Trace inequality

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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 denote the space of Hermitian matrices, denote the set consisting of positive semi-definite Hermitian matrices and 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 on an interval one may define a matrix function for any operator with eigenvalues in by defining it on the eigenvalues and corresponding projectors as

given the spectral decomposition

Operator monotone[edit]

A function defined on an interval is said to be operator monotone if for all and all with eigenvalues in the following holds,

where the inequality means that the operator is positive semi-definite. One may check that is, in fact, not operator monotone!

Operator convex[edit]

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

Note that the operator has eigenvalues in since and have eigenvalues in

A function is operator concave if is operator convex;=, that is, 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 is jointly concave if − is jointly convex, i.e. the inequality above for is reversed.

Trace function[edit]

Given a function the associated trace function on is given by

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

Convexity and monotonicity of the trace function[edit]

Let 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 with derivative f ' , or for all positive-definite Hermitian n×n matrices A and B, and all differentiable convex functions f:(0,∞) → , 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.


Let so that, for ,


varies from to .



By convexity and monotonicity of trace functions, is convex, and so for all ,


which is,


and, in fact, the right hand side is monotone decreasing in .

Taking the limit yields,


which with rearrangement and substitution is Klein's inequality:

Note that if is strictly convex and , then is strictly convex. The final assertion follows from this and the fact that is monotone decreasing in .

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 + gI.[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 Freeman 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 .


The map is jointly convex.


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] 1976: For any and

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

for and

There are several other inequalities close to the Lieb–Thirring inequality, such as the following:[21] for any and

and even more generally:[22] for any and
The above inequality generalizes the previous one, as can be seen by exchanging by and by with and using the cyclicity of the trace, leading to

Additionally, building upon the Lieb-Thirring inequality the following inequality was derived: [23] For any and all with , it holds that

Effros's theorem and its extension[edit]

E. Effros in [24] 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 .[25]

Von Neumann's trace inequality and related results[edit]

Von Neumann's trace inequality, named after its originator John von Neumann, states that for any complex matrices and with singular values and respectively,[26]

with equality if and only if and share singular vectors.[27]

A simple corollary to this is the following result:[28] For Hermitian positive semi-definite complex matrices and where now the eigenvalues are sorted decreasingly ( and respectively),

See also[edit]


  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. ^ Löwner, Karl (1934). "Über monotone Matrixfunktionen". Mathematische Zeitschrift (in German). 38 (1). Springer Science and Business Media LLC: 177–216. doi:10.1007/bf01170633. ISSN 0025-5874. S2CID 121439134.
  6. ^ W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer, (1974).
  7. ^ Golden, Sidney (1965-02-22). "Lower Bounds for the Helmholtz Function". Physical Review. 137 (4B). American Physical Society (APS): B1127–B1128. Bibcode:1965PhRv..137.1127G. doi:10.1103/physrev.137.b1127. ISSN 0031-899X.
  8. ^ Thompson, Colin J. (1965). "Inequality with Applications in Statistical Mechanics". Journal of Mathematical Physics. 6 (11). AIP Publishing: 1812–1813. Bibcode:1965JMP.....6.1812T. doi:10.1063/1.1704727. ISSN 0022-2488.
  9. ^ a b c Lieb, Elliott H (1973). "Convex trace functions and the Wigner-Yanase-Dyson conjecture". Advances in Mathematics. 11 (3): 267–288. doi:10.1016/0001-8708(73)90011-x. ISSN 0001-8708.
  10. ^ D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).
  11. ^ Wigner, Eugene P.; Yanase, Mutsuo M. (1964). "On the Positive Semidefinite Nature of a Certain Matrix Expression". Canadian Journal of Mathematics. 16. Canadian Mathematical Society: 397–406. doi:10.4153/cjm-1964-041-x. ISSN 0008-414X. S2CID 124032721.
  12. ^ a b Ando, T. (1979). "Concavity of certain maps on positive definite matrices and applications to Hadamard products". Linear Algebra and Its Applications. 26. Elsevier BV: 203–241. doi:10.1016/0024-3795(79)90179-4. ISSN 0024-3795.
  13. ^ Epstein, H. (1973). "Remarks on two theorems of E. Lieb". Communications in Mathematical Physics. 31 (4). Springer Science and Business Media LLC: 317–325. Bibcode:1973CMaPh..31..317E. doi:10.1007/bf01646492. ISSN 0010-3616. S2CID 120096681.
  14. ^ Ruskai, Mary Beth (2002). "Inequalities for quantum entropy: A review with conditions for equality". Journal of Mathematical Physics. 43 (9). AIP Publishing: 4358–4375. arXiv:quant-ph/0205064. Bibcode:2002JMP....43.4358R. doi:10.1063/1.1497701. ISSN 0022-2488. S2CID 3051292.
  15. ^ Ruskai, Mary Beth (2007). "Another short and elementary proof of strong subadditivity of quantum entropy". Reports on Mathematical Physics. 60 (1). Elsevier BV: 1–12. arXiv:quant-ph/0604206. Bibcode:2007RpMP...60....1R. doi:10.1016/s0034-4877(07)00019-5. ISSN 0034-4877. S2CID 1432137.
  16. ^ Lindblad, Göran (1974). "Expectations and entropy inequalities for finite quantum systems". Communications in Mathematical Physics. 39 (2). Springer Science and Business Media LLC: 111–119. Bibcode:1974CMaPh..39..111L. doi:10.1007/bf01608390. ISSN 0010-3616. S2CID 120760667.
  17. ^ a b C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).
  18. ^ Hansen, Frank; Pedersen, Gert K. (2003-06-09). "Jensen's Operator Inequality". Bulletin of the London Mathematical Society. 35 (4): 553–564. arXiv:math/0204049. doi:10.1112/s0024609303002200. ISSN 0024-6093. S2CID 16581168.
  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. ^ Araki, Huzihiro (1990). "On an inequality of Lieb and Thirring". Letters in Mathematical Physics. 19 (2). Springer Science and Business Media LLC: 167–170. Bibcode:1990LMaPh..19..167A. doi:10.1007/bf01045887. ISSN 0377-9017. S2CID 119649822.
  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. ^ L. Lafleche, C. Saffirio, Strong Semiclassical Limit from Hartree and Hartree-Fock to Vlasov-Poisson Equation, arXiv:2003.02926 [math-ph].
  23. ^ V. Bosboom, M. Schlottbom, F. L. Schwenninger, On the unique solvability of radiative transfer equations with polarization, in Journal of Differential Equations, (2024).
  24. ^ Effros, E. G. (2009-01-21). "A matrix convexity approach to some celebrated quantum inequalities". Proceedings of the National Academy of Sciences USA. 106 (4). Proceedings of the National Academy of Sciences: 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. ISSN 0027-8424. PMC 2633548. PMID 19164582.
  25. ^ Ebadian, A.; Nikoufar, I.; Eshaghi Gordji, M. (2011-04-18). "Perspectives of matrix convex functions". Proceedings of the National Academy of Sciences. 108 (18). Proceedings of the National Academy of Sciences USA: 7313–7314. Bibcode:2011PNAS..108.7313E. doi:10.1073/pnas.1102518108. ISSN 0027-8424. PMC 3088602.
  26. ^ Mirsky, L. (December 1975). "A trace inequality of John von Neumann". Monatshefte für Mathematik. 79 (4): 303–306. doi:10.1007/BF01647331. S2CID 122252038.
  27. ^ Carlsson, Marcus (2021). "von Neumann's trace inequality for Hilbert-Schmidt operators". Expositiones Mathematicae. 39 (1): 149–157. doi:10.1016/j.exmath.2020.05.001.
  28. ^ 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.