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Trace inequality

  (Redirected from Von Neumann's trace inequality)

Basic definitionsEdit

Let Hn denote the space of Hermitian n×n matrices, Hn+ denote the set consisting of positive semi-definite n×n Hermitian matrices and Hn++ denote the set of positive definite Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint, in which case similar definitions apply, but we discuss only matrices, for simplicity.

For any real-valued function f on an interval I ⊂ ℝ, one may define a matrix function f(A) for any operator AHn with eigenvalues λ in I by defining it on the eigenvalues and corresponding projectors P as

  given the spectral decomposition  

Operator monotoneEdit

A function f: I → ℝ defined on an interval I ⊂ ℝ is said to be operator monotone if ∀n, and all A,BHn with eigenvalues in I, the following holds,

 

where the inequality A ≥ B means that the operator AB ≥ 0 is positive semi-definite. One may check that f(A)=A2 is, in fact, not operator monotone!

Operator convexEdit

A function   is said to be operator convex if for all   and all A,BHn with eigenvalues in I, and  , the following holds

 

Note that the operator   has eigenvalues in  , since   and   have eigenvalues in I.

A function   is operator concave if   is operator convex, i.e. the inequality above for   is reversed.

Joint convexityEdit

A function  , defined on intervals   is said to be jointly convex if for all   and all   with eigenvalues in   and all   with eigenvalues in  , and any   the following holds

 

A function g is jointly concave if −g is jointly convex, i.e. the inequality above for g is reversed.

Trace functionEdit

Given a function f: ℝ → ℝ, the associated trace function on Hn is given by

 

where A has eigenvalues λ and Tr stands for a trace of the operator.

Convexity and monotonicity of the trace functionEdit

Let f: ℝ → ℝ be continuous, and let n be any integer. Then, if   is monotone increasing, so is   on Hn.

Likewise, if   is convex, so is   on Hn, and it is strictly convex if f is strictly convex.

See proof and discussion in,[1] for example.

Löwner–Heinz theoremEdit

For  , the function   is operator monotone and operator concave.

For  , the function   is operator monotone and operator concave.

For  , the function   is operator convex. Furthermore,

  is operator concave and operator monotone, while
  is operator convex.

The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for f to be operator monotone.[5] An elementary proof of the theorem is discussed in [1] and a more general version of it in.[6]

Klein's inequalityEdit

For all Hermitian n×n matrices A and B and all differentiable convex functions f: ℝ → ℝ with derivative f ' , or for all positive-definite Hermitian n×n matrices A and B, and all differentiable convex functions f:(0,∞) → ℝ, the following inequality holds,

 

In either case, if f is strictly convex, equality holds if and only if A = B. A popular choice in applications is f(t) = t log t, see below.

ProofEdit

Let C = AB so that, for 0 < t < 1,

 

Define

 

By convexity and monotonicity of trace functions, φ is convex, and so for all 0 < t < 1,

 

and, in fact, the right hand side is monotone decreasing in t. Taking the limit t→0 yields Klein's inequality.

Note that if f is strictly convex and C ≠ 0, then φ is strictly convex. The final assertion follows from this and the fact that   is monotone decreasing in t.

Golden–Thompson inequalityEdit

In 1965, S. Golden [7] and C.J. Thompson [8] independently discovered that

For any matrices  ,

 

This inequality can be generalized for three operators:[9] for non-negative operators  ,

 

Peierls–Bogoliubov inequalityEdit

Let   be such that Tr eR = 1. Defining g = Tr FeR, we have

 

The proof of this inequality follows from the above combined with Klein's inequality. Take f(x) = exp(x), A=R + F, and B = R + gI.[10]

Gibbs variational principleEdit

Let   be a self-adjoint operator such that   is trace class. Then for any   with  

 

with equality if and only if  

Lieb's concavity theoremEdit

The following theorem was proved by E. H. Lieb in.[9] It proves and generalizes a conjecture of E. P. Wigner, M. M. Yanase and F. J. Dyson.[11] Six years later other proofs were given by T. Ando [12] and B. Simon,[3] and several more have been given since then.

For all   matrices  , and all   and   such that   and  , with   the real valued map on   given by

 
  • is jointly concave in  
  • is convex in  .

Here   stands for the adjoint operator of  

Lieb's theoremEdit

For a fixed Hermitian matrix  , the function

 

is concave on  .

The theorem and proof are due to E. H. Lieb,[9] Thm 6, where he obtains this theorem as a corollary of Lieb's concavity Theorem. The most direct proof is due to H. Epstein;[13] see M.B. Ruskai papers,[14][15] for a review of this argument.

Ando's convexity theoremEdit

T. Ando's proof [12] of Lieb's concavity theorem led to the following significant complement to it:

For all   matrices  , and all   and   with  , the real valued map on   given by

 

is convex.

Joint convexity of relative entropyEdit

For two operators   define the following map

 

For density matrices   and  , the map   is the Umegaki's quantum relative entropy.

Note that the non-negativity of   follows from Klein's inequality with  .

StatementEdit

The map   is jointly convex.

ProofEdit

For all  ,   is jointly concave, by Lieb's concavity theorem, and thus

 

is convex. But

 

and convexity is preserved in the limit.

The proof is due to G. Lindblad.[16]

Jensen's operator and trace inequalitiesEdit

The operator version of Jensen's inequality is due to C. Davis.[17]

A continuous, real function   on an interval   satisfies Jensen's Operator Inequality if the following holds

 

for operators   with   and for self-adjoint operators   with spectrum on  .

See,[17][18] for the proof of the following two theorems.

Jensen's trace inequalityEdit

Let f be a continuous function defined on an interval I and let m and n be natural numbers. If f is convex, we then have the inequality

 

for all (X1, ... , Xn) self-adjoint m × m matrices with spectra contained in I and all (A1, ... , An) of m × m matrices with

 

Conversely, if the above inequality is satisfied for some n and m, where n > 1, then f is convex.

Jensen's operator inequalityEdit

For a continuous function   defined on an interval   the following conditions are equivalent:

  •   is operator convex.
  • For each natural number   we have the inequality
 

for all   bounded, self-adjoint operators on an arbitrary Hilbert space   with spectra contained in   and all   on   with  

  •   for each isometry   on an infinite-dimensional Hilbert space   and

every self-adjoint operator   with spectrum in  .

  •   for each projection   on an infinite-dimensional Hilbert space  , every self-adjoint operator   with spectrum in   and every   in  .

Araki–Lieb–Thirring inequalityEdit

E. H. Lieb and W. E. Thirring proved the following inequality in [19] in 1976: For any  ,   and  

 

In 1990 [20] H. Araki generalized the above inequality to the following one: For any  ,   and  

  for  

and

  for  

The Lieb–Thirring inequality also enjoys the following generalization:[21] for any  ,   and  

 

Effros's theorem and its extensionEdit

E. Effros in [22] proved the following theorem.

If   is an operator convex function, and   and   are commuting bounded linear operators, i.e. the commutator  , the perspective

 

is jointly convex, i.e. if   and   with   (i=1,2),  ,

 

Ebadian et al. later extended the inequality to the case where   and   do not commute . [23]

Von Neumann's trace inequality and related resultsEdit

Von Neumann's trace inequality, named after its originator John von Neumann, states that for any n × n complex matrices AB with singular values   and   respectively,[24]

 

A simple corollary to this is the following result[25]: For hermitian n × n complex matrices AB where now the eigenvalues are sorted decreasingly (  and  , respectively),

 

See alsoEdit

ReferencesEdit

  1. ^ a b c E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2010) 73–140 doi:10.1090/conm/529/10428
  2. ^ R. Bhatia, Matrix Analysis, Springer, (1997).
  3. ^ a b B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005).
  4. ^ M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, (1993).
  5. ^ K. Löwner, "Uber monotone Matrix funktionen", Math. Z. 38, 177–216, (1934).
  6. ^ W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer, (1974).
  7. ^ S. Golden, Lower Bounds for Helmholtz Functions, Phys. Rev. 137, B 1127–1128 (1965)
  8. ^ C.J. Thompson, Inequality with Applications in Statistical Mechanics, J. Math. Phys. 6, 1812–1813, (1965).
  9. ^ a b c E. H. Lieb, Convex Trace Functions and the Wigner–Yanase–Dyson Conjecture, Advances in Math. 11, 267–288 (1973).
  10. ^ D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).
  11. ^ E. P. Wigner, M. M. Yanase, On the Positive Semi-Definite Nature of a Certain Matrix Expression, Can. J. Math. 16, 397–406, (1964).
  12. ^ a b . Ando, Convexity of Certain Maps on Positive Definite Matrices and Applications to Hadamard Products, Lin. Alg. Appl. 26, 203–241 (1979).
  13. ^ H. Epstein, Remarks on Two Theorems of E. Lieb, Comm. Math. Phys., 31:317–325, (1973).
  14. ^ M. B. Ruskai, Inequalities for Quantum Entropy: A Review With Conditions for Equality, J. Math. Phys., 43(9):4358–4375, (2002).
  15. ^ M. B. Ruskai, Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy, Reports Math. Phys. 60, 1–12 (2007).
  16. ^ G. Lindblad, Expectations and Entropy Inequalities, Commun. Math. Phys. 39, 111–119 (1974).
  17. ^ a b C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).
  18. ^ F. Hansen, G. K. Pedersen, Jensen's Operator Inequality, Bull. London Math. Soc. 35 (4): 553–564, (2003).
  19. ^ E. H. Lieb, W. E. Thirring, Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities, in Studies in Mathematical Physics, edited E. Lieb, B. Simon, and A. Wightman, Princeton University Press, 269–303 (1976).
  20. ^ H. Araki, On an Inequality of Lieb and Thirring, Lett. Math. Phys. 19, 167–170 (1990).
  21. ^ Z. Allen-Zhu, Y. Lee, L. Orecchia, Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver, in ACM-SIAM Symposium on Discrete Algorithms, 1824–1831 (2016).
  22. ^ E. Effros, A Matrix Convexity Approach to Some Celebrated Quantum Inequalities, Proc. Natl. Acad. Sci. USA, 106, n.4, 1006–1008 (2009).
  23. ^ A. Ebadian, I. Nikoufar, and M. Gordjic, "Perspectives of matrix convex functions," Proc. Natl Acad. Sci. USA, 108(18), 7313–7314 (2011)
  24. ^ Mirsky, L. (December 1975). "A trace inequality of John von Neumann". Monatshefte für Mathematik. 79 (4): 303–306. doi:10.1007/BF01647331.
  25. ^ Marshall, Albert W.; Olkin, Ingram; Arnold, Barry (2011). Inequalities: Theory of Majorization and Its Applications (2nd ed.). New York: Springer. p. 340-341. ISBN 978-0-387-68276-1.