Trace inequalities

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In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.[1][2][3][4]

Basic definitions[edit]

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 monotone[edit]

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 A − B ≥ 0 is positive semi-definite. One may check that f(A)=A2 is, in fact, not monotone!

Operator convex[edit]

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 convexity[edit]

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 function[edit]

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 function[edit]

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 theorem[edit]

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 inequality[edit]

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 logt, see below.

Proof[edit]

Let C = A − B 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 inequality[edit]

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 inequality[edit]

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+g I.[10]

Gibbs variational principle[edit]

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

with equality if and only if .

Lieb's concavity theorem[edit]

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 theorem[edit]

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 theorem[edit]

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 entropy[edit]

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 .

Statement[edit]

The map is jointly convex.

Proof[edit]

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 inequalities[edit]

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 inequality[edit]

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 inequality[edit]

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 inequality[edit]

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

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

Effros's theorem and its extension[edit]

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[edit]

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]

The equality is achieved when and are simultaneously unitarily diagonalizable (see trace).

See also[edit]

References[edit]

  1. ^ a b c E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2009).
  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.