Article A Generalization Of Powers-størmer

Lett Math Phys (2011) 97:339–346 DOI 10.1007/s11005-011-0504-y

A Generalization of Powers–Størmer Inequality YOSHIKO OGATA Graduate School of Mathematics, University of Tokyo, Tokyo, Japan. e-mail: [email protected] Received: 15 December 2010 / Accepted: 5 June 2011 Published online: 29 June 2011 – © Springer 2011 s

2 ξϕ 2 ≥ ϕ(1) + η(1) − Abstract. In this note, we prove the following inequality: 2ηϕ |ϕ − η|(1), where ϕ and η are positive normal linear functionals over a von Neumann algebra. This is a generalization of the famous Powers–Størmer inequality (Powers and Størmer proved the inequality for L(H) in Commun Math Phys 16:1–33, 1970; Takesaki in Theory of Operator Algebras II, 2001). For matrices, this inequality was proven by Audenaert et al. (Phys Rev Lett 98:160501, 2007). We extend their result to general von Neumann algebras.

Mathematics Subject Classification (2010). 46L10. Keywords. Powers–Størmer inequality, Chernoff bound.

1. Introduction Let A, B be positive matrices and 0 ≤ s ≤ 1. Then an inequality 2T r As B 1−s ≥ T r (A + B − |A − B|)

(1)

holds. This is a key inequality to prove the upper bound of Chernoff bound, in quantum hypothesis testing theory. This inequality was first proven in [3], using an integral representation of the function t s . Recently, N. Ozawa gave a much simpler proof for the same inequality. In this note, based on his proof, we extend the inequality to general von Neumann algebras. More precisely, we prove the following: Let {M, H, J, P} be a standard form associated with a von Neumann algebra M, i.e., H is a Hilbert space where M acts on, J is the modular conjugation, and P is the natural positive cone (see [6]). Let M∗+ be the set of all positive normal linear functionals over M. For each ϕ ∈ M∗+ , ξϕ is the unique element in the natural positive cone P which satisfies ϕ(x) = (xξϕ , ξϕ ) for all x ∈ M. We denote the relative modular operator associated with ϕ, ψ ∈ M∗+ by ϕψ (see Appendix). The main result in this note is the following: PROPOSITION 1.1. Let ϕ, η be positive normal linear functionals on a von Neumann algebra M. Then, for any 0 ≤ s ≤ 1, s

2 2ηϕ ξϕ 2 ≥ ϕ(1) + η(1) − |ϕ − η|(1).

(2)

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YOSHIKO OGATA

The equality holds iff η = (η − ϕ)+ + ψ and ϕ = (η − ϕ)− + ψ for some ψ ∈ M∗+ whose support is orthogonal to the support of |η − ϕ|. If s = 21 , this is the Powers–Størmer inequality [5]. Applications of this inequality for hypothesis testing problem can be found in [4].

2. Proof of Proposition 1.1 We first prove the following lemma which we need in the proof of Proposition 1.1: LEMMA 2.1. Let ϕ1 , ϕ2 , ψ, η be faithful normal positive linear functionals over a von Neumann algebra M. Assume that ϕ1 ≤ ϕ2 and η ≤ ψ. Then for all 0 < s < 1, s

s

s

s

ϕ22 η ξη 2 − ϕ21 η ξη 2 ≤ ϕ22 ψ ξψ 2 − ϕ21 ψ ξψ 2 . Proof. First we consider the case ϕ2 ≤ ψ. In this case, by Lemma A.1, (Dϕ1 : Dψ)t , (Dϕ2 : Dψ)t , (Dη : Dψ)t have continuations (Dϕ1 : Dψ)z , (Dϕ2 : Dψ)z , (Dη : Dψ)z ∈ M, analytic on I− 1 := {z ∈ C : − 21 < z < 0} and bounded continuous on 2

I− 1 , with norm less than or equal to 1. 2

We define a positive operator T := (Dϕ2 : Dψ)∗−i s (Dϕ2 : Dψ)−i 2s − (Dϕ1 : Dψ)∗−i s (Dϕ1 : Dψ)−i 2s ∈ M. 2

(3)

2

−s

To see that T is positive, recall from Lemma A.1 that for any ξ ∈ D(ψψ2 ), we have −s

s

ψψ2 ξ ∈ D(ϕ2k ψ ), and s

−s

ϕ2k ψ ψψ2 ξ = (Dϕk : Dψ)−i 2s ξ,

k = 1, 2.

From this, we obtain s

−s

s

−s

(T ξ, ξ ) = (Dϕ2 : Dψ)−i 2s ξ 2 −(Dϕ1 : Dψ)−i 2s ξ 2 =ϕ22 ψ ψψ2 ξ 2 − ϕ21 ψ ψψ2 ξ 2 . (4) −s ψψ2 ξ

s 2

As is in D(ϕ2 ψ ), the last term is positive from Lemma A.2. This proves T ≥ 0. Next, we define x  := J ((Dη : Dψ)−i 1−s )J ∈ M . From x   ≤ 1 and 0 ≤ T ∈ M, we 2 have ∗

∗

1

1

(x  T x  ξψ , ξψ ) = (T 2 x  x  T 2 ξψ , ξψ ) ≤ (T ξψ , ξψ ).

(5)

− 1−s

As ξψ ∈ D(ψψ2 ), from Lemma A.1, we have 1

− 1 + 2s

−s

2 2 x  ξψ = J (Dη : Dψ)−i 1−s ξψ = J ηψ ψψ2 1 s 2−2

2 s 2

1 2

s 2

ξψ −s

= J ηψ ξψ = ψη J ηψ ξψ = ψη ξη ∈ D(ψη2 ).

(6)

341

A GENERALIZATION OF POWERS–STØRMER INEQUALITY −s

s

By this and Lemma A.1, we have ψη2 x  ξψ ∈ D(ϕ2k η ) and s

−s

s

(Dϕk : Dψ)−i 2s x  ξψ = ϕ2k η ψη2 x  ξψ = ϕ2k η ξη

(7)

for k = 1, 2. Hence, we obtain s

∗

s

(x  T x  ξψ , ξψ ) = ϕ22 η ξη 2 − ϕ21 η ξη 2 .

(8) − 2s

On the other hand, substituting ξ = ξψ ∈ D(ψψ ) to (4), we have s

s

(T ξψ , ξψ ) = ϕ22 ψ ξψ 2 − ϕ21 ψ ξψ 2 .

(9)

From (5), (8), (9), we obtain the result for the ϕ2 ≤ ψ case. To extend the result to a general case, we use Lemma A.3. For any ε > 0, we have εϕ1 ≤ εϕ2 ≤ ψ + εϕ2 ,

η ≤ ψ + εϕ2 .

(10)

Therefore, for any 0 < s < 1, we have s

s

s

s

2 2 2 2 2 2 2 2 εϕ 2 ,η ξη  − εϕ1 ,η ξη  ≤ εϕ2 ,ψ+εϕ2 ξψ+εϕ2  − εϕ1 ,ψ+εϕ2 ξψ+εϕ2  . s

s

s

2 2 2 Using relations εϕ 2 ,η = ε ϕ2 ,η etc, we have s

s

s

s

ϕ22 ,η ξη 2 − ϕ21 ,η ξη 2 ≤ ϕ22 ,ψ+εϕ2 ξψ+εϕ2 2 − ϕ21 ,ψ+εϕ2 ξψ+εϕ2 2 1−s

1−s

2 2 = ψ+εϕ ξϕ2 2 − ψ+εϕ ξϕ1 2 2 ,ϕ2 2 ,ϕ1

In the second line, we used Lemma A.5. Taking ε → 0 and applying Lemmas A.3 and A.5, we obtain the result. 2 Proof of Proposition 1.1. It is trivial for s = 0, 1. We prove the claim for 0 < s < 1. We first consider faithful ϕ, η. From ϕ ≤ ϕ + (η − ϕ)+ and Lemma A.2, we have s

s

s

s

2 2 2 2 ξϕ 2 − η,ϕ ξϕ 2 ≤ ϕ+(η−ϕ) ξϕ 2 − η,ϕ ξϕ 2 . ϕ,ϕ + ,ϕ

(11)

By Lemma 2.1 and inequalities η ≤ ϕ + (η − ϕ)+ , ϕ ≤ ϕ + (η − ϕ)+ , the last term is bounded as s

s

2 2 ξ 2 − η,ϕ+(η−ϕ) ξ 2 ≤ ϕ+(η−ϕ) + ,ϕ+(η−ϕ)+ ϕ+(η−ϕ)+ + ϕ+(η−ϕ)+ s

2 = ξϕ+(η−ϕ)+ 2 − η,ϕ+(η−ϕ) ξ 2 + ϕ+(η−ϕ)+ 1−s

2 = ξϕ+(η−ϕ)+ 2 − ϕ+(η−ϕ) ξη 2 . + ,η

(12)

By ϕ + (η − ϕ)+ ≥ η and Lemma A.2, we have 1−s

2 (12) ≤ ξϕ+(η−ϕ)+ 2 − η,η ξη 2 = ϕ(1) + (η − ϕ)+ (1) − η(1).

(13)

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YOSHIKO OGATA

Hence, we obtain s

2 ϕ(1) − η,ϕ ξϕ 2 ≤ ϕ(1) + (η − ϕ)+ (1) − η(1),

(14)

which is equal to s

2 ξϕ 2 . η(1) − (η − ϕ)+ (1) ≤ η,ϕ

(15)

We now prove this inequality for general ϕ, η. By considering a von Neumann  algebra Me := eMe with e := s(η) s(ϕ) instead of M if it is necessary, we may assume ϕ + εη, η + δϕ are faithful on M for any ε, δ > 0. We then have s

2 ξϕ+εη 2 . (η + δϕ)(1) − (η + δϕ − (ϕ + εη))+ (1) ≤ η+δϕ,ϕ+εη

(16)

Taking the limit ε → 0 and then the limit δ → 0, and using Lemmas A.3 and A.5 we obtain the inequality (15) for general ϕ, η. To check the condition for the equality, by approximating ϕ and η by ϕ + εη, η + δϕ in (11), (12), and (13), just as in (16), and taking the limit ε → 0 and δ → 0, we obtain s

s

s

s

2 2 2 2 ξϕ 2 − η,ϕ ξϕ 2 ≤ ϕ+(η−ϕ) ξϕ 2 − ηϕ ξϕ 2 ϕ,ϕ + ,ϕ s

s

2 2 ≤ ϕ+(η−ϕ) ξ 2 − η,ϕ+(η−ϕ) ξ 2 + ,ϕ+(η−ϕ)+ ϕ+(η−ϕ)+ + ϕ+(η−ϕ)+ 1−s

1−s

2 2 = ξϕ+(η−ϕ)+ 2 − ϕ+(η−ϕ) ξη 2 ≤ ξϕ+(η−ϕ)+ 2 − η,η ξη 2 + ,η

= ϕ(1) + (η − ϕ)+ (1) − η(1).

(17)

By Lemma A.4, the first inequality in (??) is an equality iff the support of (η − ϕ)+ is orthogonal to ϕ and the third inequality is an equality iff the support of (η − ϕ)− is orthogonal to η. Therefore, if the equality in (17) holds, then (η − ϕ)+ is orthogonal to ϕ and (η − ϕ)− is orthogonal to η. Conversely, if (η − ϕ)+ is orthogonal to ϕ and (η − ϕ)− is orthogonal to η. Then we have ϕ + (η − ϕ)+ = η + (η − ϕ)− , where both sides of the equality are sum of orthogonal elements. Therefore, we have s

s

2 2 ξ 2 − η,ϕ+(η−ϕ) ξ 2 ϕ+(η−ϕ) + ,ϕ+(η−ϕ)+ ϕ+(η−ϕ)+ + ϕ+(η−ϕ)+ 1−s

s

2 2 = (η−ϕ) ξ 2 = ϕ+(η−ϕ) ξ 2 − ,ϕ+(η−ϕ)+ ϕ+(η−ϕ)+ + ,(η−ϕ)− (η−ϕ)− 1−s

1−s

1−s

s

2 2 = ϕ,(η−ϕ) ξ 2 + (η−ϕ) ξ 2 − (η−ϕ)− + ,(η−ϕ)− (η−ϕ)− 2 2 = ϕ,(η−ϕ) ξ 2 = (η−ϕ) ξϕ 2 . − (η−ϕ)− − ,ϕ

Furthermore, we have s

s

s

2 2 2 ϕ+(η−ϕ) ξϕ 2 − ηϕ ξϕ 2 = (η−ϕ) ξϕ 2 . + ,ϕ − ,ϕ

(18)

A GENERALIZATION OF POWERS–STØRMER INEQUALITY

343

Hence, the second inequality in (17) is an equality in this case. As the first and third inequalities are equalities from the orthogonality of (η − ϕ)+ with ϕ and (η − ϕ)− with η, respectively, the equality holds in (17). Therefore, the equality in (17) holds iff (η − ϕ)+ is orthogonal to ϕ and (η − ϕ)− is orthogonal to η. However, the latter condition means η = (η − ϕ)+ + ψ and ϕ = (η − ϕ)− + ψ for some ψ ∈ M∗+ whose support is orthogonal to the support of |η − ϕ|. Replacing η, ϕ, s in (15) with ϕ, η, 1 − s, respectively, we obtain 1−s

s

2 2 ξη 2 = η,ϕ ξϕ 2 . ϕ(1) − (ϕ − η)+ (1) ≤ ϕ,η

(19) 2

Summing (15) and (19), we obtain (2).

Acknowledgements The author is grateful for Professor N. Ozawa, for giving her the simple proof of the matrix inequality (1). She also thanks Prof. Jakˇsi´c and Prof. Seiringer for kind advices on this note, and Prof. Y. Kawahigashi, Prof. H. Kosaki, and Prof. C.-A. Pillet for kind discussion. The present research is supported by JSPS Grantin-Aid for Young Scientists (B), Hayashi Memorial Foundation for Female Natural Scientists, Sumitomo Foundation, and Inoue Foundation (Inoue Science Research Award).

A. Appendix Let {M, H, J, P} be a standard form associated with a von Neumann algebra M, i.e., H is a Hilbert space where M acts on, J is the modular conjugation, and P is the natural positive cone. Let M∗+ be the set of all positive normal linear functionals over M. For each ϕ ∈ M∗+ , ξϕ is the unique element in the natural positive cone P which satisfies ϕ(x) = (xξϕ , ξϕ ) for all x ∈ M. For ϕ, ψ ∈ M∗+ , we define an operator Sϕψ as the closure of the operator Sϕψ (xξψ + (1 − j (s(ψ)))ζ ) := s(ψ)x ∗ ξϕ ,

x ∈ M, ζ ∈ H,

where s(ψ) ∈ M is the support projection of ψ and j (y) := J y J . The polar decom1

2 position of Sϕψ is given by Sϕψ = J ϕψ where ϕψ is the relative modular operator associated with ϕ, ψ ∈ M∗+ . The subspace Mξψ + (1 − j (s(ψ)))H of H is a 1

2 . The support projection of the positive operator ϕψ is s(ϕ) j (s(ψ)). core of ϕψ z by For a complex number z ∈ C, we define a closed operator ϕψ

z := (exp[z(log ϕψ )s(ϕ) j (s(ψ))])s(ϕ) j (s(ψ)). ϕψ

For an operator A on a Hilbert space H, we denote by D(A) its domain.

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YOSHIKO OGATA

LEMMA A.1. Let ϕ, ψ be faithful normal positive linear functionals over a von Neumann algebra M. Suppose that there exists a constant λ > 0 such that λϕ ≤ ψ. Then the cocyle R t → (Dϕ : Dψ)t ∈ M has an extension (Dϕ : Dψ)z ∈ M analytic on I− 1 := {z ∈ C : − 21 < z < 0} and bounded continuous on I− 1 with the bound 2

2

(Dϕ : Dψ)z  ≤ λz for all z ∈ I− 1 . Furthermore, for any faithful ζ ∈ M∗+ , 0 < s < 21 , 2

−s s and any element ξ in D(−s ψζ ), ψζ ξ is in the domain of ϕζ , and

sϕζ −s ψζ ξ = (Dϕ : Dψ)−is ξ.

(20)

Proof. The existence and boundedness of (Dϕ : Dψ)z is proven in [1]. To show the latter part of the Lemma, let ζ ∈ M∗+ be faithful. We define the region I−s in the complex plane by I−s := {z ∈ C : −s < z < 0} for each 0 < s < 21 . For any s ξ ∈ D(−s ψζ ) and ξ1 ∈ D(ϕζ ), we consider two functions on I−s given by F(z) := −i z −i z¯ (ψζ ξ, ϕζ ξ1 ), and G(z) := ((Dϕ : Dψ)z ξ, ξ1 ). Both of these functions are bounded continuous on I−s and analytic on I−s . Furthermore, they are equal on R: F(t) = (itϕζ −it ψζ ξ, ξ1 ) = ((Dϕ : Dψ)t ξ, ξ1 ) = G(t),

∀t ∈ R.

This means F(z) = G(z) for all z ∈ I−s . In particular, we have F(−is) = G(−is), i.e., s (−s ψζ ξ, ϕζ ξ1 ) = ((Dϕ : Dψ)−is ξ, ξ1 ). s As this holds for all ξ1 ∈ D(sϕζ ), −s ψζ ξ is in the domain of ϕζ , and (20) holds. 2

LEMMA A.2. Let ϕ, η, ψ be normal positive linear functionals over a von Neumann s s 2 2 ) and algebra M such that ϕ ≤ η. Then for any 0 ≤ s ≤ 1, we have D(η,ψ ) ⊂ D(ϕ,ψ s

s

s

2 2 ϕ,ψ ξ  ≤ η,ψ ξ ,

2 ∀ξ ∈ D(η,ψ ).

Proof. This is proven in [2].

(21) 2

LEMMA A.3. Let ϕ and η be elements in M∗+ and ϕn a sequence in M∗+ such that limn→∞ ϕn − ϕ = 0. Then for any and 0 < s < 1, s

s

2 ξη . lim ϕ2n ,η ξη  = ϕη

n→∞

Proof. By the integral representation of t s , we have s

s

2 ξη 2 ϕ2n ,η ξη 2 − ϕη ∞  sin sπ = dλλs−1 ((ϕn ,η (ϕn ,η + λ)−1 − ϕ,η (ϕ,η + λ)−1 )ξη , ξη ). π

0

(22)

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A GENERALIZATION OF POWERS–STØRMER INEQUALITY

We denote the term inside of the integral by f n (λ). It is easy to see | f n (λ)| ≤ λs−1 η(1),  1    1 2 ξη 2 ≤ λs−2 ϕ(1) + sup ϕn (1) . | f n (λ)| ≤ λs−2 ϕ2n ,η ξη 2 + ϕη n

Hence, | f n (λ)| is bounded from above by an integrable function independent of n. Next, we show limn→∞ f n (λ) = 0 for all λ > 0. To do so, we first observe that 1

1

2 ϕ2n ,η converges to ϕη in the strong resolvent sense: For all xξη + (1 − j (s(η)))ζ ∈ Mξη + (1 − j (s(η)))H, using Powers–Størmer inequality, we have 1

1

2 ϕ2n ,η (xξη + (1 − j (s(η)))ζ ) − ϕ,η (xξη +(1 − j (s(η)))ζ )2 =s(η)x ∗ ξϕn − s(η)x ∗ ξϕ 2

≤ x ∗ 2 ξϕn − ξϕ 2 ≤ x ∗ 2 ϕn − ϕ → 0, as n → ∞. 1

1

2 As Mξη + (1 − j (s(η)))H is a common core for all ϕ2n ,η and ϕη , this means 1

1

2 in the strong resolvent sense. Therefore, for a bounded conϕ2n ,η converges to ϕη 1

1

2 ) strongly. Hence, tinuous function g(t) = t 2 (t 2 + λ)−1 , g(ϕ2n ,η ) converges to g(ϕη we have limn→∞ f n (λ) = 0. By the Lebesgue’s theorem, we obtain the result. 2

LEMMA A.4. For any ϕ, η ∈ M∗+ with ϕ ≤ η and 0 < s < 1, s

2 ξϕ  = ξϕ  ηϕ

(23)

if and only if η − ϕ is orthogonal to ϕ. s

2 Proof. First we prove if ηϕ ξϕ  = ξϕ , then η − ϕ is orthogonal to ϕ. From

−s

−s

s

2 Lemma A.2, for any ζ ∈ D(ηϕ2 ), ηϕ2 ζ is in D(ϕϕ ) and s

−s

s

−s

2 2 ϕϕ ηϕ2 ζ  ≤ ηϕ ηϕ2 ζ  ≤ ζ . s

−s

−s

2 Therefore, ϕϕ ηϕ2 defined on D(ηϕ2 ) can be uniquely extended to a bounded operator A on H, with norm A ≤ 1. We define an operator 0 ≤ T ≤ 1 by T := A∗ A. Note that s

s

−s

s

s

s

2 2 2 2 2 Aηϕ ξϕ = ϕϕ ηϕ2 ηϕ ξϕ = ϕϕ s(η)ξϕ = ϕϕ ξϕ = ξϕ .

From this, and the assumption, we have s

s

s

s

2 2 2 2 (T ηϕ ξϕ , ηϕ ξϕ ) = Aηϕ ξϕ 2 = ξϕ 2 = ηϕ ξϕ 2 .

As the spectrum of T is included in [0, 1], this equality means s

s

2 2 T ηϕ ξϕ = ηϕ ξϕ .

(24)

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YOSHIKO OGATA s

2 For any ζ ∈ D(ηϕ ), we have s

s

s

s

s

s

2 2 2 2 2 2 ξϕ , ηϕ ζ ) = (T ηϕ ξϕ , ηϕ ζ ) = (Aηϕ ξϕ , Aηϕ ζ ) = (ξϕ , ζ ), (ηϕ

(25) 1

2 from (24). Therefore, ξϕ ∈ D(sηϕ ) and sηϕ ξϕ = ξϕ . Hence, we obtain ηϕ ξϕ = ξϕ . From this, we have 1

2 s(ϕ)ξη = J ηϕ ξϕ = ξϕ .

We then obtain (η − ϕ)(s(ϕ)) = 0, i.e., the support of η − ϕ is orthogonal to the support of ϕ. Conversely, if the support of η − ϕ is orthogonal to ϕ, then we have s

s

s

2 2 2 ηϕ ξϕ 2 = η−ϕ,ϕ ξϕ 2 + ϕϕ ξϕ 2 = ξϕ 2 .

(26) 2

LEMMA A.5. For all normal positive linear functionals ψ1 , ψ2 over a von Neumann algebra M, and 0 ≤ s ≤ 1, s

1−s

ψ2 1 ,ψ2 ξψ2  = ψ22 ,ψ1 ξψ1 .

(27) z

Proof. Functions 1−¯z

z¯

F(z) := (ψ2 1 ,ψ2 ξψ2 , ψ2 1 ,ψ2 ξψ2 ) and

1−z

G(z) := (ψ22 ,ψ1 ξψ1 ,

ψ22 ,ψ1 ξψ1 ) are bounded continuous on 0 ≤ z ≤ 1 and analytic on 0 < z < 1. It is easy to check F(it) = G(it) for t ∈ R. Hence, we obtain F(z) = G(z) on 0 ≤ z ≤ 1. 2

References 1. Araki, H.: Relative entropy of states of von Neumann algebras. Pub. R.I.M.S., Kyoto Univ. 11, 809–833 (1976) 2. Araki, H., Masuda, T.: Positive cones and L p -spaces for von Neumann algebras. Pub. R.I.M.S., Kyoto Univ. 18, 339–411 (1982) 3. Audenaert, K.M.R., Calsamiglia, J., Masanes, Ll., Munoz-Tapia, R., Acin, A., Bagan, E., Verstraete, F.: The quantum Chernoff bound. Phys. Rev. Lett. 98, 160501 (2007) 4. Jakˇsi´c, V., Ogata, Y., Pillet, C.-A., Seiringer, R.: (in preparation) 5. Powers, R.T., Størmer, E.: Free states of canonical anticommutation relations. Commun. Math. Phys. 16, 1–33 (1970) 6. Takesaki, M.: Theory of Operator Algebras II, Springer Encyclopedia of Mathematical Sciences, vol. 125. Springer, Berlin (2001)