aa r X i v : . [ m a t h - ph ] O c t Golden-Thompson’s inequalityfordeformed exponentials
Frank HansenSeptember 3, 2014
Abstract
Deformed logarithms and their inverse functions, the deformed ex-ponentials, are important tools in the theory of non-additive entropiesand non-extensive statistical mechanics. We formulate and provecounterparts of Golden-Thompson’s trace inequality for q -exponentialswith parameter q in the interval [1 , . MSC2010
Key words and phrases: deformed exponentials; Golden-Thompson’strace inequality.
Tsallis [7] generalised in 1988 the standard Bolzmann-Gibbs entropy to a non-extensive quantity S q depending on a parameter q. In the quantum versionit is given by S q ( ρ ) = 1 − Tr ρ q q − q = 1 , where ρ is a density matrix. It has the property that S q ( ρ ) → S ( ρ ) for q → , where S ( ρ ) = − Tr ρ log ρ is the von Neumann entropy. The Tsallis entropymay be written on a similar form S q ( ρ ) = − Tr ρ log q ( ρ ) , q is given bylog q x = Z x t q − dt = x q − − q − q > x q = 1for x > . The deformed logarithm is also denoted the q -logarithm. Theinverse function exp q is called the q -exponential and is given byexp q ( x ) = ( x ( q −
1) + 1) / ( q − for x > − q − . The q -logarithm and the q -exponential functions converge, respectively, tothe logarithmic and the exponential functions for q → . The aim of this article is to generalise Golden-Thompson’s trace inequal-ity [2, 6] to deformed exponentials. The main result is the following:
Theorem 1.1.
Let A and B be positive definite matrices.(i) If ≤ q < then Tr exp q ( A + B ) ≤ Tr exp q ( A ) − q (cid:0) A ( q −
1) + exp q B (cid:1) . (ii) If ≤ q ≤ then Tr exp q ( A + B ) ≥ Tr exp q ( A ) − q (cid:0) A ( q −
1) + exp q B (cid:1) . Notice that we for q = 1 recovers Golden-Thomson’s trace inequalityTr exp( A + B ) ≤ Tr exp( A ) exp( B ) . This inequality is valid for arbitrary self-adjoint matrices A and B. However,it is sufficient to know the inequality for positive definite matrices, since thegeneral form follows by multiplication with positive numbers.
We collect a few well-known results that we are going to use in the proof ofthe main theorem. 2he q -logarithm is a bijection of the positive half-line onto the open inter-val ( − ( q − − , ∞ ) , and the q -exponential is consequently a bijection of theinterval ( − ( q − − , ∞ ) onto the positive half-line. For q > q -logarithm and the q -exponential to positive definiteoperators. We also notice that(1) ddx log q ( x ) = x q − and ddx exp q ( x ) = exp q ( x ) − q . The proof of the following lemma is rather easy and may be found in [4,Lemma 5].
Lemma 2.1.
Let ϕ : D → A sa be a map defined in a convex cone D in aBanach space X with values in the self-adjoint part of a C ∗ -algebra A . If ϕ is Fr´echet differentiable, convex and positively homogeneous then dϕ ( x ) h ≤ ϕ ( h ) . for x, h ∈ D . Let H be any n × n matrix. The map A → Tr ( H ∗ A p H ) /p , defined in positive definite n × n matrices, is concave for 0 < p ≤ ≤ p ≤ , cf. [1, Theorem 1.1]. By a slight modification ofthe construction given in Remark 3.2 in the same reference, cf. also [3], weobtain that the mapping(2) ( A , . . . , A k ) → Tr ( H ∗ A p A + · · · + H ∗ k A k H k ) /p , defined in k -tuples of positive definite n × n matrices, is concave for 0 < p ≤ ≤ p ≤
2; for arbitrary n × n matrices H , . . . , H k . Theorem 3.1.
Let H , . . . , H k be matrices with H ∗ H + · · · + H ∗ k H k = 1 anddefine the function (3) ϕ ( A , . . . , A k ) = Tr exp q (cid:16) k X i =1 H ∗ i log q ( A i ) H i (cid:17) in k -tuples of positive definite matrices. Then ϕ is positively homogeneous ofdegree one. It is concave for ≤ q ≤ and convex for ≤ q ≤ . roof. For q > ϕ ( A , . . . , A k ) = Tr exp q (cid:16) k X i =1 H ∗ i log q ( A i ) H i (cid:17) = Tr (cid:16) ( q − (cid:16) k X i =1 H ∗ i log q ( A i ) H i (cid:17) + 1 (cid:17) / ( q − = Tr (cid:16) ( q − (cid:16) k X i =1 H ∗ i A q − i − q − H i (cid:17) + 1 (cid:17) / ( q − = Tr (cid:16) k X i =1 H ∗ i ( A q − i − H i + 1 (cid:17) / ( q − = Tr (cid:0) H ∗ A q − H + · · · + H ∗ k A q − k H k (cid:1) / ( q − . From this identity it follows that ϕ is positively homogeneous of degree one.The concavity for 1 < q ≤ ≤ q ≤ q = 1 follows by letting q tend to one. QEDCorollary 3.2.
Let L be positive definite, and let H , . . . , H k be matricessuch that H ∗ H + · · · + H ∗ k H k ≤ . Then the function ϕ ( A , . . . , A k ) = Tr exp q (cid:0) L + H ∗ log q ( A ) H + · · · + H ∗ k log q ( A k ) H k (cid:1) , defined in k -tuples of positive definite matrices, is concave for ≤ q ≤ andconvex for ≤ q ≤ . Proof.
We may without loss of generality assume H ∗ H + · · · + H ∗ k H k < H k +1 = (cid:0) − ( H ∗ H + · · · + H ∗ k H k ) (cid:1) / . We then have H ∗ H + · · · + H ∗ k H k + H ∗ k +1 H k +1 = 1and may use the preceding theorem to conclude that the function( A , . . . , A k +1 ) → Tr exp q (cid:0) H ∗ log q ( A ) H + · · · + H ∗ k +1 log q ( A k +1 ) H k +1 (cid:1) of k + 1 variables is concave for 1 ≤ q ≤ ≤ q ≤ . Since H k +1 is invertible we may choose A k +1 = exp q (cid:0) H − k +1 LH − k +1 (cid:1) which makes sense since H − k +1 LH − k +1 is positive definite. Concavity for 1 ≤ q ≤ ≤ q ≤ k variables of the abovefunction then yields the result. QED q = 1 we recover in particular [5, Theorem 3]. Corollary 3.3.
Let H , . . . , H k be matrices with H ∗ H + · · · + H ∗ k H k ≤ , and let L be self-adjoint. The trace function ( A , . . . , A k ) → Tr exp (cid:0) L + H ∗ log( A ) H + · · · + H ∗ k log( A k ) H k (cid:1) is concave in positive definite matrices. Corollary 3.4.
The trace function ϕ defined in (3) satisfies ϕ ( B , . . . , B k ) ≤ Tr exp q (cid:16) k X i =1 H ∗ i log q ( A i ) H i (cid:17) − q k X j =1 H ∗ j ( d log q ( A j ) B j ) H j for ≤ q ≤ and ϕ ( B , . . . , B k ) ≥ Tr exp q (cid:16) k X i =1 H ∗ i log q ( A i ) H i (cid:17) − q k X j =1 H ∗ j ( d log q ( A j ) B j ) H j for ≤ q ≤ , where A , . . . , A k and B , . . . , B k are positive definite matrices.Proof. For 1 ≤ q ≤ d ϕ ( A , . . . , A k )( B , . . . , B k ) ≥ ϕ ( B , . . . , B k )by Lemma 2.1. By the chain rule for Fr´echet differentiable mappings betweenBanach spaces we therefore obtain ϕ ( B , . . . , B k ) ≤ k X j =1 d j ϕ ( A , . . . , A k ) B j = k X j =1 Tr d exp q (cid:16) k X i =1 H ∗ i log q ( A i ) H i (cid:17) H ∗ j ( d log q ( A j ) B j ) H j = k X j =1 Tr exp q (cid:16) k X i =1 H ∗ i log q ( A i ) H i (cid:17) − q H ∗ j ( d log q ( A j ) B j ) H j where we used the identity Tr d f ( A ) B = Tr f ′ ( A ) B valid for differentiablefunctions. This proves the first assertion. The result for 2 ≤ q ≤ QED Proof of the main theorem
In order to prove Theorem 1.1 (i) we set k = 2 in Corollary 3.4 and obtain ϕ ( B , B ) ≤ Tr exp q ( X ) − q (cid:0) H ∗ ( d log q ( A ) B ) H + H ∗ ( d log q ( A ) B ) H (cid:1) for 1 ≤ q ≤ A , A and B , B where X = H ∗ log q ( A ) H + H ∗ log q ( A ) H . If we set A = B and A = 1 the inequality reduces to ϕ ( B , B ) ≤ Tr exp q ( H ∗ log q ( B ) H ) − q (cid:0) H ∗ B q − H + H ∗ B H (cid:1) . We now set H = ε / for 0 < ε < , and to fixed positive definite matrices L and L we choose B and B such that L = H ∗ log q ( B ) H = ε log q ( B ) L = H ∗ log q ( B ) H = (1 − ε ) log q ( B ) . It follows that B = exp q ( ε − L ) and B = exp q ((1 − ε ) − L ) . Inserting in the inequality we now obtainTr exp q ( L + L ) ≤ Tr exp q ( L ) − q (cid:0) ε exp q ( ε − L ) q − + (1 − ε ) exp q ((1 − ε ) − L ) (cid:1) = Tr exp q ( L ) − q (cid:0) L ( q −
1) + ε + (1 − ε ) exp q ((1 − ε ) − L ) (cid:1) . This expression decouble L and L and reduces the minimisation problemover ε to the commutative case. We furthermore realise that minimum isobtained by letting ε tend to zero and thatlim ε → (1 − ε ) exp q (cid:0) (1 − ε ) − L (cid:1) = exp q ( L ) . We finally replace L and L with A and B. This proves the first statementin Theorem 1.1.The proof of the second statement is virtually identical to the proof of the6rst. Since now 2 ≤ q ≤ k = 2 and applying the same substitutions as in the proof of the firststatement we arrive at the inequalityTr exp q ( L + L ) ≥ Tr exp q ( L ) − q (cid:0) L ( q −
1) + ε + (1 − ε ) exp q ((1 − ε ) − L ) (cid:1) . Since 2 ≤ q ≤ ε → ε + (1 − ε ) exp q ((1 − ε ) − L )is now decreasing, and we thus maximise the right hand side in the aboveinequality by letting ε tend to zero. This proves the second statement inTheorem 1.1. References [1] E.A. Carlen and E.H. Lieb. A Minkowsky type trace inequality and strongsubadditivity of quantum entropy II: Convexity and concavity.
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Brazilian Journal of Physics , 39(2A):337–356, 2009.Frank Hansen: Institute for Excellence in Higher Education, Tohoku University,Japan.Email: [email protected]., 39(2A):337–356, 2009.Frank Hansen: Institute for Excellence in Higher Education, Tohoku University,Japan.Email: [email protected].