. We hope that the notation is selfexplanatory, otherwise see [4] 13] or p. 23 in [7]. If the second subsystem is lacking, 20) reduces to (19) 19) was proven rst by Lieb and Ruskai in 1973 [4] The measurement conditional expectation was introduced by von Neumann as the basic irreversible state change, and it is of the form P i DP i ; 21) where P i are pairwise ....

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M. Ohya, D. Petz, Quantum Entropy and its Use, Springer, 1993.

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Ohya, M., Petz, D.: Quantum Entropy and its Use. Berlin: Springer 1993.

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Ohya, M., Petz, D.: Quantum Entropy and its Use, Springer-Verlag, Berlin (1993).

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Ohya, M., Petz, D.: Quantum Entropy and its Use, Springer-Verlag, Berlin (1993).

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Ohya, M. and Petz, D., Quantum Entropy and Its Use, Springer-Verlag (1993).

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Ohya, M., Petz, D.: Quantum Entropy and its Use. Springer-Verlag, Berlin (1993).

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M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer-Verlag, Heidelberg, 1993.

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Ohya, M., Petz, D.: Quantum Entropy and its Use. Springer-Verlag, Berlin (1993).

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Ohya, M., Petz, D.: Quantum Entropy and its Use. Springer-Verlag, Berlin (1993).

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M. Ohya and D. Petz, \Quantum Entropy and its Use," Springer-Verlag, Berlin, 1993.

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M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer-Verlag, Berlin - Heidelberg, 1993).

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