Trace inequality – Wikipedia
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 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 log t, see below.
Proof[edit]
Let
so that, for
,
- ,
varies from
to
.
Define
- .
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 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
.
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
for
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
Effros’s theorem and its extension[edit]
E. Effros in [23] 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 .[24]
Von Neumann’s trace inequality, named after its originator John von Neumann, states that for any complex matrices and with singular values and respectively,[25]
with equality if and only if
and
share singular vectors.[26]
A simple corollary to this is the following result:[27] For Hermitian
positive semi-definite complex matrices
and
where now the eigenvalues are sorted decreasingly (
and
respectively),
See also[edit]
References[edit]
- ^ 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
- ^ R. Bhatia, Matrix Analysis, Springer, (1997).
- ^ a b B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005).
- ^ M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, (1993).
- ^ Löwner, Karl (1934). “Über monotone Matrixfunktionen”. Mathematische Zeitschrift (in German). Springer Science and Business Media LLC. 38 (1): 177–216. doi:10.1007/bf01170633. ISSN 0025-5874. S2CID 121439134.
- ^ W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer, (1974).
- ^ Golden, Sidney (1965-02-22). “Lower Bounds for the Helmholtz Function”. Physical Review. American Physical Society (APS). 137 (4B): B1127–B1128. Bibcode:1965PhRv..137.1127G. doi:10.1103/physrev.137.b1127. ISSN 0031-899X.
- ^ Thompson, Colin J. (1965). “Inequality with Applications in Statistical Mechanics”. Journal of Mathematical Physics. AIP Publishing. 6 (11): 1812–1813. Bibcode:1965JMP…..6.1812T. doi:10.1063/1.1704727. ISSN 0022-2488.
- ^ 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.
- ^ D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).
- ^ Wigner, Eugene P.; Yanase, Mutsuo M. (1964). “On the Positive Semidefinite Nature of a Certain Matrix Expression”. Canadian Journal of Mathematics. Canadian Mathematical Society. 16: 397–406. doi:10.4153/cjm-1964-041-x. ISSN 0008-414X. S2CID 124032721.
- ^ a b Ando, T. (1979). “Concavity of certain maps on positive definite matrices and applications to Hadamard products”. Linear Algebra and Its Applications. Elsevier BV. 26: 203–241. doi:10.1016/0024-3795(79)90179-4. ISSN 0024-3795.
- ^ Epstein, H. (1973). “Remarks on two theorems of E. Lieb”. Communications in Mathematical Physics. Springer Science and Business Media LLC. 31 (4): 317–325. Bibcode:1973CMaPh..31..317E. doi:10.1007/bf01646492. ISSN 0010-3616. S2CID 120096681.
- ^ Ruskai, Mary Beth (2002). “Inequalities for quantum entropy: A review with conditions for equality”. Journal of Mathematical Physics. AIP Publishing. 43 (9): 4358–4375. arXiv:quant-ph/0205064. Bibcode:2002JMP….43.4358R. doi:10.1063/1.1497701. ISSN 0022-2488. S2CID 3051292.
- ^ Ruskai, Mary Beth (2007). “Another short and elementary proof of strong subadditivity of quantum entropy”. Reports on Mathematical Physics. Elsevier BV. 60 (1): 1–12. arXiv:quant-ph/0604206. Bibcode:2007RpMP…60….1R. doi:10.1016/s0034-4877(07)00019-5. ISSN 0034-4877. S2CID 1432137.
- ^ Lindblad, Göran (1974). “Expectations and entropy inequalities for finite quantum systems”. Communications in Mathematical Physics. Springer Science and Business Media LLC. 39 (2): 111–119. Bibcode:1974CMaPh..39..111L. doi:10.1007/bf01608390. ISSN 0010-3616. S2CID 120760667.
- ^ a b C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).
- ^ 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.
- ^ 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).
- ^ Araki, Huzihiro (1990). “On an inequality of Lieb and Thirring”. Letters in Mathematical Physics. Springer Science and Business Media LLC. 19 (2): 167–170. Bibcode:1990LMaPh..19..167A. doi:10.1007/bf01045887. ISSN 0377-9017. S2CID 119649822.
- ^ 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).
- ^ L. Lafleche, C. Saffirio, Strong Semiclassical Limit from Hartree and
Hartree-Fock to Vlasov-Poisson Equation, arXiv:2003.02926 [math-ph]. - ^ Effros, E. G. (2009-01-21). “A matrix convexity approach to some celebrated quantum inequalities”. Proceedings of the National Academy of Sciences USA. Proceedings of the National Academy of Sciences. 106 (4): 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. ISSN 0027-8424. PMC 2633548. PMID 19164582.
- ^ Ebadian, A.; Nikoufar, I.; Eshaghi Gordji, M. (2011-04-18). “Perspectives of matrix convex functions”. Proceedings of the National Academy of Sciences. Proceedings of the National Academy of Sciences USA. 108 (18): 7313–7314. Bibcode:2011PNAS..108.7313E. doi:10.1073/pnas.1102518108. ISSN 0027-8424. PMC 3088602.
- ^ Mirsky, L. (December 1975). “A trace inequality of John von Neumann”. Monatshefte für Mathematik. 79 (4): 303–306. doi:10.1007/BF01647331. S2CID 122252038.
- ^ 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.
- ^ 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.
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