数理解析研究所講究録
第 1839 巻 2013 年 17-24

17

ON GENERALIZED POWERS-STORMER’SINEQUALITY
HIROYUKI OSAKAA
ABSTRACT.

generalization of Powers-Strmer’s inequality for
operator monotone functions on $[0, +\infty)$ and for positive linear
functional on general-algebras will be proved. It also will beshown that the generalized Powers-Strmer inequality characterizes the tracial functionals on -algebras.
$A$$C^{*}$$C^{*}$

1. INTRODUCTION

Powers-St’s inequality (see, for example, [16, Lemma 2.4], [4,Theorem 11.19] asserts that for $s\in[0,1]$ the following inequality$\emptyset rmer$

$)$

(1)

$2 Tr(A^{s}B^{1-s})\geq Tr(A+B-|A-B|)$holds for any pair of positive matrices $A,$ $B$ . This is a key inequality
to prove the upper bound of Chernoff bound, in quantum hypothesis
testing theory [1]. This inequality was first proven in [1], using an
integral representation of the function . After that, N. Ozawa gavea much simpler proof for the same inequality, using fact that for $s\in$$[0,1]$ function $f(t)=t^{S}(t\in[0, +\infty))$ is an operator monotone ([11,Proposition 1.1] . Recently, Y. Ogata in [13] extended this inequality
to standard von Neumann algebras. The motivation of this paper isthat if the function $f(t)=t^{s}$ is replaced by another operator monotonefunction (this class is intensively studied, see [8][14]), then Tr $(A+B-$$|A-B|)$ may get smaller upper bound than what is used in quantum
hypothesis testing. Based on N. Ozawa’s proof we formulate PowersSt’s inequality for an arbitrary operator monotone function on$[0, +\infty)$ in the context of general-algebras.
$t^{S}$$)$$\emptyset rmer$$C^{*}$

2. DOUBLE PILING STRUCTURES FOR MATRIX FUNCTIONS
Throughout this note,
stands for the algebra of all
matrices,
denote the set of positive semi-definite matrices. We call a function$M_{n}$$n\cross n$$M_{n}^{+}$

Research partially supported by the JSPS grant for Scientic Research No.
20540220.
Date: 13, Feb., 2013.
2000 Mathematics Subject Classification. $46L30,15A45.$$1A$$C^{*}$

Key words and phrases. Powers-Strmer’s inequality, trace, positive functional,-algebras.

18
H. OSAKA

matrix convex of order or -convex in short (resp. matrix concaveof order or -concave) whenever the inequality

$f$

$n$

$n$

$n$

$n$

$f(\lambda A+(1-\lambda)B)\leq\lambda f(A)+(1-\lambda)f(B), \lambda\in[0,1]$

holds forresp. $f(\lambda A+(1-\lambda)B)\geq\lambda f(A)+(1-\lambda)f(B),$
$A,$
$B\in
M_{n}$
such that all eigenvalues
every pair of selfadjoint matrices
of and are contained in I. Matrix monotone functions on are
similarly defined as the inequality$\lambda\in[0,1])$

$($

$I$

$B$

$A$

$A\leq B\Rightarrow f(A)\leq f(B)$for any pair of selfadjoint matrices $A,$ $B\in M_{n}$ such that $A\leq B$ andare contained in . We call a function
all eigenvalues of andis koperator convex (resp. opemtor concave) if for eachisconvex (resp. -concave) and opemtor monotone if for each-monotone.
In [15] Tomiyama and the author discussed about the following 3
assertions at each level among them in order to see clear insight ofthe double piling structure of matrix monotone functions and of matrix
convex functions:
$B$

$A$

$I$

$f$

$k\in \mathbb{N},$

$k$

$f$

$k\in \mathbb{N}f$

$k$

$n$

and consider the following
and :Theorem 2.1. Let
assertions.
,(i) $f(O)\leq 0$ and is -convex in
and a contmction(ii) For each matrix with its spectrum inin the matrix algebm$f$$n\in \mathbb{N}$$f$$[0, \alpha)arrow \mathbb{R}$$[0, \alpha)$$n$$[0, \alpha)$$a$$M_{n},$$c$$f(c^{\star}ac)\leq c^{\star}f(a)c,$(iii) The function $f\Delta^{t}t(=g(t))$ is -monotone in
Then we have$n$$(0, \alpha)$.

$(i)_{n+1}\prec(ii)_{n}\sim(iii)_{n}\prec(i)_{[\frac{n}{2}]},$

where the denotation $(A)_{m}\prec(B)_{n}$ means that “if $(A)$ holds for the”, then $(B)$ holds for the matrix algebm
matrix algebm$M_{n}$$M_{m}$

The following result is proved in [5].
.
Lemma 2.1. Let $f$ be a strictly positive, continuous function on$2n$any-monotone,
positivethen forsemi-definite
If the function is
we haveand a contraction $C$ in$[0, \infty)$

$A$

$f$

$M_{n}$$C^{*}f(A)C\leq f(C^{*}AC)$.

The following result is essentially proved in [7, Theorem 2.4], but for
the reader’s convenience we will include a proof.

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GENERALIZED $POWERS-ST\emptyset RMER^{)}S$ INEQUALITY

Proposition 2.1. Let be a strictly positive, continuous function on. If is $2n$ -monotone, the functionis -monotone on$f$$[0, \infty)$$[0, \infty)$$f$$g(t)= \frac{t}{f(t)}$$n$.

Proof. Let

be positive matrices insuch that $0<A\leq B.$
Let. Then. Since is $2n$ -monotone, $-f$satisfies the Jensen type inequality from Lemma 2.1, that is,$A,$ $B$$M_{n}$$C=B^{-\frac{1}{2}}A^{\frac{1}{2}}$$\Vert C\Vert\leq 1$$f$$-f(A)=-f(C^{*}BC)\leq-C^{*}f(B)C$$-f(A)\leq-A^{\frac{1}{2}}B^{-\frac{1}{2}}f(B)B^{-\frac{1}{2}}A^{\frac{1}{2}}$$-A^{-\frac{1}{2}}f(A)A^{-\frac{1}{2}}\leq-B^{-\frac{1}{2}}f(B)B^{-\frac{1}{2}}$$-A^{-1}f(A)\leq-B^{-1}f(B)$Therefore, since-monotone.

-

$1/t$

is operator monotone, $-1/(-f(t)/t)=t/f(t)$ is$n$$\square$Remark 1. The condition of $2n$-monotonicity of is needed to guarantee the -monotonicity of . Indeed, it is well-known that is monotone, but not 2-monotone. In this case the functionisobviously not 1-monotone.
$f$

$n$

$t^{3}$$g$$g(t)= \frac{t}{t^{3}}=\eta t1$

Corollary 2.1. $Le$ $tf$ be a $2n$ -monotone, continuous function onsuch that $f((O, \infty)$, and let be a Borel function on$)$

defined by $g(t)=\{$$\subset(0, \infty)$$g$$(t\in(0, \infty))$$\frac{t}{f(t),0}$$(t=0)$$[0, \infty)$$[0, \infty)$

Then for any pair of positive

matrices $A,$ $B\in M_{n}$ with $A\leq B,$ $g(A)\leq g(B)$ .

Similarly, we can get the concave version of the above observation[10].

Theorem 2.2. Forand :we consider the following
assertions:(iv) $f(O)\geq 0$ and is -concave in,(v) For each matrix with spectrum inand a contmctionin the matrix algebm$n\in \mathbb{N}$

$f$

$f$

$[0, \alpha)arrow \mathbb{R}$$[0, \alpha)$$n$$[0, \alpha)$$a$$M_{n},$$f(c^{\star}ac)\geq c^{\star}f(a)c,$(vi) The function$\frac{t}{f(t)}$

is

$n$

-monotone in$(0, \alpha)$.

We can show that$(iv)_{n+1}\prec(v)_{n}\sim(vi)_{n}\prec(iv)_{[\frac{n}{2}]}.$

$c$

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H. OSAKA

3. GENERALIZED POWERS-STRMER’S INEQUALITYinIn this section we investigate the generalized Powers-St$2n$-monotonicity
equality from the point of matrix functions. Note that the
implies the -concavity of by [2, Theoof a function on
rem V.2.5].
$\emptyset rmer$$[0, \infty)$

$f$

$f$

$n$

and be $a(n+1)-$
Theorem 3.1. Let Tr be the canonical tmce on$2n$such that $f((O, \infty))\subset$
concave (or -monotone) function on. Then for any pair of positive matrices $A,$ $B\in M_{n}$$M_{n}$$f$$[0, \infty)$$(0, \infty)$(2)$R(A)+$ Tr $(B)-$ Tr $(|A-B|)\leq 2$ Tr $(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})$ ,where $g(t)=\{$$(t\in(0, \infty))$$\frac{t}{f(t),0}$$(t=0)$

Proof. Note that we know that a function$g$is -monotone from Corol$n$

lary 2.1 and Theorem 2.2.
Let $A,$ be any positive matrices in
For operator $(A-B)$ let us denote by $P=(A-B)^{+}$ and $Q=$$(A-B)^{-}$ its positive and negative part, respectively. Then we have$B$

$M_{n}.$$A-B=P-Q$ and $|A-B|=P+Q,$(3)

from that it follows that

$A+Q=B+P.$(4)

On account of (4) the inequality (2) is equivalent to the following
Tr $(A)-$ Tr$n$$(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})\leq$

Tr $(P)$ .

Since $B+P\geq B\geq 0$ and $B+P=A+Q\geq A\geq 0$ and-monotone, we have $g(A)\leq g(B+P)$ and
Tr $(A)-$ Tr$(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})$

Tr

$=$

Tr

$\leq$Tr $(f(A)^{\frac{1}{2}}g(B+P)f(A)^{\frac{1}{2}})-$ Tr$(f(A)^{\frac{1}{2}}g(A)f(A)^{\frac{1}{2}})-$$(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})$$(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})$$=Tr(f(A)^{\frac{1}{2}}(g(B+P)-g(B))f(A)^{\frac{1}{2}})$$\leq Tr(f(B+P)^{\frac{1}{2}}(g(B+P)-g(B))f(B+P)^{\frac{1}{2}})$$=Tr(f(B+P)^{\frac{1}{2}}g(B+P)f(B+P)^{\frac{1}{2}})$-Tr $(f(B+P)^{\frac{1}{2}}g(B)f(B+P)^{\frac{1}{2}})$$\leq$$=$Tr $(B+P)-$ Tr
Tr $(B+P)-$ Tr $(B)$$=Tr(P)$ .

$(f(B)^{\frac{1}{2}}g(B)f(B)^{\frac{1}{2}})$

$g$

is

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GENERALIZED $POWERS-ST\emptyset RMER’ S$ INEQUALITY

Hence, we have the conclusion.
$\square$Corollary 3.1. Let
be an opemtor monotone function onsuch that $f((O, \infty))\subset(0, \infty)$ . Then for any pair of positive matrices$f$$[0, \infty)$$A,$ $B\in M_{n}$(5)

Tr $(A)+Tr(B)-Tr(|A-B|)\leq 2$ Tr

where $g(t)=\{$$(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})$,$(t\in(0, \infty))$$\frac{t}{f(t),0}$$(t=0)$

Note that the operator monotonicity is equivalent to the operator
concavity in the case that $f([O, \infty))\subset[0, \infty)$ [ , Theorem V. 2.5].
$2$Corollary 3.2. [1, Theorem 1] Let$A$

for all $s\in[0,1]$

and

$B$

be positive matrices, then

TN $(A+B-|A-B|)\leq$ Tr $(A^{s}B^{1-s})$ .

Remark 2. As pointed in Proposition 2.1, 2-monotonicity of

$f$

is

needed to guarantee the inequality (2). Indeed, let $f(t)=t^{3}$ and$b\in(0, \infty)$ , the inequality (2) $wo$ uld imply$n=1$ . Then, for any$a,$$a\leq f(a)^{\frac{1}{2}}g(b)f(a)^{\frac{1}{2}},$that is,$\frac{a}{f(a)}\leq\frac{b}{f(b)}.$Sincesible.

$\frac{t}{f(t)}$is, however, not 1-monotone, the latter inequality is impos-

Remark 3. For matrices $A,$(6)

$B\in M_{n}^{+}$

let us denote$Q(A, B)= \min_{s\in[0,1]}Tr(A^{(1-s)/2}B^{s}A^{(1-s)/2})$

and
(7)

$Q_{\mathcal{F}_{2n}}(A, B)= \inf_{f\in \mathcal{F}_{2n}}$Tr$(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})$

,

where
is the set of all $2n$-monotone functions on $[0, +\infty)$ satisfy
condition of the Theorem 3.1 and $g(t)=tf(t)^{-1}(t\in[0, +\infty))$ .
Since the class of 2 -monotone functions is large enough [14], weknow that $Q_{\mathcal{F}_{2n}}(A, B)\leq Q(A, B)$ . Hence, we hope on finding another$2n$ -monotone function
on $[0, +\infty)$ such that$\mathcal{F}_{2n}$

$n$

$f$

(8)

$Tr(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})<Q(A, B)$.

22
H. OSAKA

If we can find such a function, then we may get smaller upper boundthan what is used in quantum hypothesis testing [1]. For example,
considering the trace distance $T(A, B)= \frac{Tr(|A-B|)}{2}$ , we might havethe following better estimate$\frac{1}{2}$

Tr $(A+B)-Q_{\mathcal{F}_{2n}}(A, B)\leq T(A, B)\leq\sqrt{\{\frac{1}{2}Tr(A+B)\}^{2}-Q_{\mathcal{F}_{2n}}(A,B)^{2}}.$(See the estimate (6) in [1].)4. CHARACTERIZATIONS OF THE TRACE PROPERTY
inequality in the preIn this section the generalized Powers-St
vious section implies the trace property for a positive linear functionalon operator algebras.
$\emptyset rmer$

beand
be a positive linear functional on
Lemma 4.1. Let$f((O, \infty))\subset$$f(O)=0$
and
that
such
on
a continuous function. If the following inequality$f$$M_{n}$$\varphi$$[0, \infty)$$(0, \infty)$(9)$\varphi(A+B)-\varphi(|A-B|)\leq 2\varphi(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})$holds true for all $A,$

$B\in M_{n}^{+}$of the canonical tmce] on$\}$, then$M_{n}$

should be a positive scalar multiple$\varphi$, where$g(t)=\{\begin{array}{ll}\frac{t}{f(t)} (t\in(0, \infty))0 (t=0)\end{array}$

Theorem 4.1. Let be a positive normal linear functional on a vonsuch
and be a continuous function on
Neumann algebm
that $f(O)=0$ and $f((O, \infty))\subset(0, \infty)$ . If the following inequality$\varphi$(10)$[0, \infty)$$f$$\mathcal{M}$$\varphi(A)+\varphi(B)-\varphi(|A-B|)\leq 2\varphi(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})$holds true for any pair $A,$

$B\in \mathcal{M}^{+}$, then$\varphi$

is a tmce, where $g(t)=$$\{\begin{array}{ll}\frac{t}{f(t)} (t\in(0, \infty))0 (t=0)\end{array}$-algebm
Corollary 4.1. Let be a positive linear functional on a$f(O)=0$
andsuch thatand be a continuous function on$f((O, \infty))\subset(0, \infty)$ .
inequality
followingthe
If$C^{*}$$\varphi$$[0, \infty)$$f$$\mathcal{A}$(11)$\varphi(A)+\varphi(B)-\varphi(|A-B|)\leq 2\varphi(f(A)^{\frac{1}{2}}g(B)f(A)^{\frac{1}{2}})$holds true for any pair $A,$

$B\in \mathcal{A}^{+}$, then$\varphi$

is a tmcial functional, where$g(t)=\{\begin{array}{ll}\frac{t}{f(t)} (t\in(0, \infty))0 (t=0)\end{array}$Remark 4. Let be a von Neumann algebra and be a positive norof all orthogonal projections
mal linear functional on . The set
from is enough as a testing space for some inequality to characterize$\mathcal{A}$$\varphi$$\mathcal{A}$$\mathcal{A}$$P(\mathcal{A})$

23
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GENERALIZED$POWERS-ST\emptyset RMER’ S$ INEQUALITY

the trace property of (see [3]). But, in the case of the inequality (10)the setis not enough as a testing set.
Indeed, let $P,$ be arbitrary orthogonal projections from a von Neumann algebra . Since $Q\geq P\wedge Q$ it follows that $PQP\geq P(P\wedge Q)P=$$P\wedge Q$ . So $PQP\geq P\wedge Q$ holds for any pairof projections.
that
it follows$\varphi$$P(\mathcal{A})$$Q$$\mathcal{M}$$\mathbb{R}om$$\varphi(P+Q-|P-Q|)=2\varphi(P\wedge Q)\leq 2\varphi(PQP)=2\varphi(f(P)^{\frac{1}{2}}g(Q)f(P)^{\frac{1}{2}})$

whenever$\varphi$

is an arbitrary positive linear functional on$\mathcal{M}.$

REFERENCES[1] K. M. R.Audenaert, J. Calsamiglia, LI. Masanes, R. Munoz-Tapia, A. Acin,
E. Bagan, F. Verstraete, The Quantum Chernoff Bound, Phys. Rev. Lett. 98(2007) 16050.
[2] R. Bhatia, Matrix analysis, Graduate texts in mathematics, Springer New York,

1997.

[3] A. M. Bikchentaev, Commutation of projections and characterization of traces
on von Neumann algebras, Siberian Math. J., 51 (2010) 971-977. [Ranslation
from
Matematicheckii Zhurnal, 51 (2010) 1228-1236][4] D. Petz, Quantum information theory and quantum statistics, Theoretical andMathematical Physics. Springer-Verlag, Berlin, 2008.
[5] F. Hansen, An operator inequality, Math. Ann. 246 (1979/80), no. 3, 249-250.
[6] F. Hansen, Some operator monotone functions, Linear Algebra and its Applications, 430 (2009) 795-99.
[7] F. Hansen, G. K. Pedersen, Jensen’s inequality for operator and Lowner’s theorem, Math. Ann., 258 (1982) 229-241.
[8] F. Hansen, G. Ji, J. Tomiyama, Gaps between classes of matrix monotone functions, Bull. London Math. Soc. 36 (2004) 53-58.
[9] D. T. Hoa, H. Osaka, H. M. Toan, On generalized Powers-Strmer’s inequality. Linear Algebra and its Applications 438 (2013), 242-249 (in press).
arxiv: 1204: 6665.
[10] D. T. Hoa, H. Osaka, J. Tomiyama, Characterization of the monotonicity by
the inequality, arXiv: 1207.5201.
[11] V. Jaksic, Y. Ogata, C. -A. Pillet, R. Seiringer, Quantum hypothesis testingand non-equilibrium statistical mechanics, arXiv:1109.3804vl [math-ph].
[12] K. Loewner, \"Uber monotone Matrixfunktionen, Math. Z. 38 (1934) 177-216.
[13] Y. Ogata, A Generalization of Powers-Strmer Inequality, Letters in Mathematical Physics, 97:3 (2011) 339-346.
[14] H. Osaka, S. Silvestrov, J. Tomiyama, Monotone operator functions, gaps andpower moment problem, Math. Scand. 100:1 (2007) 161-183.
[15] H. Osaka, J. Tomiyama, Double piling structure of matrix monotone functions
and of matrix convex functions. Linear Algebra and its Applications 431(2009),$Sibirski\check{i}$1825-1832.

[16] R. T. Powers, E. Strmer, Free States of the Canonical Anticommutatibn Relations, Commun. math. Phys., 16 (1970) 1-33.
[17] O. E. Tikhonov, Subadditivity ineqalities in von Neumann algebras and characterization of tracial functional, Positivity. 9 (2005) 259-264.

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H. OSAKADEPARTMENT OF MATHEMATICAL SCIENCES, RITSUMEIKAN UNIVERSITY, KUSATSU,SHIGA 525-8577, JAPAN,-mail address: osaka@se.ritsumei.ac.jp$E$