Trace inequality#Von Neumann's trace inequality

Let $\backslash mathbf\{H\}\_n$ denote the space of Hermitian $n\; \backslash times\; n$ matrices, $\backslash mathbf\{H\}\_n^+$ denote the set consisting of positive semi-definite $n\; \backslash times\; n$ Hermitian matrices and $\backslash mathbf\{H\}\_n^\{++\}$ 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 $I\; \backslash subseteq\; \backslash Reals,$ one may define a matrix function $f(A)$ for any operator $A\; \backslash in\; \backslash mathbf\{H\}\_n$ with eigenvalues $\backslash lambda$ in $I$ by defining it on the eigenvalues and corresponding projectors $P$ as

$$f(A)\; \backslash equiv\; \backslash sum\_j\; f(\backslash lambda\_j)P\_j\; ~,$$

given the spectral decomposition $A\; =\; \backslash sum\_j\; \backslash lambda\_j\; P\_j.$

{{Main|Operator monotone function}}

A function $f\; :\; I\; \backslash to\; \backslash Reals$ defined on an interval $I\; \backslash subseteq\; \backslash Reals$ is said to be operator monotone if for all $n,$ and all $A,\; B\; \backslash in\; \backslash mathbf\{H\}\_n$ with eigenvalues in $I,$ the following holds,

$$A\; \backslash geq\; B\; \backslash implies\; f(A)\; \backslash geq\; f(B),$$

where the inequality $A\; \backslash geq\; B$ means that the operator $A\; -\; B\; \backslash geq\; 0$ is positive semi-definite. One may check that $f(A)\; =\; A^2$ is, in fact, not operator monotone!

A function $f\; :\; I\; \backslash to\; \backslash Reals$ is said to be operator convex if for all $n$ and all $A,\; B\; \backslash in\; \backslash mathbf\{H\}\_n$ with eigenvalues in $I,$ and , the following holds

$$f(\backslash lambda\; A\; +\; (1-\backslash lambda)B)\; \backslash leq\; \backslash lambda\; f(A)\; +\; (1\; -\backslash lambda)f(B).$$

Note that the operator $\backslash lambda\; A\; +\; (1-\backslash lambda)B$ has eigenvalues in $I,$ since $A$ and $B$ have eigenvalues in $I.$

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

A function $g\; :\; I\; \backslash times\; J\; \backslash to\; \backslash Reals,$ defined on intervals $I,\; J\; \backslash subseteq\; \backslash Reals$ is said to be {{visible anchor|jointly convex}} if for all $n$ and all

$A\_1,\; A\_2\; \backslash in\; \backslash mathbf\{H\}\_n$ with eigenvalues in $I$ and all $B\_1,\; B\_2\; \backslash in\; \backslash mathbf\{H\}\_n$ with eigenvalues in $J,$ and any $0\; \backslash leq\; \backslash lambda\; \backslash leq\; 1$ the following holds

$$g(\backslash lambda\; A\_1\; +\; (1-\backslash lambda)\; A\_2,\; \backslash lambda\; B\_1\; +\; (1-\backslash lambda)\; B\_2)\; ~\backslash leq~\; \backslash lambda\; g(A\_1,\; B\_1)\; +\; (1\; -\backslash lambda)\; g(A\_2,\; B\_2).$$

A function $g$ is {{visible anchor|jointly concave}} if −$g$ is jointly convex, i.e. the inequality above for $g$ is reversed.

Given a function $f\; :\; \backslash Reals\; \backslash to\; \backslash Reals,$ the associated trace function on $\backslash mathbf\{H\}\_n$ is given by

$$A\; \backslash mapsto\; \backslash operatorname\{Tr\}\; f(A)\; =\; \backslash sum\_j\; f(\backslash lambda\_j),$$

where $A$ has eigenvalues $\backslash lambda$ and $\backslash operatorname\{Tr\}$ stands for a trace of the operator.

Let $f:\; \backslash mathbb\{R\}\; \backslash rarr\; \backslash mathbb\{R\}$ be continuous, and let {{mvar|n}} be any integer. Then, if $t\backslash mapsto\; f(t)$ is monotone increasing, so

is $A\; \backslash mapsto\; \backslash operatorname\{Tr\}\; f(A)$ on H_n.

Likewise, if $t\; \backslash mapsto\; f(t)$ is convex, so is $A\; \backslash mapsto\; \backslash operatorname\{Tr\}\; f(A)$ on H_n, and

it is strictly convex if {{mvar|f}} is strictly convex.

See proof and discussion in, for example.

For $-1\backslash leq\; p\; \backslash leq\; 0$, the function $f(t)\; =\; -t^p$ is operator monotone and operator concave.

For $0\; \backslash leq\; p\; \backslash leq\; 1$, the function $f(t)\; =\; t^p$ is operator monotone and operator concave.

For $1\; \backslash leq\; p\; \backslash leq\; 2$, the function $f(t)\; =\; t^p$ is operator convex. Furthermore,

:$f(t)\; =\; \backslash log(t)$ is operator concave and operator monotone, while

:$f(t)\; =\; t\; \backslash log(t)$ is operator convex.

The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for {{mvar|f}} to be operator monotone.{{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}} An elementary proof of the theorem is discussed in and a more general version of it in.W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer, (1974).

For all Hermitian {{mvar|n}}×{{mvar|n}} matrices {{mvar|A}} and {{mvar|B}} and all differentiable convex functions

$f:\; \backslash mathbb\{R\}\; \backslash rarr\; \backslash mathbb\{R\}$

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,∞) → $\backslash mathbb\{R\}$, the following inequality holds,

{{Equation box 1

|indent =:

|equation = $\backslash operatorname\{Tr\}[f(A)-\; f(B)-\; (A\; -\; B)f\text{'}(B)]\; \backslash geq\; 0~.$

|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.

Let $C=A-B$ so that, for $t\backslash in\; (0,1)$,

:$B\; +\; tC\; =\; (1\; -t)B\; +\; tA$,

varies from $B$ to $A$.

Define

:$F(t)\; =\; \backslash operatorname\{Tr\}[f(B\; +\; tC)]$.

By convexity and monotonicity of trace functions, $F(t)$ is convex, and so for all $t\backslash in\; (0,1)$,

:$F(0)\; +\; t(F(1)-F(0))\backslash geq\; F(t)$,

which is,

:$F(1)\; -\; F(0)\; \backslash geq\; \backslash frac\{F(t)-F(0)\}\{t\}$,

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

Taking the limit $t\backslash to\; 0$ yields,

:$F(1)\; -\; F(0)\; \backslash geq\; F\text{'}(0)$,

which with rearrangement and substitution is Klein's inequality:

:$\backslash mathrm\{tr\}[f(A)-f(B)-(A-B)f\text{'}(B)]\; \backslash geq\; 0$

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

{{main|Golden–Thompson inequality}}

In 1965, S. Golden {{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 }} and C.J. Thompson {{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 }} independently discovered that

For any matrices $A,\; B\backslash in\backslash mathbf\{H\}\_n$,

:$\backslash operatorname\{Tr\}\; e^\{A+B\}\backslash leq\; \backslash operatorname\{Tr\}\; e^A\; e^B.$

This inequality can be generalized for three operators:{{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/ }} for non-negative operators $A,\; B,\; C\backslash in\backslash mathbf\{H\}\_n^+$,

:$\backslash operatorname\{Tr\}\; e^\{\backslash ln\; A\; -\backslash ln\; B+\backslash ln\; C\}\backslash leq\; \backslash int\_0^\backslash infty\; \backslash operatorname\{Tr\}\; A(B+t)^\{-1\}C(B+t)^\{-1\}\backslash ,\backslash operatorname\{d\}t.$

Let $R,\; F\backslash in\; \backslash mathbf\{H\}\_n$ be such that Tr e^R = 1.

Defining {{math|g {{=}} Tr Fe^R}}, we have

:$\backslash operatorname\{Tr\}\; e^F\; e^R\; \backslash geq\; \backslash operatorname\{Tr\}\; e^\{F+R\}\backslash geq\; e^g.$

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

Let $H$ be a self-adjoint operator such that $e^\{-H\}$ is trace class. Then for any $\backslash gamma\backslash geq\; 0$ with $\backslash operatorname\{Tr\}\backslash gamma=1,$

:$\backslash operatorname\{Tr\}\backslash gamma\; H+\backslash operatorname\{Tr\}\backslash gamma\backslash ln\backslash gamma\backslash geq\; -\backslash ln\; \backslash operatorname\{Tr\}\; e^\{-H\},$

with equality if and only if $\backslash gamma=\backslash exp(-H)/\backslash operatorname\{Tr\}\; \backslash exp(-H).$

The following theorem was proved by E. H. Lieb in. It proves and generalizes a conjecture of E. P. Wigner, M. M. Yanase, and Freeman Dyson.{{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 }} Six years later other proofs were given by T. Ando {{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 }} and B. Simon, and several more have been given since then.

For all $m\backslash times\; n$ matrices $K$, and all $q$ and $r$ such that $0\; \backslash leq\; q\backslash leq\; 1$ and $0\backslash leq\; r\; \backslash leq\; 1$, with $q\; +\; r\; \backslash leq\; 1$ the real valued map on $\backslash mathbf\{H\}^+\_m\; \backslash times\; \backslash mathbf\{H\}^+\_n$ given by

:$$

F(A,B,K) = \operatorname{Tr}(K^*A^qKB^r)

• is jointly concave in $(A,B)$
• is convex in $K$.

Here $K^*$ stands for the adjoint operator of $K.$

For a fixed Hermitian matrix $L\backslash in\backslash mathbf\{H\}\_n$, the function

:$f(A)=\backslash operatorname\{Tr\}\; \backslash exp\backslash \{L+\backslash ln\; A\backslash \}$

is concave on $\backslash mathbf\{H\}\_n^\{++\}$.

The theorem and proof are due to E. H. Lieb, Thm 6, where he obtains this theorem as a corollary of Lieb's concavity Theorem.

The most direct proof is due to H. Epstein;{{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 }} see M.B. Ruskai papers,{{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 }}{{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 }} for a review of this argument.

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

For all $m\; \backslash times\; n$ matrices $K$, and all $1\; \backslash leq\; q\; \backslash leq\; 2$ and $0\; \backslash leq\; r\; \backslash leq\; 1$ with $q-r\; \backslash geq\; 1$, the real valued map on $\backslash mathbf\{H\}^\{++\}\_m\; \backslash times\; \backslash mathbf\{H\}^\{++\}\_n$ given by

:$(A,B)\; \backslash mapsto\; \backslash operatorname\{Tr\}(K^*A^qKB^\{-r\})$

is convex.

For two operators $A,\; B\backslash in\backslash mathbf\{H\}^\{++\}\_n$ define the following map

:$R(A\backslash parallel\; B):=\; \backslash operatorname\{Tr\}(A\backslash log\; A)\; -\; \backslash operatorname\{Tr\}(A\backslash log\; B).$

For density matrices $\backslash rho$ and $\backslash sigma$, the map $R(\backslash rho\backslash parallel\backslash sigma)=S(\backslash rho\backslash parallel\backslash sigma)$ is the Umegaki's quantum relative entropy.

Note that the non-negativity of $R(A\backslash parallel\; B)$ follows from Klein's inequality with $f(t)=t\backslash log\; t$.

The map $R(A\backslash parallel\; B):\; \backslash mathbf\{H\}^\{++\}\_n\; \backslash times\; \backslash mathbf\{H\}^\{++\}\_n\; \backslash rightarrow\; \backslash mathbf\{R\}$ is jointly convex.

For all , $(A,B)\; \backslash mapsto\; \backslash operatorname\{Tr\}(B^\{1-p\}A^p)$ is jointly concave, by Lieb's concavity theorem, and thus

:$(A,B)\backslash mapsto\; \backslash frac\{1\}\{p-1\}(\backslash operatorname\{Tr\}(B^\{1-p\}A^p)-\backslash operatorname\{Tr\}A)$

is convex. But

:$\backslash lim\_\{p\backslash rightarrow\; 1\}\backslash frac\{1\}\{p-1\}(\backslash operatorname\{Tr\}(B^\{1-p\}A^p)-\backslash operatorname\{Tr\}A)=R(A\backslash parallel\; B),$

and convexity is preserved in the limit.

The proof is due to G. Lindblad.{{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 }}

The operator version of Jensen's inequality is due to C. Davis.C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).

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

:$f\backslash left(\backslash sum\_kA\_k^*X\_kA\_k\backslash right)\backslash leq\backslash sum\_k\; A\_k^*f(X\_k)A\_k,$

for operators $\backslash \{A\_k\backslash \}\_k$ with $\backslash sum\_k\; A^*\_kA\_k=1$ and for self-adjoint operators $\backslash \{X\_k\backslash \}\_k$ with spectrum on $I$.

See,{{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 }} for the proof of the following two theorems.

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

:$\backslash operatorname\{Tr\}\backslash Bigl(f\backslash Bigl(\backslash sum\_\{k=1\}^nA\_k^*X\_kA\_k\backslash Bigr)\backslash Bigr)\backslash leq\; \backslash operatorname\{Tr\}\backslash Bigl(\backslash sum\_\{k=1\}^n\; A\_k^*f(X\_k)A\_k\backslash Bigr),$

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

:$\backslash sum\_\{k=1\}^nA\_k^*A\_k=1.$

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

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

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

:$f\backslash Bigl(\backslash sum\_\{k=1\}^nA\_k^*X\_kA\_k\backslash Bigr)\backslash leq\backslash sum\_\{k=1\}^n\; A\_k^*f(X\_k)A\_k,$

for all $(X\_1,\; \backslash ldots\; ,\; X\_n)$ bounded, self-adjoint operators on an arbitrary Hilbert space $\backslash mathcal\{H\}$ with

spectra contained in $I$ and all $(A\_1,\; \backslash ldots\; ,\; A\_n)$ on $\backslash mathcal\{H\}$ with $\backslash sum\_\{k=1\}^n\; A^*\_kA\_k=1.$

• $f(V^*XV)\; \backslash leq\; V^*f(X)V$ for each isometry $V$ on an infinite-dimensional Hilbert space $\backslash mathcal\{H\}$ and

every self-adjoint operator $X$ with spectrum in $I$.

• $Pf(PXP\; +\; \backslash lambda(1\; -P))P\; \backslash leq\; Pf(X)P$ for each projection $P$ on an infinite-dimensional Hilbert space $\backslash mathcal\{H\}$, every self-adjoint operator $X$ with spectrum in $I$ and every $\backslash lambda$ in $I$.

{{distinguish|text=the Lieb–Thirring inequality}}

E. H. Lieb and W. E. Thirring proved the following inequality in 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). 1976: For any $A\; \backslash geq\; 0,$ $B\; \backslash geq\; 0$ and $r\; \backslash geq\; 1,$

$$\backslash operatorname\{Tr\}\; ((BAB)^r)\; ~\backslash leq~\; \backslash operatorname\{Tr\}\; (B^r\; A^r\; B^r).$$

In 1990 {{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 }} H. Araki generalized the above inequality to the following one: For any $A\; \backslash geq\; 0,$ $B\; \backslash geq\; 0$ and $q\; \backslash geq\; 0,$

$$\backslash operatorname\{Tr\}((BAB)^\{rq\})\; ~\backslash leq~\; \backslash operatorname\{Tr\}((B^r\; A^r\; B^r)^q),$$

for $r\; \backslash geq\; 1,$ and

$$\backslash operatorname\{Tr\}((B^r\; A^r\; B^r)^q)\; ~\backslash leq~\; \backslash operatorname\{Tr\}((BAB)^\{rq\}),$$

for $0\; \backslash leq\; r\; \backslash leq\; 1.$

There are several other inequalities close to the Lieb–Thirring inequality, such as the following: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). for any $A\; \backslash geq\; 0,$ $B\; \backslash geq\; 0$ and $\backslash alpha\; \backslash in\; [0,\; 1],$

$$\backslash operatorname\{Tr\}\; (B\; A^\backslash alpha\; B\; B\; A^\{1-\backslash alpha\}\; B)\; ~\backslash leq~\; \backslash operatorname\{Tr\}\; (B^2\; A\; B^2),$$

and even more generally:L. Lafleche, C. Saffirio, Strong Semiclassical Limit from Hartree and

Hartree-Fock to Vlasov-Poisson Equation, arXiv:2003.02926 [math-ph]. for any $A\; \backslash geq\; 0,$ $B\; \backslash geq\; 0,$ $r\; \backslash geq\; 1/2$ and $c\; \backslash geq\; 0,$

$$\backslash operatorname\{Tr\}((B\; A\; B^\{2c\}\; A\; B)^r)\; ~\backslash leq~\; \backslash operatorname\{Tr\}((B^\{c+1\}\; A^2\; B^\{c+1\})^r).$$

The above inequality generalizes the previous one, as can be seen by exchanging $A$ by $B^2$ and $B$ by $A^\{(1-\backslash alpha)/2\}$ with $\backslash alpha\; =\; 2\; c\; /\; (2\; c\; +\; 2)$ and using the cyclicity of the trace, leading to

$$\backslash operatorname\{Tr\}((B\; A^\backslash alpha\; B\; B\; A^\{1-\backslash alpha\}\; B)^r)\; ~\backslash leq~\; \backslash operatorname\{Tr\}((B^2\; A\; B^2)^r).$$

Additionally, building upon the Lieb-Thirring inequality the following inequality was derived: V. Bosboom, M. Schlottbom, F. L. Schwenninger, On the unique solvability of radiative transfer equations with polarization, in Journal of Differential Equations, (2024). For any $A,B\backslash in\; \backslash mathbf\{H\}\_n,\; T\backslash in\; \backslash mathbb\{C\}^\{n\backslash times\; n\}$ and all $1\backslash leq\; p,q\backslash leq\; \backslash infty$ with $1/p+1/q\; =\; 1$, it holds that

$$|\backslash operatorname\{Tr\}(TAT^*B)|\; ~\backslash leq~\; \backslash operatorname\{Tr\}(T^*T|A|^p)^\backslash frac\{1\}\{p\}\backslash operatorname\{Tr\}(TT^*|B|^q)^\backslash frac\{1\}\{q\}.$$

E. Effros in {{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}} proved the following theorem.

If $f(x)$ is an operator convex function, and $L$ and $R$ are commuting bounded linear operators, i.e. the commutator $[L,R]=LR-RL=0$, the perspective

:$g(L,\; R):=f(LR^\{-1\})R$

is jointly convex, i.e. if $L=\backslash lambda\; L\_1+(1-\backslash lambda)L\_2$ and $R=\backslash lambda\; R\_1+(1-\backslash lambda)R\_2$ with $[L\_i,\; R\_i]=0$ (i=1,2), $0\backslash leq\backslash lambda\backslash leq\; 1$,

:$g(L,R)\backslash leq\; \backslash lambda\; g(L\_1,R\_1)+(1-\backslash lambda)g(L\_2,R\_2).$

Ebadian et al. later extended the inequality to the case where $L$ and $R$ do not commute .{{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 }}

{{visible anchor|Von Neumann's trace inequality}}, named after its originator John von Neumann, states that for any $n\; \backslash times\; n$ complex matrices $A$ and $B$ with singular values $\backslash alpha\_1\; \backslash geq\; \backslash alpha\_2\; \backslash geq\; \backslash cdots\; \backslash geq\; \backslash alpha\_n$ and $\backslash beta\_1\; \backslash geq\; \backslash beta\_2\; \backslash geq\; \backslash cdots\; \backslash geq\; \backslash beta\_n$ respectively,{{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}}

$$|\backslash operatorname\{Tr\}(A\; B)|\; ~\backslash leq~\; \backslash sum\_\{i=1\}^n\; \backslash alpha\_i\; \backslash beta\_i\backslash ,,$$

with equality if and only if $A$ and $B^\{\backslash dagger\}$ share singular vectors.{{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}}

A simple corollary to this is the following result:{{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}} For Hermitian $n\; \backslash times\; n$ positive semi-definite complex matrices $A$ and $B$ where now the eigenvalues are sorted decreasingly ($a\_1\; \backslash geq\; a\_2\; \backslash geq\; \backslash cdots\; \backslash geq\; a\_n$ and $b\_1\; \backslash geq\; b\_2\; \backslash geq\; \backslash cdots\; \backslash geq\; b\_n,$ respectively),

$$\backslash sum\_\{i=1\}^n\; a\_i\; b\_\{n-i+1\}\; ~\backslash leq~\; \backslash operatorname\{Tr\}(A\; B)\; ~\backslash leq~\; \backslash sum\_\{i=1\}^n\; a\_i\; b\_i\backslash ,.$$

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

{{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)