Wehrl, A. General properties of entropy. Rev. Mod. Phys. 50, 2 (April 1978), 221-260.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... measurement is given by H(P) tr## i ) log (tr## i ) 78) A natural question to ask is: With respect to a given density operator #, which measurement will give the most predictability over its outcomes As it turns out, the answer is any that forms a set of eigenprojectors for # [82]. When this obtains, the Shannon entropy of the measurement outcomes reduces to simply the von Neumann entropy of the density operator. The von Neumann entropy, then, signifies the amount of impredictability one achieves by way of a standard measurement in a best case scenario. Indeed, true to ....

A. Wehrl, "General Properties of Entropy," Rev. Mod. Phys. 50, 221--259 (1978).

....of entropy We give a very brief overview of the concept of entropy. Most of the results are just stated without proofs. The reader is encouraged to consult the book by Gray [53] 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 ....

A. Wehrl, \General properties of entropy," Rev. Mod. Phys. 50, 221 (1978).

....Subspaces 71 The entropy of finite systems is robust against small changes. This continuity of S over the space of finite dimensional density matrices # is also called insensitivity, and is expressed by the following lemma. Fact 21 (Insensitivity of Von Neumann entropy (see Section II.A in [105]) If a sequence # 1 , # 2 , has lim k## # k = #, then also lim k## S(# k ) S(#) Proof: The convergence of # 1 , # 2 , to # is understood to use some kind of norm for the density matrices that is continuous in the matrix entries #i # j#. The operator norm = tr(## # ) for ....

....norm = tr(## # ) for example. The entropy S(#) is a continuous function of the finite set of eigenvalues of #. These eigenvalues are also continuous in the entries of #. Further background on these measures of quantum information and their properties can be found in [78, Chapter 5] and [105]. Another good source is Nielsen s thesis [73] 7.5 Symmetric Subspaces We use the symmetric subspace of the Hilbert space to prove some of our results on copies of quantum states. Let d be a Hilbert space of dimension d with the basis states labeled d#. #m d or d of the m fold ....

Alfred Wehrl. General properties of entropy. Reviews of Modern Physics, 50(2):221--260, 1978.

....P i i log i , where f i g is the multiset of all the eigenvalues of ae. In other words, S(ae) is the Shannon entropy of the distribution induced by the eigenvalues of ae on the corresponding eigenvectors. For a comprehensive introduction to this concept and its properties, see, for instance, [15, 12, 13]. The density matrix corresponding to a mixed state with superpositions drawn from a Hilbert space H is said to have support in H. First, we note the following. Fact 3.1 If ae is a density matrix with support in a Hilbert space of dimension d, then S(ae) log d. This is because the probability ....

A. Wehrl. General properties of entropy. Reviews of Modern Physics 50(2), 1978, pp. 221--260.

.... by H(P) Gamma d X i=1 (trae Pi i ) log (trae Pi i ) 43) A natural question to ask is: With respect to a given density operator ae, which measurement P will give the most predictability over its outcomes As it turns out, the answer is any P that forms a set of eigenprojectors for ae [45]. When this obtains, the Shannon entropy of the measurement outcomes reduces to simply the von Neumann entropy of the density operator. The von Neumann entropy, then, signifies the amount of impredictability one achieves by way of a standard measurement in a best case scenario. Indeed, true to ....

A. Wehrl, "General Properties of Entropy," Rev. Mod. Phys. 50, 221--259 (1978).

.... by a diagonal matrix diag(p 1 ; p 2 ; pm ) which is quite clearly itself, a nonnegative de nite Hermitian matrix with unit trace) Given two density matrices 1 and 2 , the quantum counterpart of the relative entropy, that is, the relative entropy of 1 with respect to 2 , is [38] [59] (cf. 41] S( 1 ; 2 ) Tr 1 (log 1 log 2 ) 1.3) where the logarithm of a matrix is de ned as P k 1 ( 1) k 1 ( I) k =k, with I the appropriate identity matrix. Alternatively, if acts diagonally on a basis fv 1 ; v 2 ; vm g of the Hilbert space by v i = i v i , ....

....C. The relative entropies of n with respect to the Bayesian density matrices n (u) We now apply the preceding results to compute the relative entropy S( n ; n (u) of n with respect to n (u) Utilizing the de nition (1. 3) of relative entropy and employing the property [38] [59] that S( n ) nS( it is given by nS( Tr n log n (u) 2.32) For the rst term, for the entropy S( of , being given by (1.4) we have, using spherical coordinates (r; #; so that r = x 2 y 2 z 2 ) 1=2 , S( 1 r) 2 log (1 r) 2 (1 r) 2 log (1 ....

A. Wehrl, \General properties of entropy," Rev. Mod. Phys., vol. 50, no. 2, pp. 221-260, Apr. 1978.

....Q ) gives S(ae RQ 0 ) Gamma S(ae Q 0 ) log d (7.27) 144 whence immediately I c (ae Q ; E) j S(ae Q 0 ) Gamma S(ae RQ 0 ) GammaS (ae RQ 0 ; I=d Omega ae Q 0 ) log d : 7. 28) The lemma then follows immediately from the joint convexity of the relative entropy [56] [57], which is itself another consequence of strong subadditivity. A simpler and closely related proof is to formulate the coherent information as a conditional entropy, and use the concavity of the latter, also an easy consequence of strong subadditivity [56] This may be generalized to: Lemma 7.3 ....

A. Wehrl, "General properties of entropy," Reviews of Modern Physics, vol. 50, pp. 221--260, 1978.

....Theorem 3 [54] The relative entropy is non negative, H(p(x)kq(x) 0, with equality if and only if p(x) q(x) for all x. As exemplified here, many of the references for elementary results in this Chapter will be to the excellent text of Cover and Thomas [54] or the review paper of Wehrl [182], to which you may refer for historical details. Proof A very useful inequality in information theory is log x ln 2 = ln x x Gamma 1, for all positive x, with equality if and only if x = 1. Here we need to rearrange the result slightly, to Gamma log x ln 2 1 Gamma x, and then note that ....

....ae) Gamma tr(ae log oe) 4.33) Conventionally, this is defined to be 1 if the kernel of oe has non trivial intersection with the support of ae, and is finite otherwise. The quantum relative entropy is non negative, a result sometimes known as Klein s inequality. Theorem 9 (Klein s inequality) [182] The relative entropy is non negative, S(aejjoe) 0; 4.34) with equality if and only if ae = oe. Proof Let ae = P i p i jiihij and oe = P j q j jjihjj be orthogonal decompositions for ae and oe. From the definition of the relative entropy we have S(aejjoe) X i p i log p i Gamma ....

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A. Wehrl. General properties of entropy. Rev. Mod. Phys., 50:221, 1978.

....i ) p i )g. The entropy of finite systems is robust against small changes. This continuity of S over the space of finite dimensional density matrices ae is also called insensitivity, and is expressed by the following lemma. Lemma 1 Insensitivity of Von Neumann entropy (see Section II.A in [21]) If a sequence ae 1 ; ae 2 ; has lim k 1 ae k = ae, then also lim k 1 S(ae k ) S(ae) Proof: The convergence of ae 1 ; ae 2 ; to ae is understood to use some kind of norm for the density matrices that is continuous in the matrix entries hijaejji. The operator norm jaej = ....

Alfred Wehrl, "General Properties of Entropy", Reviews of Modern Physics, Volume 50, No. 2, pp. 221-- 260 (1978)

....This S(ae) can be interpreted as the minimal Shannon entropy of the measurement outcome, minimized over all possible complete measurements. Note that S(ae) only depends on the eigenvalues of ae. The following properties of Von Neumann entropy will be useful later (for proofs see for instance [9]) 1. S(jOEi hOEj) 0, for every pure state jOEi. 2. S(ae 1 Omega ae 2 ) S(ae 1 ) S(ae 2 ) 3. S(U ae U y ) S(ae) 4. S( 1 ae 1 2 ae 2 Delta Delta Delta n ae n ) 1 S(ae 1 ) 2 S(ae 2 ) Delta Delta Delta nS(ae n ) if i 0 and P i i = 1. 5. If ae = P N i=1 p i ....

A. Wehrl. General properties of entropy. Review of Modern Physics, 50(2):221--260, 1978.

....dominant eigenspaces of i n (u) 1. Introduction A theorem has recently been proven [30, 47] cf. 7, 19, 35] in the context of quantum information theory [7, 40] that is analogous to the noiseless coding theorem of classical information theory. In the quantum result, the von Neumann entropy [39, 58], S(ae) Gamma Tr ae log ae (1.1) equalling the Shannon entropy of the probability distribution formed by the eigenvalues of ae) of the density matrix, ae = X a p(a) a ; 1.2) Key words and phrases. Quantum information theory, two level quantum systems, universal data compression, ....

....parameters. In this investigation, instead of probability densities as in [16, 17, 18] we employ density matrices (nonnegative definite Hermitian matrices of unit trace) and instead of the classical form of the relative entropy (the Kullback Leibler information measure) its quantum counterpart [39, 58] (cf. 44] S(ae 1 ; ae 2 ) Tr ae 1 (log ae 1 Gamma log ae 2 ) 1.5) that is, the relative entropy of the density matrix ae 1 with respect to ae 2 . The three dimensional convex set of 2 Theta 2 density matrices that will be the focus of our study has members representable in the form, ae ....

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A. Wehrl, "General properties of entropy," Rev. Mod. Phys., vol. 50, no. 2, pp. 221-260, Apr. 1978. ASYMPTOTIC REDUNDANCIES 47

.... r ) j r j C Delta jffi (a)j: 156) On the other hand by subadditivity (155) implies S( j ) S j S 2A ;r r; Gamma X 2A ;r GammaA Gamma ;r S i j ( r; n) j S j S 2A ;r r; Gamma C Delta jffi (a)j: 157) We use strong subadditivity [51] S Gamma j I [ II [ III Delta Gamma S Gamma j I [ II Delta S Gamma j I [ III Delta Gamma S Gamma j I Delta (158) of von Neumann entropy for obtaining a lower bound for the first term in (157) We denote by Gamma j (j = 1; jA ;r j) the union ....

Wehrl, A.: General Properties of Entropy. Review of Modern Physics 50, 221--260 (1978)

.... respect to additional such arguments gets also lost, which is actually against any physical intuition if the entropy H respectively N should be a sensible measure of information in some sense (side remark: the non monotonicity of the quantum mechanical von Neumann entropy S( cf. [27, 5], of a state 2 SA with respect to restriction to subalgebras B ae A is not a good excuse at this point, as it refers to a rather different aspect of quantum information ; see again also [14] in preparation) 5. Finally, we should add the following remarks: Although Thomsen s entropy [19] or ....

....This work is dedicated to the memory of Alfred Wehrl, whose untimely death struck me while being here in Berkeley. Fredl 54 THOMAS HUDETZ had been my first guiding teacher in mathematical physics, then my co operative colleague and helpful friend. In his most frequently cited, excellent review [27], he wrote in section IV (Related concepts) after part A on dynamical entropies (not topological, but only measure theoretic; also mentioning Emch, Lindblad and Connes St rmer in the quantum case ) at the end of the short section B: Some concepts measuring the amount of information have been ....

Wehrl, A.: General Properties of Entropy, Rev. Mod. Phys. 50 (2), 221--260 (1978).

.... Hence (ae (m) X now be written as: ae (m) X = X ff 0 ; ff m Gamma1 Gamma T m Gamma1 (C ff m Gamma1 ) Delta Delta Delta C ff 0 Delta j Psi ff 0 Omega Delta Delta Delta Psi ff m Gamma1 i h Psi ff 0 Omega Delta Delta Delta Psi ff m Gamma1 j : By [9], formula (2.4) we have: S(ae (m) X ) X ff 0 ; ff m Gamma1 j Gamma Gamma T m Gamma1 (C ff m Gamma1 ) Delta Delta Delta C ff 0 Delta Delta = h Gamma T m Gamma1 (C) Delta Delta Delta T (C) C Delta : Dividing both sides by m, and taking limes suprema and ....

A. Wehrl: "General properties of entropy", Rev. Mod. Phys. 50, 221--260 (1978)

....a density matrix on a tensor product H 1 Omega H 2 Omega H 3 of three Hilbert spaces. The partial trace of ae 123 over the third space will be denoted by ae 12 , with similar notations for the other partial traces. The following inequality is known as the strong subadditivity of the entropy [LR, Weh] S Gamma ae 123 Delta S Gamma ae 2 Delta S Gamma ae 12 Delta S Gamma ae 23 Delta : 3:8) First consider the case of an atomic probability measure supported in n points and apply strong subadditivity to the density matrix n Gamma1 M j=0 (j) P[F 1 ] j) Omega ....

.... ffi Theta(F ) ffi F : We now write ae[F (m) Z X d P[F (m) X C ff 2C (m) Z C ff d P[F (m) X C ff 2C (m) Z C ff d P[ Theta m Gamma1 (F) Omega Delta Delta Delta Omega P [F ] In order to get an upper bound for H[F (m) we will use the estimates [Weh] S iX i i ae i j X i i S(ae i ) X i j( i ) 3:9) for density matrices ae i and 0 i with P i i = 1 and S(ae 12 ) S(ae 1 ) S(ae 2 ) where ae 12 is a density matrix on H 1 Omega H 2 and ae 1 and ae 2 denote the partial traces of ae 12 over H 2 and H 1 . Then H[F (m) ....

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A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50, 221--260 (1978)

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A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50(1978), 221--260. 10

....not provide any heuristic argument to support the law. This fact might partly be responsible for the mystery surrounding entropy for a long time. As an extreme, we can cite Alfred Wehrl who had the opinion in 1978 that the second law of thermodynamics does not appear to be fully understood yet [13]. The concept of entropy was really clari ed by Ludwig Boltzmann. His scienti c program was to deal with the mechanical theory of heat in connection with probabilities. Assume that a macroscopic system consists of a large number of microscopic ones, we simply call them particles. Since we have ....

....not dicult to state 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 ....

A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50(1978), 221-260. 10

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Wehrl, A. General properties of entropy. Rev. Mod. Phys. 50, 2 (April 1978), 221-260.

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Wehrl, A. General Properties of entropy. ReV. Mod. Phys. 1978, 50, 221-260.

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A. Wehrl, \General Properties of Entropy", Rev. Mod. Phys. 50, 221 - 260 (1978). 17

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