Trace inequality

{{Short description|Concept in Hlibert spaces mathematics}} In [[mathematics]], there are many kinds of [[inequality (mathematics)|inequalities]] involving [[matrix (mathematics)|matrices]] and [[linear operator]]s on [[Hilbert space]]s. This article covers some important operator inequalities connected with [[Trace (linear algebra)|traces]] of matrices.<ref name="C09">E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2010) 73–140 {{doi|10.1090/conm/529/10428}}</ref><ref>R. Bhatia, Matrix Analysis, Springer, (1997).</ref><ref name="B05">B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005).</ref><ref>M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, (1993).</ref>

==Basic definitions== Let <math>\mathbf{H}_n</math> denote the space of [[Hermitian matrix|Hermitian]] <math>n \times n</math> matrices, <math>\mathbf{H}_n^+</math> denote the set consisting of [[Positive-definite matrix#Negative-definite, semidefinite and indefinite matrices|positive semi-definite]] <math>n \times n</math> Hermitian matrices and <math>\mathbf{H}_n^{++}</math> denote the set of [[Positive-definite matrix#Negative-definite, semidefinite and indefinite matrices|positive definite]] Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that they be [[trace class]] and [[self-adjoint operator|self-adjoint]], in which case similar definitions apply, but we discuss only matrices, for simplicity.

For any real-valued function <math>f</math> on an interval <math>I \subseteq \Reals,</math> one may define a [[matrix function]] <math>f(A)</math> for any operator <math>A \in \mathbf{H}_n</math> with [[eigenvalues and eigenvectors|eigenvalues]] <math>\lambda</math> in <math>I</math> by defining it on the eigenvalues and corresponding [[Projection (linear algebra)|projectors]] <math>P</math> as <math display=block>f(A) \equiv \sum_j f(\lambda_j)P_j ~,</math> given the [[Spectral theorem|spectral decomposition]] <math>A = \sum_j \lambda_j P_j.</math>

===Operator monotone=== {{Main|Operator monotone function}}

A function <math>f : I \to \Reals</math> defined on an interval <math>I \subseteq \Reals</math> is said to be '''operator monotone''' if for all <math>n,</math> and all <math>A, B \in \mathbf{H}_n</math> with eigenvalues in <math>I,</math> the following holds, <math display=block>A \geq B \implies f(A) \geq f(B),</math> where the inequality <math>A \geq B</math> means that the operator <math>A - B \geq 0</math> is positive semi-definite. One may check that <math>f(A) = A^2</math> is, in fact, ''not'' operator monotone!

===Operator convex===

A function <math>f : I \to \Reals</math> is said to be '''operator convex''' if for all <math>n</math> and all <math>A, B \in \mathbf{H}_n</math> with eigenvalues in <math>I,</math> and <math>0 < \lambda < 1</math>, the following holds <math display=block>f(\lambda A + (1-\lambda)B) \leq \lambda f(A) + (1 -\lambda)f(B).</math> Note that the operator <math>\lambda A + (1-\lambda)B </math> has eigenvalues in <math>I,</math> since <math> A</math> and <math>B </math> have eigenvalues in <math>I.</math>

A function <math>f</math> is '''{{visible anchor|operator concave}}''' if <math>-f</math> is operator convex;=, that is, the inequality above for <math>f</math> is reversed.

==={{anchor|Joint_convexity_function_2016_10}}Joint convexity===

A function <math>g : I \times J \to \Reals,</math> defined on intervals <math>I, J \subseteq \Reals</math> is said to be '''{{visible anchor|jointly convex}}''' if for all <math>n</math> and all <math>A_1, A_2 \in \mathbf{H}_n</math> with eigenvalues in <math>I</math> and all <math>B_1, B_2 \in \mathbf{H}_n</math> with eigenvalues in <math>J,</math> and any <math>0 \leq \lambda \leq 1</math> the following holds <math display=block>g(\lambda A_1 + (1-\lambda) A_2, \lambda B_1 + (1-\lambda) B_2) ~\leq~ \lambda g(A_1, B_1) + (1 -\lambda) g(A_2, B_2).</math>

A function <math>g</math> is '''{{visible anchor|jointly concave}}''' if −<math>g</math> is jointly convex, i.e. the inequality above for <math>g</math> is reversed.

===Trace function===

Given a function <math>f : \Reals \to \Reals,</math> the associated '''trace function''' on <math>\mathbf{H}_n</math> is given by <math display=block>A \mapsto \operatorname{Tr} f(A) = \sum_j f(\lambda_j),</math> where <math>A</math> has eigenvalues <math>\lambda</math> and <math>\operatorname{Tr}</math> stands for a [[Trace (linear algebra)|trace]] of the operator.

==Convexity and monotonicity of the trace function== Let <math>f: \mathbb{R} \rarr \mathbb{R}</math> be continuous, and let {{mvar|n}} be any [[integer]]. Then, if <math>t\mapsto f(t)</math> is monotone increasing, so is <math>A \mapsto \operatorname{Tr} f(A)</math> on '''H'''_''n''.

Likewise, if <math>t \mapsto f(t)</math> is [[Convex function|convex]], so is <math>A \mapsto \operatorname{Tr} f(A)</math> on '''H'''_''n'', and it is strictly convex if {{mvar|f}} is strictly convex.

See proof and discussion in,<ref name="C09" /> for example.

==Löwner–Heinz theorem== For <math>-1\leq p \leq 0</math>, the function <math>f(t) = -t^p</math> is operator monotone and operator concave.

For <math>0 \leq p \leq 1</math>, the function <math>f(t) = t^p</math> is operator monotone and operator concave.

For <math>1 \leq p \leq 2</math>, the function <math>f(t) = t^p</math> is operator convex. Furthermore, :<math>f(t) = \log(t)</math> is operator concave and operator monotone, while :<math>f(t) = t \log(t)</math> is operator convex.

The original proof of this theorem is due to [[Karl Löwner|K. Löwner]] who gave a necessary and sufficient condition for {{mvar|f}} to be operator monotone.<ref>{{cite journal | last=Löwner | first=Karl | title=Über monotone Matrixfunktionen | journal=Mathematische Zeitschrift | publisher=Springer Science and Business Media LLC | volume=38 | issue=1 | year=1934 | issn=0025-5874 | doi=10.1007/bf01170633 | pages=177–216 | s2cid=121439134 | language=de}}</ref> An [[elementary proof]] of the theorem is discussed in <ref name="C09" /> and a more general version of it in.<ref>[[William F. Donoghue Jr.|W.F. Donoghue, Jr.]], Monotone Matrix Functions and Analytic Continuation, Springer, (1974).</ref>

== {{anchor|Klein2016_10}}Klein's inequality == For all Hermitian {{mvar|n}}×{{mvar|n}} matrices {{mvar|A}} and {{mvar|B}} and all differentiable [[convex function]]s <math>f: \mathbb{R} \rarr \mathbb{R}</math> with [[derivative]] {{math|''f ' ''}}, or for all positive-definite Hermitian {{mvar|n}}×{{mvar|n}} matrices {{mvar|A}} and {{mvar|B}}, and all differentiable convex functions {{mvar|f}}:(0,∞) → <math>\mathbb{R}</math>, the following inequality holds, {{Equation box 1 |indent =: |equation = <math> \operatorname{Tr}[f(A)- f(B)- (A - B)f'(B)] \geq 0~.</math> |cellpadding= 6 |border |border colour = #0073CF |bgcolor=#F9FFF7}} In either case, if {{mvar|f}} is strictly convex, equality holds if and only if {{mvar|A}} = {{mvar|B}}. A popular choice in applications is {{math|''f''(''t'') {{=}} ''t'' log ''t''}}, see below.

===Proof=== Let <math>C=A-B</math> so that, for <math>t\in (0,1)</math>, :<math>B + tC = (1 -t)B + tA</math>, varies from <math>B</math> to <math>A</math>.

Define :<math>F(t) = \operatorname{Tr}[f(B + tC)]</math>. By convexity and monotonicity of trace functions, <math>F(t)</math> is convex, and so for all <math>t\in (0,1)</math>, :<math> F(0) + t(F(1)-F(0))\geq F(t) </math>, which is, :<math> F(1) - F(0) \geq \frac{F(t)-F(0)}{t} </math>, and, in fact, the right hand side is monotone decreasing in <math>t</math>.

Taking the limit <math> t\to 0 </math> yields, :<math> F(1) - F(0) \geq F'(0) </math>, which with rearrangement and substitution is Klein's inequality: :<math> \mathrm{tr}[f(A)-f(B)-(A-B)f'(B)] \geq 0 </math>

Note that if <math> f(t)</math> is strictly convex and <math> C\neq 0 </math>, then <math> F(t) </math> is strictly convex. The final assertion follows from this and the fact that <math>\tfrac{F(t) -F(0)}{t}</math> is monotone decreasing in <math>t</math>.

==Golden–Thompson inequality==

{{main|Golden–Thompson inequality}}

In 1965, S. Golden <ref>{{cite journal | last=Golden | first=Sidney | title=Lower Bounds for the Helmholtz Function | journal=Physical Review | publisher=American Physical Society (APS) | volume=137 | issue=4B | date=1965-02-22 | issn=0031-899X | doi=10.1103/physrev.137.b1127 | pages=B1127–B1128| bibcode=1965PhRv..137.1127G }}</ref> and C.J. Thompson <ref>{{cite journal | last=Thompson | first=Colin J. | title=Inequality with Applications in Statistical Mechanics | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=6 | issue=11 | year=1965 | issn=0022-2488 | doi=10.1063/1.1704727 | pages=1812–1813| bibcode=1965JMP.....6.1812T }}</ref> independently discovered that

For any matrices <math>A, B\in\mathbf{H}_n</math>, :<math>\operatorname{Tr} e^{A+B}\leq \operatorname{Tr} e^A e^B.</math>

This inequality can be generalized for three operators:<ref name="L73">{{cite journal | last=Lieb | first=Elliott H | title=Convex trace functions and the Wigner-Yanase-Dyson conjecture | journal=[[Advances in Mathematics]] | volume=11 | issue=3 | year=1973 | issn=0001-8708 | doi=10.1016/0001-8708(73)90011-x | doi-access=free | pages=267–288| url=http://www.numdam.org/item/RCP25_1973__19__A4_0/ }}</ref> for non-negative operators <math>A, B, C\in\mathbf{H}_n^+</math>, :<math>\operatorname{Tr} e^{\ln A -\ln B+\ln C}\leq \int_0^\infty \operatorname{Tr} A(B+t)^{-1}C(B+t)^{-1}\,\operatorname{d}t.</math>

==Peierls–Bogoliubov inequality== Let <math>R, F\in \mathbf{H}_n</math> be such that Tr e^''R'' = 1. Defining {{math|''g'' {{=}} Tr ''Fe^R''}}, we have :<math>\operatorname{Tr} e^F e^R \geq \operatorname{Tr} e^{F+R}\geq e^g.</math>

The proof of this inequality follows from the above combined with [[#Klein's inequality|Klein's inequality]]. Take {{math|''f''(''x'') {{=}} exp(''x''), ''A''{{=}}''R'' + ''F'', and ''B'' {{=}} ''R'' + ''gI''}}.<ref name="R69">D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).</ref>

==Gibbs variational principle== Let <math>H</math> be a self-adjoint operator such that <math>e^{-H}</math> is [[trace class]]. Then for any <math>\gamma\geq 0 </math> with <math>\operatorname{Tr}\gamma=1,</math> :<math>\operatorname{Tr}\gamma H+\operatorname{Tr}\gamma\ln\gamma\geq -\ln \operatorname{Tr} e^{-H},</math> with equality if and only if <math>\gamma=\exp(-H)/\operatorname{Tr} \exp(-H).</math>

==Lieb's concavity theorem== The following theorem was proved by [[Elliott Lieb|E. H. Lieb]] in.<ref name="L73" /> It proves and generalizes a conjecture of [[Eugene Wigner|E. P. Wigner]], [[Mutsuo Yanase|M. M. Yanase]], and [[Freeman Dyson]].<ref name="WY64">{{cite journal | last1=Wigner | first1=Eugene P. | last2=Yanase | first2=Mutsuo M. | title=On the Positive Semidefinite Nature of a Certain Matrix Expression | journal=Canadian Journal of Mathematics | publisher=Canadian Mathematical Society | volume=16 | year=1964 | issn=0008-414X | doi=10.4153/cjm-1964-041-x | pages=397–406| s2cid=124032721 }}</ref> Six years later other proofs were given by T. Ando <ref name="A79">{{cite journal | last=Ando | first=T. | title=Concavity of certain maps on positive definite matrices and applications to Hadamard products | journal=Linear Algebra and Its Applications | publisher=Elsevier BV | volume=26 | year=1979 | issn=0024-3795 | doi=10.1016/0024-3795(79)90179-4 | pages=203–241| doi-access=free }}</ref> and B. Simon,<ref name="B05" /> and several more have been given since then.

For all <math>m\times n</math> matrices <math>K</math>, and all <math>q </math> and <math>r</math> such that <math>0 \leq q\leq 1</math> and <math>0\leq r \leq 1</math>, with <math>q + r \leq 1</math> the real valued map on <math>\mathbf{H}^+_m \times \mathbf{H}^+_n</math> given by :<math> F(A,B,K) = \operatorname{Tr}(K^*A^qKB^r) </math> * is jointly concave in <math>(A,B)</math> * is convex in <math>K</math>.

Here <math>K^* </math> stands for the [[Hermitian adjoint|adjoint operator]] of <math>K.</math>

==Lieb's theorem== For a fixed Hermitian matrix <math>L\in\mathbf{H}_n</math>, the function :<math> f(A)=\operatorname{Tr} \exp\{L+\ln A\} </math> is concave on <math>\mathbf{H}_n^{++}</math>.

The theorem and proof are due to E. H. Lieb,<ref name="L73" /> Thm 6, where he obtains this theorem as a corollary of Lieb's concavity Theorem. The most direct proof is due to H. Epstein;<ref name="E73">{{cite journal | last=Epstein | first=H. | title=Remarks on two theorems of E. Lieb | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=31 | issue=4 | year=1973 | issn=0010-3616 | doi=10.1007/bf01646492 | pages=317–325| bibcode=1973CMaPh..31..317E | s2cid=120096681 | url=http://projecteuclid.org/euclid.cmp/1103859039 }}</ref> see [[Mary Beth Ruskai|M.B. Ruskai]] papers,<ref name="R02">{{cite journal | last=Ruskai | first=Mary Beth | title=Inequalities for quantum entropy: A review with conditions for equality | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=43 | issue=9 | year=2002 | issn=0022-2488 | doi=10.1063/1.1497701 | pages=4358–4375| arxiv=quant-ph/0205064 | bibcode=2002JMP....43.4358R | s2cid=3051292 }}</ref><ref name="R06">{{cite journal | last=Ruskai | first=Mary Beth | title=Another short and elementary proof of strong subadditivity of quantum entropy | journal=Reports on Mathematical Physics | publisher=Elsevier BV | volume=60 | issue=1 | year=2007 | issn=0034-4877 | doi=10.1016/s0034-4877(07)00019-5 | pages=1–12| arxiv=quant-ph/0604206 | bibcode=2007RpMP...60....1R | s2cid=1432137 }}</ref> for a review of this argument.

==Ando's convexity theorem== T. Ando's proof <ref name="A79" /> of [[#Lieb's concavity theorem|Lieb's concavity theorem]] led to the following significant complement to it:

For all <math>m \times n</math> matrices <math>K</math>, and all <math>1 \leq q \leq 2</math> and <math>0 \leq r \leq 1</math> with <math>q-r \geq 1</math>, the real valued map on <math>\mathbf{H}^{++}_m \times \mathbf{H}^{++}_n</math> given by :<math> (A,B) \mapsto \operatorname{Tr}(K^*A^qKB^{-r})</math> is convex.

== {{anchor|Joint_convexity_2016_10}}Joint convexity of relative entropy == For two operators <math>A, B\in\mathbf{H}^{++}_n </math> define the following map :<math> R(A\parallel B):= \operatorname{Tr}(A\log A) - \operatorname{Tr}(A\log B).</math>

For [[Density matrix|density matrices]] <math>\rho</math> and <math>\sigma</math>, the map <math>R(\rho\parallel\sigma)=S(\rho\parallel\sigma)</math> is the Umegaki's [[quantum relative entropy]].

Note that the non-negativity of <math>R(A\parallel B)</math> follows from Klein's inequality with <math>f(t)=t\log t</math>.

===Statement=== The map <math>R(A\parallel B): \mathbf{H}^{++}_n \times \mathbf{H}^{++}_n \rightarrow \mathbf{R}</math> is jointly convex.

===Proof=== For all <math>0 < p < 1</math>, <math>(A,B) \mapsto \operatorname{Tr}(B^{1-p}A^p)</math> is jointly concave, by [[#Lieb's concavity theorem|Lieb's concavity theorem]], and thus :<math>(A,B)\mapsto \frac{1}{p-1}(\operatorname{Tr}(B^{1-p}A^p)-\operatorname{Tr}A)</math> is convex. But :<math>\lim_{p\rightarrow 1}\frac{1}{p-1}(\operatorname{Tr}(B^{1-p}A^p)-\operatorname{Tr}A)=R(A\parallel B),</math> and convexity is preserved in the limit.

The proof is due to G. Lindblad.<ref name="Ldb74">{{cite journal | last=Lindblad | first=Göran | title=Expectations and entropy inequalities for finite quantum systems | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=39 | issue=2 | year=1974 | issn=0010-3616 | doi=10.1007/bf01608390 | pages=111–119| bibcode=1974CMaPh..39..111L | s2cid=120760667 | url=http://projecteuclid.org/euclid.cmp/1103860161 }}</ref>

==Jensen's operator and trace inequalities== The operator version of [[Jensen's inequality]] is due to C. Davis.<ref name="D57">C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).</ref>

A continuous, real function <math>f</math> on an interval <math>I</math> satisfies '''Jensen's Operator Inequality''' if the following holds :<math> f\left(\sum_kA_k^*X_kA_k\right)\leq\sum_k A_k^*f(X_k)A_k, </math> for operators <math>\{A_k\}_k</math> with <math>\sum_k A^*_kA_k=1</math> and for [[self-adjoint operator]]s <math>\{X_k\}_k</math> with [[Spectrum (functional analysis)|spectrum]] on <math>I</math>.

See,<ref name="D57" /><ref name="HP02">{{cite journal | last1=Hansen | first1=Frank | last2=Pedersen | first2=Gert K. | title=Jensen's Operator Inequality | journal=Bulletin of the London Mathematical Society | volume=35 | issue=4 | date=2003-06-09 | issn=0024-6093 | doi=10.1112/s0024609303002200 | pages=553–564|arxiv=math/0204049| s2cid=16581168 }}</ref> for the proof of the following two theorems.

===Jensen's trace inequality=== Let {{mvar|f}} be a [[continuous function]] defined on an interval {{mvar|I}} and let {{mvar|m}} and {{mvar|n}} be natural numbers. If {{mvar|f}} is convex, we then have the inequality :<math> \operatorname{Tr}\Bigl(f\Bigl(\sum_{k=1}^nA_k^*X_kA_k\Bigr)\Bigr)\leq \operatorname{Tr}\Bigl(\sum_{k=1}^n A_k^*f(X_k)A_k\Bigr),</math> for all ({{mvar|X}}₁, ... , {{mvar|X}}_''n'') self-adjoint {{mvar|m}} × {{mvar|m}} matrices with spectra contained in {{mvar|I}} and all ({{mvar|A}}₁, ... , {{mvar|A}}_''n'') of {{mvar|m}} × {{mvar|m}} matrices with :<math>\sum_{k=1}^nA_k^*A_k=1.</math>

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

===Jensen's operator inequality=== For a continuous function <math>f</math> defined on an interval <math>I</math> the following conditions are equivalent: * <math>f</math> is operator convex. * For each natural number <math>n</math> we have the inequality :<math> f\Bigl(\sum_{k=1}^nA_k^*X_kA_k\Bigr)\leq\sum_{k=1}^n A_k^*f(X_k)A_k, </math> for all <math>(X_1, \ldots , X_n)</math> bounded, self-adjoint operators on an arbitrary [[Hilbert space]] <math>\mathcal{H}</math> with spectra contained in <math>I</math> and all <math>(A_1, \ldots , A_n)</math> on <math>\mathcal{H}</math> with <math>\sum_{k=1}^n A^*_kA_k=1.</math> * <math>f(V^*XV) \leq V^*f(X)V</math> for each isometry <math>V</math> on an infinite-dimensional Hilbert space <math>\mathcal{H}</math> and every self-adjoint operator <math>X</math> with spectrum in <math>I</math>. * <math>Pf(PXP + \lambda(1 -P))P \leq Pf(X)P</math> for each projection <math>P</math> on an infinite-dimensional Hilbert space <math>\mathcal{H}</math>, every self-adjoint operator <math>X</math> with spectrum in <math>I</math> and every <math>\lambda</math> in <math>I</math>.

==Araki–Lieb–Thirring inequality== {{distinguish|text=the [[Lieb–Thirring inequality]]}}

E. H. Lieb and W. E. Thirring proved the following inequality in <ref>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).</ref> 1976: For any <math>A \geq 0,</math> <math>B \geq 0</math> and <math>r \geq 1,</math> <math display=block>\operatorname{Tr} ((BAB)^r) ~\leq~ \operatorname{Tr} (B^r A^r B^r).</math>

In 1990 <ref>{{cite journal | last=Araki | first=Huzihiro | title=On an inequality of Lieb and Thirring | journal=Letters in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=19 | issue=2 | year=1990 | issn=0377-9017 | doi=10.1007/bf01045887 | pages=167–170| bibcode=1990LMaPh..19..167A | s2cid=119649822 }}</ref> H. Araki generalized the above inequality to the following one: For any <math>A \geq 0,</math> <math>B \geq 0</math> and <math>q \geq 0,</math> <math display=block>\operatorname{Tr}((BAB)^{rq}) ~\leq~ \operatorname{Tr}((B^r A^r B^r)^q),</math> for <math>r \geq 1,</math> and <math display=block>\operatorname{Tr}((B^r A^r B^r)^q) ~\leq~ \operatorname{Tr}((BAB)^{rq}),</math> for <math>0 \leq r \leq 1.</math>

There are several other inequalities close to the Lieb–Thirring inequality, such as the following:<ref>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).</ref> for any <math>A \geq 0,</math> <math>B \geq 0</math> and <math>\alpha \in [0, 1],</math> <math display=block>\operatorname{Tr} (B A^\alpha B B A^{1-\alpha} B) ~\leq~ \operatorname{Tr} (B^2 A B^2),</math> and even more generally:<ref>{{cite journal | last1=Lafleche | first1=L. | last2=Saffirio | first2=C. | title=Strong Semiclassical Limits from Hartree and Hartree–Fock to Vlasov–Poisson Equations | journal=Analysis and PDE | publisher=Mathematical Sciences Publishers | volume=16 | issue=4 | date=2023-06-15 | doi=10.2140/apde.2023.16.891 | pages=891–926 | arxiv=2003.02926 }}</ref> for any <math>A \geq 0,</math> <math>B \geq 0,</math> <math>r \geq 1/2</math> and <math>c \geq 0,</math> <math display=block>\operatorname{Tr}((B A B^{2c} A B)^r) ~\leq~ \operatorname{Tr}((B^{c+1} A^2 B^{c+1})^r).</math> The above inequality generalizes the previous one, as can be seen by exchanging <math>A</math> by <math>B^2</math> and <math>B</math> by <math>A^{(1-\alpha)/2}</math> with <math>\alpha = 2 c / (2 c + 2)</math> and using the cyclicity of the trace, leading to <math display=block>\operatorname{Tr}((B A^\alpha B B A^{1-\alpha} B)^r) ~\leq~ \operatorname{Tr}((B^2 A B^2)^r).</math>

Additionally, building upon the Lieb–Thirring inequality the following inequality was derived: <ref> V. Bosboom, M. Schlottbom, F. L. Schwenninger, On the unique solvability of radiative transfer equations with polarization, in Journal of Differential Equations, (2024).</ref> For any <math> A,B\in \mathbf{H}_n, T\in \mathbb{C}^{n\times n}</math> and all <math> 1\leq p,q\leq \infty</math> with <math>1/p+1/q = 1</math>, it holds that <math display=block>|\operatorname{Tr}(TAT^*B)| ~\leq~ \operatorname{Tr}(T^*T|A|^p)^\frac{1}{p}\operatorname{Tr}(TT^*|B|^q)^\frac{1}{q}.</math>

==Effros's theorem and its extension== E. Effros in <ref name="E09">{{cite journal | last=Effros | first=E. G. | title=A matrix convexity approach to some celebrated quantum inequalities | journal=Proceedings of the National Academy of Sciences USA| publisher=Proceedings of the National Academy of Sciences | volume=106 | issue=4 | date=2009-01-21 | issn=0027-8424 | doi=10.1073/pnas.0807965106 | pages=1006–1008| pmid=19164582 | pmc=2633548 |arxiv=0802.1234| bibcode=2009PNAS..106.1006E |doi-access=free}}</ref> proved the following theorem.

If <math>f(x)</math> is an operator convex function, and <math>L</math> and <math>R</math> are commuting bounded linear operators, i.e. the commutator <math>[L,R]=LR-RL=0</math>, the ''perspective'' :<math>g(L, R):=f(LR^{-1})R </math> is jointly convex, i.e. if <math>L=\lambda L_1+(1-\lambda)L_2</math> and <math>R=\lambda R_1+(1-\lambda)R_2</math> with <math>[L_i, R_i]=0</math> (i=1,2), <math>0\leq\lambda\leq 1</math>, :<math>g(L,R)\leq \lambda g(L_1,R_1)+(1-\lambda)g(L_2,R_2).</math>

Ebadian et al. later extended the inequality to the case where <math>L</math> and <math>R</math> do not commute .<ref>{{cite journal | last1=Ebadian | first1=A. | last2=Nikoufar | first2=I. | last3=Eshaghi Gordji | first3=M. | title=Perspectives of matrix convex functions | journal=Proceedings of the National Academy of Sciences | publisher=Proceedings of the National Academy of Sciences USA| volume=108 | issue=18 | date=2011-04-18 | issn=0027-8424 | doi=10.1073/pnas.1102518108| pmc=3088602 | pages=7313–7314| bibcode=2011PNAS..108.7313E | doi-access=free }}</ref>

==Von Neumann's trace inequality and related results==

{{visible anchor|Von Neumann's trace inequality}}, named after its originator [[John von Neumann]], states that for any <math>n \times n</math> complex matrices <math>A</math> and <math>B</math> with [[singular value]]s <math>\alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_n</math> and <math>\beta_1 \geq \beta_2 \geq \cdots \geq \beta_n</math> respectively,<ref>{{cite journal|last1=Mirsky|first1=L.|title=A trace inequality of John von Neumann|journal=Monatshefte für Mathematik|date=December 1975|volume=79|issue=4|pages=303–306|doi=10.1007/BF01647331|s2cid=122252038}}</ref> <math display=block>|\operatorname{Tr}(A B)| ~\leq~ \sum_{i=1}^n \alpha_i \beta_i\,,</math> with equality if and only if <math>A</math> and <math>B^{\dagger}</math> share singular vectors.<ref>{{cite journal|last1=Carlsson|first1=Marcus|title=von Neumann's trace inequality for Hilbert-Schmidt operators|journal=Expositiones Mathematicae|date=2021|volume=39|issue=1|pages=149–157|doi=10.1016/j.exmath.2020.05.001|doi-access=free}}</ref>

A simple corollary to this is the following result:<ref>{{cite book|last1=Marshall|first1=Albert W.|last2=Olkin|first2=Ingram|last3=Arnold|first3=Barry|title=Inequalities: Theory of Majorization and Its Applications|url=https://archive.org/details/inequalitiestheo00mars_587|url-access=limited|date=2011|edition=2nd|location=New York |publisher=Springer|page=[https://archive.org/details/inequalitiestheo00mars_587/page/n370 340]-341|isbn=978-0-387-68276-1}}</ref> For [[Hermitian matrix|Hermitian]] <math>n \times n</math> positive semi-definite complex matrices <math>A</math> and <math>B</math> where now the [[eigenvalue]]s are sorted decreasingly (<math> a_1 \geq a_2 \geq \cdots \geq a_n</math> and <math> b_1 \geq b_2 \geq \cdots \geq b_n,</math> respectively), <math display=block>\sum_{i=1}^n a_i b_{n-i+1} ~\leq~ \operatorname{Tr}(A B) ~\leq~ \sum_{i=1}^n a_i b_i\,.</math>

==See also==

* {{annotated link|Lieb–Thirring inequality}} * {{annotated link|Schur–Horn theorem}} * {{annotated link|Trace identity}} * {{annotated link|von Neumann entropy}}

==References== {{reflist}} *[http://www.scholarpedia.org/article/Matrix_and_Operator_Trace_Inequalities#Gibbs_variational_principle Scholarpedia] primary source.

[[Category:Operator theory]] [[Category:Matrix theory]] [[Category:Inequalities (mathematics)]]