Trace- and improved data processing inequalities for von Neumann algebras
aa r X i v : . [ m a t h - ph ] F e b Trace- and improved data processing inequalities forvon Neumann algebras
Stefan Hollands ∗ ITP, Universität Leipzig and MPI-MiS Leipzig
February 23, 2021
Abstract
We prove a version of the data-processing inequality for the relative entropy forgeneral von Neumann algebras with an explicit lower bound involving the measuredrelative entropy. The inequality, which generalizes previous work by Sutter et al. onfinite dimensional density matrices, yields a bound how well a quantum state can berecovered after it has been passed through a channel. The natural applications ofour results are in quantum field theory where the von Neumann algebras are knownto be of type III. Along the way we generalize various multi-trace inequalities togeneral von Neumann algebras.
The relative entropy S p ρ | σ q “ Tr p ρ ln ρ ´ ρ ln σ q is an important operationally definedmeasure for the distinguishability of two statistical operators ρ, σ . A fundamental prop-erty of S is that S p ρ | σ q ´ S p T p ρ q| T p σ qq ě (1)for a quantum channel T , i.e. completely positive linear trace preserving map . Theabove difference represents the loss of distinguishability between σ, ρ if these are passedthrough the channel T .An important general question that can be abstracted from concrete settings suchas quantum communication or quantum error correction is to what extent the action ofa quantum channel can be reversed, i.e. to what extent it may be possible to recover ρ from T p ρ q . It was understood already a long time ago by Petz that the questionof recoverability is intimately linked to the case of saturation of the data processinginequality (DPI) (1), see e.g. [28]. As was understood by [17] – and has subsequently ∗ [email protected] In the body of the paper, we use the slightly different notation ˜ T for the action of a channel on adensity matrix (Schrödinger picture), while T denotes the dual action (Heisenberg picture) of the channelon the observables. ρ in a suitable information theoretic measure provided thedifference in the DPI is also small. The recovery channel α σ,T is called “explicit” becauseit is given by a concrete expression involving only reference state σ and T (not the state ρ that is to be recovered), and always perfectly recovers σ , i.e. α σ,T p T p σ qq “ σ . In fact,it is closely related – though not precisely equal – to the channel originally proposed byPetz [29, 30, 31, 28].The above mentioned works (though not [29, 30, 31, 28]) establish their results onlyfor type I von Neumann algebras – in particular [35] assumes a finite-dimensional Hilbertspace. While this is well-motivated by applications in quantum computing, there arecases of interest when the algebras are not of this type. A notable example of this arequantum field theoretic applications related to the “quantum null energy condition” (seee.g. [12]) where the algebras are of type III [9, 19]. With this application in mind weproved in [15] a generalization of [25] in the case when the channel T corresponds to aninclusion of general von Neumann algebras. This result has been generalized to arbitrary2-positive channels T in [16], where the following improved DPI has been demonstrated: S p ρ | σ q ´ S p T p ρ q| T p σ qq ě ´ ss ż R dt β p t q D s p α tσ,T p T p ρ qq| ρ q . (2)Here, s P r { , q and D s are the so-called “sandwiched Renyi entropies” [27, 40], whichfor s “ { become the negative log squared fidelity. β p t q dt is a certain explicit prob-ability density and α tη,T is an explicit 1-parameter family of recovery channels that isa disintegration of α η,T in the sense ş dt β p t q α tη,T “ α η,T . Using convexity of D s andJensen’s inequality, the bound implies S p ρ | σ q ´ S p T p ρ q| T p σ qq ě ´ ss D s p α σ,T p T p ρ qq| ρ q . (3)A qualitatively similar result has been proved for general von Neumann algebras by Jungeand LaRacuente [26]. In their result, the sandwiched Renyi entropies are now replacedby some other information theoretic quantity with an operational meaning. Both [16, 26]lead to the same inequality for s “ { . For type I algebras and s “ { (2) is the resultby [25], but the relation for general s is unclear to the author. We also mention recentresults by Gao and Wilde [18] of a roughly similar flavor but different emphasis, whichapply to von Neumann algebras with a trace though not type III.In the present paper, we provide a generalization of [35] to arbitrary (sigma-finite)von Neumann algebras. This version of the improved DPI is qualitatively similar to (3).The definition of the recovery channel is in fact identical to that in (3), but we have yetanother information theoretic quantity on the right side, namely (thm. 1) S p ρ | σ q ´ S p T p ρ q| T p σ qq ě S meas p α σ,T p T p ρ qq| ρ q . (4)Here, S meas is the “measured relative entropy”, defined as the maximum possible value ofthe relative entropy restricted to a commutative subalgebra. We show below (prop. 1)2hat for s “ { , this inequality is sharper than (3) – though not in general the inequality(2) with the integral outside – for all ρ, σ . A conceptual advantage of (4) over both(2) and (3) (and likewise to the inequalities proven in [26]) is that it is saturated in thecommutative case, as noted already by [35]. So in this respect (4) is sharp unlike itspredecessors.Our proof technique is similar in several respects to that in [35] and related antecedentssuch as [25] in that we also use interpolation arguments for L p -spaces. However, there arealso some key differences requiring technical modifications: For instance, the operators ln ρ or ln σ no longer exist for general von Neumann algebras or the use of ordinary L p (Schatten)-spaces is prohibited since a general von Neumann algebra does not have atrace. As in our previous papers [15, 16] – referred to as papers I,II – our solution to thefirst problem is to work entirely with Araki’s relative modular operator, the log of whichcan roughly be viewed as a difference between ln ρ and ln σ . Likewise, as in [15, 16],our solution to the second problem is to work with the Araki-Masuda non-commutative L p -spaces [3] which are very closely related to the sandwiched relative Renyi entropies .For these norms, we require a complex interpolation theory, see lem. 1, which generalizesa result in [15]. This result is then applied to a specially constructed analytic family ofvectors and combined with certain cutoff-techiques for appropriately extended domainsof analyticity in a similar way as in [15]. However, in [15, 16], such cutoff techniqueswere needed to control the limit of the Araki-Masuda norms as p Ñ , whereas in thepresent paper, it is the limit p Ñ 8 which is relevant. The regularization is necessaryhere to apply the powerful technique of bounded perturbations of normal states of a vonNeumann algebra, and a (somewhat modified) version of the Lie-Trotter product formulafor von Neumann algebras [6]. These ideas go beyond [15, 16] and also yield various new“trace” inequalities for von Neumann algebras which could be of independent interest.This paper is organized as follows. In sec. 2 we review some prerequisite notions fromthe theory of von Neumann algebras. In sec. 3 we establish an interpolation theoremfor the Araki-Masuda L p -norms, which we apply in sec. 4 to obtain generalizations ofvarious known mutli-trace inequalities to von Neumann algebras. In sec. 5 we establishour main result, thm. 1. The definition of the L p -norms is relegated to the appendix. Let A “ M n p C q . The fundamental representation of this algebra is on C n , but one canalso work in the “standard” Hilbert space ( H » M n p C q » C n b C n ). Vectors | ζ y in H are thus identified with matrices ζ P M n p C q . H » M n p C q is both a left and right modulefor A , l p a q | ζ i “ | aζ i r p b q | ζ i “ | ζ b i , (5)and the inner product on H is the Hilbert-Schmidt inner product x ζ | ζ y “ Tr p ζ ˚ ζ q . Amixed state, represented by a density matrix ω , gives rise to a linear functional on A by ω p a q “ Tr p ωa q , (6) [26] use a somewhat different approach to L p spaces to circumvent the absence of a tracial state inthe general von Neumann algebra setting. Their approach appears to us less natural for the purposes ofthis paper. ω p a ˚ a q ě , ω p q “ .A ( σ -finite) von Neumann algebra in standard form M is an ultra-weakly closed linearsubspace of the bounded operators on a Hilbert space H . M should contain , be closedunder products and the ˚ -operation should have a cyclic and separating vector | ψ y P H .Cyclic and separating means that M | ψ y is dense in H and m | ψ y “ implies m “ . Inthe matrix example, ψ should therefore be invertible. The set of ultra-weakly continuouspositive linear functionals (thus satisfying ω p a ˚ a q ě , ω p q “ ) is called S p M q . For adetailed account of von Neumann algebras see [36].Associated with a von Neumann algebra in standard form is a convex cone P M andan anti-linear involution J , called “modular conjugation” leaving this cone invariant. Apossible choice of this non-unique “natural cone” for A “ M n p C q is the subset of positivesemi-definite matrices in H , and in this case, J ˇˇ ζ (cid:11) “ ˇˇ ζ ˚ (cid:11) . A general property of J whichis easily verified in this example is that J M J “ M , the latter meaning the commutantof M on H . Given vectors | ψ y , | η y , | ζ y P H and m P M , one defines following Araki [1](see also app. C of [3] for many more details) S η,ψ ´ m | ψ y ` p ´ π M p ψ qq| ζ y ¯ “ π M p ψ q m ˚ | η y . (7)Here π M p ψ q P M is the orthogonal projection onto the closure of the subspace M | ψ y and π M p ψ q P M that onto the closure of M | ψ y . The definition is consistent because mπ M p ψ q “ if m | ψ y “ . One shows that S η,ψ is a closable operator and that if | ψ y P P M , then S η,ψ “ J ∆ { η,ψ , S ˚ η,ψ ¯ S η,ψ “ ∆ η,ψ , (8)One calls the self-adjoint, non-negative operator ∆ η,ψ the “relative modular operator”. Itssupport is π M p η q π M p ψ q and complex powers ∆ zη,ψ are understood as on the orthogonalcomplement of the support. The modular conjugation and relative modular operators of A “ M n p C q with the above choice of natural cone are: J ˇˇ ζ (cid:11) “ ˇˇ ζ ˚ (cid:11) ∆ η,ψ “ l p ω η q r p ω ´ ψ q , (9)where we invert the density matrix ω ψ on the range of π M p ψ q which in the case at handis the orthogonal projector onto the complement of the null space of ω ψ .For a general von Neumann algebra, every positive linear functional ω P S p M q corresponds to one and only one vector | ξ ω y in the natural cone P M such that ω p a q “x ξ ω | aξ ω y . Vice versa, any vector | ψ y (in the natural cone or not) gives rise to a linearfunctional ω ψ p a q “ x ψ | aψ y , for all a P A . (10)For A “ M n p C q , this linear functional is identified with the density matrix ω ψ “ ψψ ˚ and the natural cone vectors correspond to the unique positive square root of the corre-sponding density matrix, now thought of as pure states in the standard Hilbert space. Sothe vector representative of a density matrix ω in the natural cone is | ξ ω y “ | ω { y . An More precisely, a standard form is actually defined by the combined structure p M , H , P M , J q ,which can be recovered if we have a cyclic and separating vector. } ξ ψ ´ ξ η } ď } ω η ´ ω ξ } ď } ξ ψ ` ξ η } } ξ ψ ´ ξ η } , (11)where the norm of a linear functional is } ω } “ sup t| ω p m q| : m P M , } m } “ u . In thecase A “ M n p C q , the latter norm is } ω } “ Tr | ω | , so the above relation expresses thePowers-Störmer inequality between the trace norm and the Hilbert-Schmidt norm.Let us finish this briefest of introduction to von Neumann algebras by summarizing(again) some of our Notations and conventions:
Calligraphic letters A , M , . . . denote von Neumann al-gebras, always assumed σ -finite. Calligraphic letters H , K , . . . denote complex Hilbertspaces, always assumed to be separable. S p M q denotes the set of all ultra-weakly con-tinuous, positive, normalized linear functionals on M (“states”), which are in one-to-onecorrespondence with density matrices if A “ M n p C q . M ` is the subset of all non-negativeself-adjoint operators in M and M s . a . the subset of all self-adjoint elements of the vonNeumann algebra M . We use the physicist’s “ket”-notation | ψ y for vectors in a Hilbertspace. The scalar product is written as p| ψ y , | ψ yq H “ : x ψ | ψ y (12)and is anti-linear in the first entry. The norm of a vector is written simply as }| ψ y} “ : } ψ } . The action of a linear operator T on a ket is sometimes written as T | φ y “ | T φ y .In this spirit, the norm of a bounded linear operator T on H is written as } T } “ sup | ψ y : } ψ }“ } T ψ } . L p norms For the algebra A “ M n p C q the standard Hilbert space H – M n p C q on which A actsby left multiplication can be equipped with various norms. We have already mentionedthat the 2-norm (or Hilbert-Schmidt norm) } ζ } “ p Tr ζ ζ ˚ q { , (13)actually defines the Hilbert space norm on H (so the subscript “2” is generally omitted).For p ą , one can generalize this to } ζ } p “ r Tr p ζ ζ ˚ q p { s { p . (14)Given a faithful vector | ψ y P H with associated linear functional ω ψ p a q “ x ψ | aψ y “ Tr p aω ψ q (Hilbert Schmidt inner product), one can also define the yet more general norms: } ζ } p,ψ “ r Tr p ζ ω { p ´ ψ ζ ˚ q p { s { p . (15)The faithful condition is relevant for p ą as it ensures that ω ψ is invertible. Thegeneralized L p -norms } ζ } p,ψ evidently reduce to usual L p -norms if ω ψ p a q “ Tr p a q{ n isthe tracial state. A general von Neumann algebra M in standard form need not have5uch a tracial state, but Araki and Masuda [3] have shown that one can still define theabove “non-commuting L p -norms” for p ě using the relative modular operators basedon | ψ y , see also [23, 24, 8]. Their basic definitions are recalled for convenience in theappendix. The following interpolation result for the Araki-Masuda L p -norms is one ofthe main workhorses of this article. Lemma 1.
Let | G p z qy be a H -valued holomorphic function on the strip S { “ t ă Re z ă { u that is uniformly bounded in the closure, and let | ψ y P H a state of a σ -finitevon Neumann algebra M in standard form acting on H . For ă θ ă { , p , p P r , s or p , p P r , , let p θ “ ´ θp ` θp . (16) Then ln } G p θ q} p θ ,ψ (17) ď ż dt ´ p ´ θ q α θ p t q ln } G p it q} p ,ψ ` p θ q β θ p t q ln } G p { ` it q} p ,ψ ¯ , where α θ p t q “ sin p πθ qp ´ θ qp cosh p πt q ´ cos p πθ qq , β θ p t q “ sin p πθ q θ p cosh p πt q ` cos p πθ qq . (18) Proof.
We may assume | ψ y P P M by invariance of the L p -norms. In parts (a1), (a2) ofthis proof we first apply that | ψ y is faithful in order to apply the results by [3].(a1) Assume that p , p P r , s . This part of the proof is taken from paper I andonly included for convenience. Denote the dual of a Hölder index p by p , defined so that { p ` { p “ . [3] have shown that the non-commutative L p p M , ψ q -norm of a vector | ζ y relative to | ψ y can be characterized by (dropping the superscript on the norm) } ζ } p,ψ “ sup t|x ζ | ζ y| : } ζ } p ,ψ ď u . (19)They have furthermore shown ([3], thm. 3) that when p ě , any vector | ζ y P L p p M , ψ q has a unique generalized polar decomposition, i.e. can be written in the form | ζ y “ u ∆ { p φ,ψ | ψ y , where u is a unitary or partial isometry from M . Furthermore, they showthat } ζ } p ,ψ “ } φ } p . We may thus choose a u and a normalized | φ y , so that } G p θ q} p θ ,ψ “ x u ∆ { p θ φ,ψ ψ | G p θ qy , (20)perhaps up to a small error which we can let go zero in the end. Now we define p θ as inthe statement, so that p θ “ ´ θp ` θp , (21)and we define an auxiliary function f p z q by f p z q “ x u ∆ z { p `p ´ z q{ p φ,ψ ψ | G p z qy , (22)noting that f p θ q “ } G p θ q} p θ ,ψ (23)6y construction. By Tomita-Takesaki-theory, f p z q is holomorphic in S { . For the valuesat the boundary of the strip S { , we estimate | f p it q| “ |x u ∆ ´ it p { p ´ { p q φ,ψ ∆ { p φ,ψ ψ | G p it qy|ď } u ∆ ´ it p { p ´ { p q φ,ψ ∆ { p φ,ψ ψ } p ,ψ } G p it q} p ,ψ ď } ∆ ´ it p { p ´ { p q φ,ψ ∆ { p φ,ψ ψ } p ,ψ } G p it q} p ,ψ ď } φ } p } G p it q} p ,ψ ď } G p it q} p ,ψ . (24)Here we used the version of Hölder’s inequality proved by [3], we used } a ˚ ζ } p ,ψ ď} a }} ζ } p ,ψ for any a P A , see [3], lem. 4.4, and we used } ∆ ´ it p { p ´ { p q φ,ψ ∆ { p φ,ψ ψ } p ,ψ ď } φ } p which we prove momentarily. A similar chain of inequalities also gives | f p { ` it q| ď } G p { ` it q} p ,ψ . (25)To prove the remaining claim, let | ζ y “ ∆ zφ,ψ | ψ y and z “ { p ` it . Then we have, usingthe variational characterization by [3] of the L p p M , ψ q -norm when p ě : } ζ } p ,ψ “ sup t} ∆ { ´ { p χ,ψ ∆ zφ,ψ ψ } : } χ } “ u“ sup t} ∆ { ´ { p ´ itχ,ψ ∆ { p ` itφ,ψ ψ } : } χ } “ u“ sup t} ∆ { ´ { p χ,ψ p Dχ : Dφ q t π M p φ q ∆ { p φ,ψ ψ } : } χ } “ uď sup t} ∆ { ´ { p χ,ψ a ∆ { p φ,ψ ψ } : } χ } “ , a P M , } a } “ uď sup t} a ∆ { p φ,ψ ψ } p ,ψ : a P M , } a } “ u . (26)Using [3], lem. 4.4, we continue this estimation as ď sup a P M , } a }“ } a }} ∆ { p φ,ψ ψ } p ,ψ “ } φ } p , (27)which gives the desired result.Next, we use the Hirschman improvement of the Hadamard three lines theorem [21]. Lemma 2.
Let g p z q be holomorphic on the strip S { , continuous and uniformly boundedat the boundary of S { . Then for θ P p , { q , ln | g p θ q| ď ż ` β θ p t q ln | g p { ` it q| θ ` α θ p t q ln | g p it q| ´ θ ˘ d t, (28) where α θ p t q , β θ p t q are as in lem. 1. Applying this to g “ f gives the statement of the theorem.(a2) Now we assume that p , p P r , . [3] have shown that for any ζ P L ` p p M , ψ q : “ L p -closure of P {p p q M , ď p ď there is φ P H such that for all ζ P L p p M , ψ q we have x ζ | ζ y “ x ∆ { φ,ψ ψ | ∆ p { p q´p { q φ,ψ ζ y (29) The cone P {p p q M is defined as the closure of ∆ {p p q ψ M ` | ψ y and its properties are discussed in [3]. } ζ } p ,ψ “ } φ } { p . Furthermore, by the non-commutative Hölder inequalityproven in [3], there exists ζ P L p θ p M , ψ q such that } G p θ q} p θ ,ψ “ x ζ | G p θ qy , } ζ } p θ ,ψ “ . (30)Thus, since by [3], thm. 3 we may write ζ “ uζ , u P M with u ˚ u ď and ζ P L ` p θ p M , ψ q , we have } G p θ q} p θ ,ψ “x ∆ { φ,ψ ψ | ∆ { p θ ´ { φ,ψ u ˚ G p θ qy“x ∆ { φ,ψ ψ | ∆ p ´ θ q{ p `p θ q{ p ´ { φ,ψ u ˚ G p θ qy (31)and } φ } “ . Similarly to the previous case we now consider the function f p z q “ x ∆ { φ,ψ ψ | ∆ p ´ z q{ p `p z q{ p ´ { φ,ψ u ˚ G p z qy , (32)which is holomorphic for z P S { and uniformly bounded on the closure. For the lowerboundary value we calculate | f p it q| “|x ∆ { φ,ψ ψ | ∆ ´ it p { p ´ { p q φ,ψ ∆ { p ´ { φ,ψ u ˚ G p it qy|ď} ∆ { φ,ψ ψ } } ∆ { p ´ { φ,ψ u ˚ G p it q}“} φ } } ∆ { ´ { p φ,ψ u ˚ G p it q}ď sup t} ∆ { ´ { p χ,ψ u ˚ G p it q} : } χ } “ u“} u ˚ G p it q} p ,ψ ď } u ˚ }} G p it q} p ,ψ “ } G p it q} p ,ψ (33)using in the last line the variational characterization of the L p -norms and [3], lem. 4.4. Asimilar chain of inequalities also gives | f p { ` it q| ď } G p { ` it q} p ,ψ . The rest followsfrom Hirschman’s improvement as in the previous case.(b) In the remaining part of the proof, we remove the faithful condition on the state | ψ y . Suppose that ω ψ is non-faithful. For σ -finite M , there exists some cyclic andseparating vector | η y for M and we put ω ψ ε “ p ´ ε q ω ψ ` ε ω η (34)so that | ψ ε y P P M is now faithful for M (and M ). The proof is then completed by thefollowing lemma because we can apply part (a1),(a2) to the faithful state | ψ ε y and obtainb) by taking the limit ε Ñ and using the dominated convergence theorem under theintegral. Lemma 3.
Let ω ψ , ω η P S p M q , and let ω ψ ε “ p ´ ε q ω ψ ` ε ω η . Then lim ε Ñ ` } ζ } p,ψ ε “} ζ } p,ψ for any p ě and | ζ y P H .Proof. (1) Case p ě : Clearly ω ψ ε ě p ´ ε q ω ψ , from which it follows that ∆ φ,ψ ε ďp ´ ε q ´ ∆ φ,ψ and thus by Löwner’s theorem [20], ∆ αφ,ψ ε ď p ´ ε q ´ α ∆ αφ,ψ for α P r , s , soby the variational definition of the L p norm (appendix): } ζ } p,ψ ε ď p ´ ε q p { p q´p { q } ζ } p,ψ for p ě . (35)8herefore, by choosing ε ą sufficiently small, we can achieve that } ζ } p,ψ ε ´ } ζ } p,ψ ă δ (36)for any given δ ą . To get a similar inequality in the reverse direction, we use thefollowing lemma. Lemma 4.
Let ω ψ , ω η , ω ψ n , ω η n P S p M q be such that lim n } ω ψ ´ ω ψ n } “ , lim n } ω η ´ ω η n } “ and such that ω η n ď Cω η , ω ψ ď Cω ψ n for some C ă 8 and all n . Then lim n }p ∆ α { η,ψ ´ ∆ α { η n ,ψ n q ζ } “ (37) for any α P r , q , | ζ y P D p ∆ α { η,ψ q .Proof. We use the shorthands ∆ “ ∆ η,ψ , ∆ n “ ∆ η n ,ψ n . Without loss of generality α ą .To deal with the powers, we employ the standard formula X α “ sin p πα q π ż dλ λ α “ λ ´ ´ p λ ` X q ´ ‰ (38)for α P p , q , X ě . We use this with X “ ∆ { and “ ∆ { n giving us that }p ∆ α { ´ ∆ α { n q ζ }ď ż dλ λ α ´ ››“ p ` λ ∆ ´ { q ´ ´ p ` λ ∆ ´ { n q ´ ‰ ζ ›› . (39)In the rest of the proof we denote by c any constant depending only on α, C . We splitthe integration domain into three parts: p , δ q , p δ, L q , p L, .(i) Range p , δ q : In this range, we use ż δ dλ λ α ´ ››“ p ` λ ∆ ´ { q ´ ´ p ` λ ∆ ´ { n q ´ ‰ ζ ›› “ ż δ dλ λ α ››“ p λ ` ∆ { q ´ ´ p λ ` ∆ { n q ´ ‰ ζ ›› ď ż δ dλ λ α ›› p λ ` ∆ { q ´ ζ ›› ` ›› p λ ` ∆ { n q ´ ζ ››( ď } ζ } ż δ dλ λ α ´ “ c } ζ } δ α (40)using that ∆ , ∆ n ě .(ii) Range p δ, L q : By [2], II, lem. 4.1, ››“ p λ ` ∆ { q ´ ´ p λ ` ∆ { n q ´ ‰ ζ ›› Ñ as n Ñ 8 , when λ ą . (41)and the convergence is uniform for λ in the compact set r δ, L s .(iii) Range p L, . The domination assumption gives ∆ n ď C ∆ . The function R ` Q x ÞÑ p λ ` x ´ { q ´ is operator monotone, thus by by Löwner’s theorem [20]: }p ` λ ∆ ´ { n q ´ ζ } “ x ζ |p ` λ ∆ ´ { n q ´ ζ y { ď x ζ |p ` λC ´ ∆ ´ { q ´ ζ y { . (42)9nd since C ě trivially }p ` λ ∆ ´ { q ´ ζ } “ x ζ |p ` λ ∆ ´ { q ´ ζ y { ď x ζ |p ` λC ´ ∆ ´ { q ´ ζ y { . (43)Using these inequalities under the integral (39) gives: ż L dλ λ α ´ ››“ p ` λ ∆ ´ { q ´ ´ p ` λ ∆ ´ { n q ´ ‰ ζ ›› ď ż L dλ λ α ´ ›› p ` λ ∆ ´ { q ´ ζ ›› ` ›› p ` λ ∆ ´ { n q ´ ζ ››( ď ż L dλ λ α ´ x ζ |p ` λC ´ ∆ ´ { q ´ ζ y { ď cL ´ α { "ż L dλ λ ´ ` α x ζ |p ` λC ´ ∆ ´ { q ´ ζ y * { “ cL ´ α { " x ζ | f p C ∆ { q ζ y * { ď cL ´ α { } ∆ α { ζ } , (44)uniformly in n . Here we have applied Jensen’s inequality to the probability measure L α λ ´ ´ α dλ on p L, in the third step. We have also defined/estimated the non-negativefunction f p x q “ ż L dλ λ ´ ` α p ` x ´ λ q ´ ď cx α . (45)Applying standard subharmonic analysis to the subharmonic function z ÞÑ ln } ∆ αz { } inthe strip ď Re z ď , we have } ∆ α { ζ } ď } ζ }} ∆ α { ζ } , giving ż L dλ λ α ´ ››“ p ` λ ∆ ´ { q ´ ´ p ` λ ∆ ´ { n q ´ ‰ ζ ›› ď c p L ´ α } ζ }} ∆ α { ζ }q { . (46)Now we choose δ, L so small/large that the contributions from (i), (iii), i.e. (40),(46) are ă ε { each (independently of n ) and then n so large that the contribution (ii)from p δ, L q is ă ε { . Then the integral (39) is ă ε by (i), (ii), (iii), and the proof iscomplete.We can now complete the proof of lem. 3. We can pick a unit | φ y such that } ζ } ψ,p ď} ∆ p { q´p { p q φ,ψ ζ } ` δ { by the variational definition of the L p norm for p ě . Lem. 4 andthe triangle inequality shows that there is an ε ą such that } ζ } p,ψ ď} ∆ p { q´p { p q φ,ψ ζ } ` δ { ď} ∆ p { q´p { p q φ,ψ ε ζ } ` }p ∆ p { q´p { p q φ,ψ ´ ∆ p { q´p { p q φ,ψ ε q ζ } ` δ { ď sup t} ∆ p { q´p { p q χ,ψ ε ζ } : | χ y P H , } χ } “ u ` δ “} ζ } p,ψ ε ` δ, (47)and this together with (36) gives | } ζ } p,ψ ´ } ζ } p,ψ ε | ă δ . Since δ is arbitrarily small, theproof of lem. 3 is complete when p ě . 102) Case ď p ď : This proof has already appeared in paper I and is only includedfor convenience. Since by (34) ω ψ ε {p ´ ε q ą ω ψ , it now follows similarly as in part (1) ofthis proof that } ζ } p,ψ ď p ´ ε q p { p q´p { q } ζ } p,ψ ε for ď p ď . (48)The L p -norms } ζ } pp,ψ may be considered for fixed | ζ y as functionals of the state ω ψ , andas such they are convex. Indeed, let D s p ω ζ | ω ψ q be the sandwiched relative Renyi entropyrelative between two functionals ω ζ , ω ψ on M induced by vectors | ζ y , | ψ y , related to the L p -norms by D s p ω ζ | ω ψ q “ p s ´ q ´ ln } ζ } s s,ψ . The data processing inequality for thisquantity (see e.g. [8], thm. 14) in combination with standard arguments as in e.g. [27],proof of prop. 1 implies joint convexity in ω ζ , ω ψ . This gives in combination with (34)that (for p “ s ) } ζ } p,ψ ε ď p ´ ε q } ζ } p,ψ ` ε } ζ } p,η . (49)Combining (48) with (49) implies the statement of lem. 3 in the case ď p ď .This completes the proof of lem. 1. As applications of lem. 2 we now prove various inequalities that reduce to ”multi-traceinequalities” in the case of finite type I factors. For simplicity, it will be assumed that ω ψ is a faithful state on the von Neumann algebra M , meaning ω ψ p m ˚ m q “ implies m “ for all m P M . Corollary 1.
Let a , . . . , a n P M ` , r P p , s , p ě . Then r ln } a r ¨ ¨ ¨ a rn ψ } p { r,ψ ď ż R dt β r { p t q ln } a ` it ¨ ¨ ¨ a ` itn ψ } p,ψ . (50) Proof.
We choose p “ p, p “ 8 , θ “ r { and G p z q “ a z ¨ ¨ ¨ a zn | ψ y (51)in lem. 2. Then } G p z q} is uniformly bounded on S { and p θ “ p { r . At the lowerboundary of the strip: } G p it q} p ,ψ “ } a it ¨ ¨ ¨ a itn ψ } ,ψ “ } a it ¨ ¨ ¨ a itn } “ (52)because a itk are unitary operators (using the isomeric identification of L p M , ψ q Q a | ψ y ÞÑ a P M proven in [3].) Thus the term from the lower boundary does not contribute andwe obtain the statement.Another corollary of a similar nature is: Corollary 2. (Araki-Lieb-Thirring inequality) For r ě , | ψ y , | ζ y P H there holds } ζ } r,ψ ď } ∆ r { ζ,ψ ψ } { r . (53)11 roof. A proof for this has already been given in [8], thm. 12, so the only point is toshow an alternative proof. We may assume that } ∆ r { ζ,ψ ψ } ă 8 , otherwise the statementis trivial. Also, we may assume without loss of generality that | ζ y is in the natural cone.In lem. 2, we take G p z q “ ∆ rz { ζ,ψ ψ , p “ , p “ 8 , θ “ { r , so p θ “ r . Then G p z q isholomorphic and uniformly bounded in S { , see e.g. lem. 3 of [6].On the left side of lem. 2 we obtain ln } ∆ { ζ,ψ ψ } rr,ψ “ ln } ζ } rr,ψ . We compute at thelower boundary of the strip: } G p it q} p ,ψ “ } ∆ irt { ζ,ψ ψ } ,ψ “ } ∆ irt { ζ,ψ ∆ ´ irt { ψ,ψ ψ } ,ψ “ } u p rt { q ψ } ,ψ “ } u p rt { q} “ . (54)Here u p t q “ ∆ itζ,ψ ∆ ´ itψ,ψ is the Connes cocycle which is a unitary from M and we usedagain the isomeric identification of L p M , ψ q Q a | ψ y ÞÑ a P M proven in [3]. Thus theterm from the lower boundary does not contribute. At the upper boundary of the strip: } G p { ` it q} p ,ψ “ } ∆ irt { ` r { ζ,ψ ψ } ,ψ “ } ∆ r { ζ,ψ ψ } , (55)which no longer depends upon t , using that the L norm is equal to the Hilbert space norm[3] and that ∆ itζ,ψ is a unitary operator. Since ş dtβ θ p t q “ we obtain the statement.Let h be a self-adjoint element of M and | ψ y P H a normalized state vector. FollowingAraki [4], the non-normalized perturbed state | ψ h y is defined by the absolutely convergentseries | ψ h y “ ÿ n “ ż { ds . . . ż s n ´ ds n ∆ s n ψ h ∆ s n ´ ´ s n ψ h . . . ∆ s ´ s ψ h | ψ y , (56)which can also be written as e p ln ∆ ψ ` h q{ | ψ y [6]. This technique of perturbations hasbeen generalized to semi-bounded – instead of bounded – operators by [14], see also [28],sec. 12. The perturbations, h that would normally be in M s . a . are in this frameworkgeneralized to so-called “extended-valued upper bounded self-adjoint operators affiliatedwith M ”, the space of which is called M ext . More precisely, h P M ext if(i) it is a linear, upper semi-continuous map S p M q Q σ ÞÑ σ p h q P R Y t8u , and(ii) the set t σ p h q : σ P S p M qu is bounded from above.For any “operator” h P M ext , one shows that it is consistent to define: Definition 1. (see [14], thm. 3.1) If h P M ext , the perturbed state σ h of a normal state σ P S p M q , is given by the unique extremizer of the convex variational problem c p σ, h q “ sup t ρ p h q ´ S p ρ | σ q : ρ P S p M qu (57) provided the sup is not ´8 . The latter is certainly the case if h P M s . a . is an ordinary self-adjoint element of thevon Neumann algebra M , and in this case the above “thermodynamic” definition of theperturbed state is up to normalizations equivalent to Araki’s “perturbative” definition(56): c p σ, h q “ ln } η h } , σ h p m q “ x η h | m | η h y{} η h } , (58)12herein | η y is a vector representer of the state σ , see [14], ex. 3.3. Furthermore, h P M ext has the spectral decomposition [14], prop. 2.13 (B) h “ ż c ´8 λE h p dλ q ´ 8 ¨ q. (59)Here, q P M is a projector onto the subspace where h is ´8 , and the measure E h p dλ q takes values in the projections in p ´ q q M p ´ q q , so it commutes with q . Corollary 3. (Generalized Golden-Thomson inequality) For h i P M ext , | ψ y P H , } ψ } “ there holds ln } ψ h `¨¨¨` h k } ď ż R dt β p t q ln } k ź j “ e p { ` it q h j ψ } } ź j “ k e p { ´ it q h j ψ } + . (60) Proof.
Case I). First we assume each h j P M s . a . , i.e. it is bounded. We let G p z q “ ∆ z { ψ e zh . . . e zh k | ψ y . (61)By standard results of Tomita-Takesaki theory, this family of vectors is analytic on S { and uniformly bounded in the norm of H on the closure, for instance by the maximumof and ś ki “ } e h i } using a standard Phragmen-Lindelöf type argument. In lem. 2, weuse this with p “ , p “ 8 , θ “ { n, n P N , so p θ “ n . At the lower boundary of S { , we get } G p it q} ,ψ “ – the L -norm is the Hilbert space norm – so this does notcontribute. Keeping therefore only the term from the upper boundary, we have ln } ∆ {p n q ψ e h { n ¨ ¨ ¨ e h k { n ψ } nψ,n ď ż R dt β { n p t q ln } ∆ { ψ e p { ` it q h ¨ ¨ ¨ e p { ` it q h k ψ } . (62)Now we consider the left side, putting a n “ e h { n ¨ ¨ ¨ e h k { n . By [3], thm. 3 (4), thereexists | φ n y P L n p H , ψ q X P {p n q M such that ∆ { nφ n ,ψ | ψ y “ ∆ {p n q ψ a n | ψ y , } φ n } “ } ∆ {p n q ψ a n ψ } nψ,n . (63)It follows that | φ n y “ J ∆ { φ n ,ψ | ψ y “ J p ∆ {p n q ψ a n ∆ {p n q ψ q n { | ψ y (64)by a straightforward repeated application of [3], lem. 7.7 (2); for the details see e.g. [22],lem. 4.1. Combining (62), (63), (64), we arrive at ln }p ∆ {p n q ψ e h { n ¨ ¨ ¨ e h k { n ∆ {p n q ψ q n { ψ } ď ż R dt β { n p t q ln } ∆ { ψ e p { ` it q h ¨ ¨ ¨ e p { ` it q h k ψ } . (65)We now take the limit n Ñ 8 on the left side. Araki’s version of the Lie-Trotter formula(suitably generalized to k operators h , . . . , h k , using that e h { n ¨ ¨ ¨ e h k { n “ ` n ´ p h `¨ ¨ ¨` h k q` O p n ´ q where } O p n ´ q} ď Cn ´ for all n ą ) see [6], rem.s 1 and 2, establishesthat s ´ lim n p ∆ {p n q ψ e h { n ¨ ¨ ¨ e h k { n ∆ {p n q ψ q n { | ψ y “ | ψ h `¨¨¨` h k y “ e p ln ∆ ψ ` h `¨¨¨` h k q{ | ψ y , (66) The cone P {p n q M is defined as the closure of ∆ {p n q ψ M ` | ψ y in H .
13o we get ln } ψ h `¨¨¨` h k } ď ż R dt β p t q ln } ∆ { ψ e p { ` it q h ¨ ¨ ¨ e p { ` it q h k ψ } . (67)On the integrand we finally use the following well-known application of the Hadamardthree lines theorem ( ď α ă { , m P M ), } ∆ αψ mψ } ď } ∆ { ψ mψ } α } mψ } ´ α “ } m ˚ ψ } α } mψ } ´ α (68)using that z ÞÑ ln } ∆ zψ mψ } is subharmonic on S { . Using this with α “ { , m “ e p { ` it q h ¨ ¨ ¨ e p { ` it q h k gives the statement of the corollary.Case II). The proof can be generalized to the case when h j P M ext by reducing tothe case I) via an approximation argument: Elements k P M ext can be approximated bybounded self-adjoint elements k n P M s . a . by introducing a cutoff in the spectral decom-position (69), as in k n “ ż c ´ n λE k p dλ q ´ n ¨ q ; (69)in fact one shows that | ψ k n y Ñ | ψ k y strongly, see [14], prop. 3.15. We perform thiscutoff for every h j obtaining a h j,n . Since the desired inequality holds for h j,n by case I),the proof is completed by the fact that e p { ` it q h j,n Ñ e p { ` it q h j as n Ñ 8 strongly anduniformly in t (as can be seen by decomposing H “ q j H ` p ´ q j q H ). Examples:
1) In the previous corollary we take k “ , h “ h . Then the norm in theintegrand no longer depends upon t and we can use that ş dtβ p t q “ to get: } ψ h } ď } e h { ψ } , (70)as shown previously by [6].2) Finite-dimensional type I algebras. Let A “ M n p C q . We will work in the standardHilbert space ( H » M n p C q » C n ˚ b C n ) and identify state functionals such as ω ψ withdensity matrices via ω ψ p a q “ Tr p aω ψ q . Vectors | ζ y in H are thus identified with matrices ζ P M n p C q . We have already mentioned that the L p p A , ψ q -norms can be computed usingthe well known correspondence between these norms and the sandwiched relative entropydiscussed in [8]: } ζ } pp,ψ “ Tr p ζ ρ { p ´ ψ ζ ˚ q p { where | ζ y P H is identified with a matrix ζ P M n p C q as described. Let a i be non-negative matrices. The multi-matrix inequalityin cor. 1 then reads, when ω ψ is the normalized tracial state ω ψ p a q “ Tr p a q{ n , ln Tr | a r ¨ ¨ ¨ a rk | p { r ď ż R dt β r { p t q ln Tr | a a ` it ¨ ¨ ¨ a ` itk ´ a k | p , (71)which generalzes the Araki-Lieb-Thirring inequality (corresponding to k “ ). Thishas been derived previously in [41, ? ], so our result can be seen as a generalization ofthese results to arbitrary von Neumann algebras. Cor. 2 is another generalization ofthis inequality which gives nothing new in the present case. Cor. 3 gives the followinginequality. Under the above identification of vectors | ψ y P H and matrices, the perturbedvector is | ψ h y “ | e ln ψ ` h { y (72)14assuming | ψ y to be in the natural cone, i.e. self-adjoint and non-negative), and thenchoosing | ψ “ n {? n y as the vector representing the tracial state on A , we have ln Tr e h `¨¨¨` h k ď ż R dt β p t q ln Tr | e p { q h e p { ` it q h ¨ ¨ ¨ e p { ` it q h k ´ e p { q h k | , (73)for any hermitian matrices h i . This reduces to the Golden-Thomson inequality for k “ , Tr e h ` h ď Tr p e h e h q , (74)using that the trace in the integrand no longer depends on t and ş dtβ p t q “ . Forarbitrary number of matrices this is due to [35], who also explain the relation with Lieb’striple matrix inequality (for k “ ). For the von Neumann algebra A “ M n p C q , the relative entropy between two states(density matrices) ω ψ , ω η is defined by: S p ω ψ | ω η q “ Tr p ω ψ ln ω ψ ´ ω ψ ln ω η q . (75)This may be expressed in terms of the logarithm of the relative modular operator in(9), and this observation is the basis for Araki’s approach [1, 2] to relative entropy forgeneral von Neumann algebras. The main technical difference in the general case isthat the individual terms in the above expression such as the von Neumann entropy ´ Tr p ω ψ ln ω ψ q are usually infinite. Thus form a mathematical viewpoint, the relative-and not the absolute entropy is the primary concept.Let p M , J, P M , H q be a von Neumann algebra in standard form acting on a Hilbertspace H , with natural cone P M and modular conjugation J . According to [1, 2], if π M p η q ě π M p ψ q , the relative entropy may be defined in terms of them by S p ψ | η q “ ´ lim α Ñ ` x ξ ψ | ∆ αη,ψ ξ ψ y ´ α , (76)otherwise, it is by definition infinite. Here, | ξ ψ y denotes the unique representer of a vector | ψ y in the natural cone. The relative entropy only depends on the functionals ω ψ , ω η on M , but not the choice of vectors | ψ y , | η y that define these functionals. We will thereforeuse interchangeably the notations S p ψ | η q “ S p ω ψ | ω η q . Araki’s definition of S p ω ψ | ω η q stillsatisfies the data processing inequality (1) [37] along with many other properties, see e.g.[28].For t P R , the Connes-cocycle p Dψ : Dη q t is the isometric operator from M satisfying p Dψ : Dη q t π M p ψ q “ ∆ itψ,ψ ∆ ´ itη,ψ . (77) The limit exists under this condition but may be equal to `8 .
15t only depends on the state functionals ω ψ , ω η . In terms of the Connes-cocycle, therelative entropy (76) may also be defined as S p ω ψ | ω η q ” S p ψ | η q “ ´ i dd t ω ψ pp Dη : Dψ q t q| t “ . (78)The last expression has the advantage that it does not require one to know the vectorrepresentative of | ψ y in the natural cone; in particular it shows that S only depends onthe state functionals. Later we will use the following variational expression for the relative entropy [33],prop. 1, S p ψ | η q “ sup h P M s . a . t ω ψ p h q ´ ln } η h } u , (79)with M s . a . the set of self-adjoint elements of M . A related variational quantity is the “measured relative entropy” , S meas , defined as S meas p ψ | η q “ sup h P M s . a . t ω ψ p h q ´ ln } e h { η } u . (80)From the Golden-Thomson inequality (70) we find S meas p ψ | η q ď S p ψ | η q . (81) S meas can also be written in terms of the classical relative entropy S p µ | ν q (Kullback-Leibler divergence) of two probability measures S p µ | ν q “ ż dµ ln dµdν (82)as follows. Let a P M s . a . be a self-adjoint element of M . Then it has a spectral decom-position a “ ż λE a p dλ q (83)with an M -valued projection measure E a p dλ q . Given | ψ y , | η y P H , we get Borel measures dµ ψ,a “ x ψ | E a p dλ q ψ y , and likewise for | η y . Physically, these correspond to the probabilitydistributions for measument outcomes of a in the states | ψ y resp. | η y . The relativeentropy between these measures is defined (but can be `8 ) if supp µ η,a Ă supp µ ψ,a ,wherein dµ ψ,a { dµ η,a means the Radon-Nikodym derivative between the measures. Wemay perform the maximization in over f p h q with f P L p R ; R q and h P M s . a . because f p h q P M s . a . . Maximizing first for fixed h over f and using ( “ eq. (75) in the commutativecase) sup "ż f dµ ´ ln ż e f dν : f P L p R ; R q * “ S p µ | ν q , (84) The derivative exists whenever S p ψ | η q ă 8 [28], thm. 5.7. More precisely, the space L is defined relative to the measure µ h,ψ relative to some faithful normalstate ψ P S p M q . Depending on the nature of this measure, “ L ” means either ℓ pt , . . . , n uq , ℓ p N q or L p R q or a combination thereof, wherein the counting measure is understood in the first two cases,whereas the Lebesgue measure is understood in the last case.
16e can write the measured relative entropy in the following way: S meas p ω ψ | ω η q “ sup t S p µ h,ψ | µ h,η q : h P M s . a . u“ sup t S p ω ψ | C | ω η | C q : C Ă M a commutative von Neumann subalgebra u . (85)This motivates the name “measured relative entropy”. The second equality holds by [28],prop. 7.13, for a related discussion see also [7], lem. 1 which corresponds to countingmeasures on the finite set t , . . . , n u .For later we would like to know the relationship between S meas and the fidelity, F .According to [38], the fidelity between two states ω η , ω ψ P S p M q on a von Neumannalgebra M in standard form may be defined as F p ω ψ | ω η q “ sup t|x η | u ψ y| : u P M , } u } “ u . (86)It is related to the L -norm relative to M by F p ω ψ | ω η q “ } η } ,ψ, M , see e.g. paper I,lem. 3 (1). We claim: Proposition 1. If ω η P S p M q is a faithful state on the von Neumann algebra M , then S meas p ω ψ | ω η q ě ´ ln F p ω ψ | ω η q .Proof. We may assume at that | η y is cyclic for M , for if not we can obtain an equivalentstandard form of M after a GNS-construction based on ω η and work with that standardform. Without loss of generality, | η y P P M . Consider in L p M , η q the polar decompo-sition | ψ y “ u | ψ ` y into a u P M such that u u “ π M p ψ q ď and | ψ ` y P P { M , see[3], thm. 3. By definition, the cone P { M is the closure of ∆ { ψ M | η y , which equals theclosure of M ` | η y , since J ∆ { ψ a | η y “ a | η y for a P M , J | η y “ | η y and J M J “ M .Thus, there exists a sequence t a n u Ă M ` such that lim n a n | η y “ u | ψ y strongly, so lim n x η | a n η y “ x η | u ψ y P R ` . (87)Then, with E a n p dλ q the spectral decomposition of a n and dµ a n ,ψ “ x ψ | E a n p dλ q ψ y , dµ a n ,η “x η | E a n p dλ q η y , the definition of the measured relative entropy and Jensen’s inequality ap-plied to the convex function ´ ln yields S meas p ω ψ | ω η q ě S p µ a n ,ψ | µ a n ,η q ě ´ ż ˆ dµ a n ,ψ dµ a n ,η ˙ { dµ a n ,η “ ´ F p µ a n ,ψ | µ a n ,η q , (88)where the Radon-Nikodym derivative is defined since | η y is faithful. The strong limit lim n a n | η y “ u | ψ y and dµ a n ,ψ “ x u ψ | E a n p dλ q u ψ y (because u P M , u u “ π M p ψ q and E a n takes values in M ) imply that } µ a n ,ψ ´ µ a n ,a n η } ď } ω ψ ´ ω a n η } ď } ψ ` a n η } } ψ ´ a n η } Ñ as n Ñ 8 . By paper I, lem. 11 and (11) applied to the commutative case, this givesthat also | F p µ a n ,ψ | µ a n ,η q ´ F p µ a n ,a n η | µ a n ,η q| ď } µ a n ,ψ ´ µ a n ,a n η } { Ñ . (89)By definition, ˆ dµ a n ,a n η p λ q dµ a n ,η p λ q ˙ { “ λ for λ P R ` , (90)17ence by (88) S meas p ω ψ | ω η q ě ´ n ż λdµ a n ,η “ ´ n ż λ x η | E a n p dλ q η y“ ´ n x η | a n η y “ ´ x η | u ψ y “ ´ |x η | u ψ y| . (91)The right side is by definition ě ´ ln F p ω ψ | ω η q as } u } “ , u P M , which concludes theproof. We now recall the definition of the Petz map in the case of general von Neumann algebras,discussed in more detail in [28], sec. 8. Let T : B Ñ A be a ˚ -preserving linear mapbetween two von Neumann algebras A , B in standard form acting on Hilbert spaces H , K . If ` x ζ | x ζ | ˘ T ˆ„ a bc d „ a ˚ c ˚ b ˚ d ˚ ˙ ˆ | ζ y| ζ y ˙ ě , @| ζ i y P H , T p B q “ A , (92)and for all a, b, c, d P B , then T is called 2-positive and unital. In the matrix inequality,we mean T applied to each matrix element. By duality between A and S p A q , T : B Ñ A gives a corresponding map ˜ T : S p A q Ñ S p B q by ω ÞÑ ˜ T p ω q : “ ω ˝ T . Forfinite dimensional von Neumann algebras A , B where state functionals are identified withdensity matrices through ω p a q “ Tr p ωa q , we can think of ˜ T as the linear operator ondensity matrices defined by Tr ωT p b q “ Tr ˜ T p ω q b @ b P B . (93)This operator ˜ T is completely positive and trace-preserving. The quantum data process-ing inequality (DPI) [37] states that S p ω ψ | ω η q ě S p ω ψ ˝ T | ω η ˝ T q , (94)where the right side could also be written as S p ˜ T p ω ψ q| ˜ T p ω η qq .We recall the definition of the Petz-map. Let | η A y be a cyclic and separating vectorin the natural cone of a von Neumann algebra A in standard form. Then the KMS scalarproduct on A is defined as x a , a y η “ x η A | a ˚ ∆ { η a η A y . (95)Let ω η be the normal state functional on A associated with | η A y . Then its pull-back ω η ˝ T to B , which is also faithful has a vector representative | η B y P K in the naturalcone. So: ω η p a q “ x η A | aη A y , ω η ˝ T p b q “ x η B | bη B y . (96) | η A y resp. | η B y give KMS scalar products for A resp. B , which we can use to define theadjoint T ` : A Ñ B (depending on the choices of these vectors) of the normal, unital This follows from Kadison’s inequality T p b ˚ b q ě T p b q ˚ T p b q . T : B Ñ A , which is again normal, unital, and 2-positive, see [28] prop.8.3. For finite dimensional matrix algebras T ` corresponds dually to the linear operator ˜ T ` acting on density matrices ρ for B given by ˜ T ` p ρ q “ σ { A T ´ σ ´ { B ρσ ´ { B ¯ σ { A , (97)wherein σ A is the density matrix of | η A y and σ B “ ˜ T p σ A q for | η B y . The rotated Petz map,which we call α tη,T : A Ñ B , is defined by conjugating this with the respective modularflows, i.e. α tη,T “ ς tη, B ˝ T ` ˝ ς ´ tη, A (98)where ς tη, A “ Ad∆ itη, A is the modular flow for A , | η A y etc. For finite dimensional matrixalgebras, α tη,T gives by duality a linear operator ˜ α tη,T acting on density matrices ρ for B ,which is ˜ α tη,T p ρ q “ σ { ´ it A T ´ σ ´ { ` it B ρσ ´ { ´ it B ¯ σ { ` it A . (99)An equivalent definition of the rotated Petz map is: Definition 2.
Let T : B Ñ A be a unital, normal, and 2-positive, linear map and | η A y P H a faithful state. Then the rotated Petz map α tη,T : A Ñ B is defined implicitlyby the identity: x bη B | J B ∆ itη B α tη,T p a q η B y “ x T p b q η A | J A ∆ itη A aη A y , (100) for all a P A , b P B . Closely related to the Petz map is the linear map V ψ : K Ñ H defined ω ψ by[32, 30] V ψ b | ξ B ψ y : “ T p b q| ξ A ψ y p b P B q . (102)It follows from Kadison’s property T p a ˚ a q ě T p a ˚ q T p a q (which is a consequence of (92))that V ψ is a contraction } V ψ } ď , see e.g. [32], proof of thm. 4.As in paper II, we introduce a vector valued function z ÞÑ | Γ ψ p z qy : “ ∆ zη A ,ψ A V ψ ∆ ´ zη B ,ψ B | ξ B ψ y p z P S { q , (103)the existence and properties of which are established in lem.s 3, 4 in paper II. In particular, | Γ ψ p z qy is holomorphic inside the strip S { and bounded in the closure S { in norm by1. Furthermore, the representation (24) of paper I shows in conjunction with Stone’stheorem that this function is strongly continuous on the boundaries of the strip S { , i.e.for Re p z q “ or Re p z q “ { , which is used implicitly below e.g. when we considerintegrals involving this quantity along these boundaries. The relation to the Petz map isas follows, paper II, lem. 2: x Γ ψ p { ` it q| a Γ ψ p { ` it qy ď ω ψ ˝ T ˝ α tη,T p a q t P R , a P A ` . (104) As it stands, the definition is actually consistent only when | ξ B ψ y is cyclic and separating. In thegeneral case, one can define [32] instead V ψ p b | ξ B ψ y ` | ζ yq : “ T p b q| ξ B ψ y p b P B , π B p ψ q| ζ y “ q . (101) .3 Improved DPI Our main theorem is:
Theorem 1.
Let T : B Ñ A be a two-positive, unital (in the sense (92) ) linear mapbetween two von Neumann algebras, and let ω ψ , ω η be normal states on A , with ω η faithful.Then S p ω ψ | ω η q ´ S p ω ψ ˝ T | ω η ˝ T q ě S meas p ω ψ | ω ψ ˝ T ˝ α T,η q . (105) with the recovery channel α T,η ” ż R dt β p t q α tT,η . (106) Remarks:
1) The theorem should generalize to non-faithful ω η by applying appropriatesupport projections in a similar way as in paper I, lem. 1.2) For finite-dimensional type I von Neumann algebras i.e. matrices, our result is dueto [35]. The recovery channel is given explicitly by (99) in this case as an operator ondensity matrices, where σ A , σ B are the density matrices corresponding to ω η , ω η ˝ T .3) By prop. 1, our bound implies that given in our previous paper II for the fidelity;in fact it is stronger in many cases. I) Proof under a majorization condition:
First we consider the special casewhere there exists c ě such that c ´ ω η ď ω ψ ď cω η . (107)Note that this implies c ´ ω η ˝ T ď ω ψ ˝ T ď cω η ˝ T as T is positive. By [28], thm.12.11 (due to Araki), there exists a h “ h ˚ P A such that | ψ y “ | η h y{} η h } such that } h } ď ln c , and vice versa. As is well known, this furthermore implies that the Connescocycle r Dη B : Dψ B s iz is holomorphic in the two-sided strip t z P C : | Re p z q| ă { u and bounded in norm (by c Re p z q ) on the closure of this strip, see e.g. paper II, lem. 5.As a consequence, we have an absolutely convergent (in the operator norm) power seriesexpansion r Dη B : Dψ B s iz “ ` ÿ l “ z l k l , (108)with bounded operators k l P B such that } k l } ď C l . We set k : “ didt T pr Dη B : Dψ B s t q| t “ P A s . a . . (109)Using [28], cor. 12.8, and the definition of the relative entropy in terms of the Connescocycle, S A p ψ | η k q “ S A p ψ | η q ´ ω ψ p k q“ S A p ψ | η q ´ x ψ A | didt T pr Dη B : Dψ B s t q ψ A y| t “ “ S A p ψ | η q ´ S B p ψ | η q , (110)which is one side of the inequality that we would like to prove. The variational expression(116) then gives: S A p ψ | η q ´ S B p ψ | η q “ sup h P A s . a . t ω ψ p h q ´ ln } η h ` k } u , (111)20here we used |p η k q h y “ | η k ` h y see [28], thm. 12.10. To get the desired DPI we willestablish an upper bound on ln } η h ` k } .In lem. 1, we take | G p z qy “ e zh | Γ ψ p z qy , p “ 8 , p “ where θ “ { n with n P N and h “ h ˚ P A . At the lower boundary we have with u B p t q : “ r Dη B : Dψ B s t P B , u A p t q : “r Dη A : Dψ A s t P A the unitary Connes cocycles, } G p it q} p ,ψ “} e ith ∆ itη A ,ψ A V ψ ∆ ´ itη B ,ψ B ξ B ψ } ,ψ “} e ith ∆ itη A ,ψ A T p u B p t qq ψ } ,ψ “} e ith ς tη r T p u B p t qqs u A p t q ˚ ψ } ,ψ “} e ith ς tη r T p u B p t qqs u A p t q ˚ }“} ς tη r T p u B p t qqs} ď , (112)where we used } ς tη r T p b qs} “ } T p b q} ď } b } (from the positivity of T and ς tη “ Ad∆ itη A ) aswell as the isomeric identification of L p A , ψ q Q a | ψ y ÞÑ a P A proven in [3]. Since p θ “ n and ln } G p it q} p ,ψ ď as just shown, we get from lem. 1 ln } e h { n Γ ψ p { n q} nψ,n ď ż R dt β { n p t q ln } G p { ` it q} p ,ψ “ ż R dt β { n p t q ln } e h { Γ ψ p { ` it q} ď ln ż R dt β { n p t q } e h { Γ ψ p { ` it q} ď ln ż R dt β { n p t q ω ψ ˝ T ˝ α tη,T p e h q , (113)using (104) in the third line and Jensen’s inequality in the second (noting that the inte-grand is continuous and uniformly bounded). Taking the lim-sup n Ñ 8 , we get usingthe definition of the recovery channel α T,η : lim sup n ln } e h { n Γ ψ p { n q} nψ,n ď ω ψ ˝ T ˝ α η,T p e h q . (114)The next lemmas give an expression for the lim-sup: Lemma 5.
We have } e h { n Γ ψ p { n q} nψ,n “ }p e h { n ∆ { nη,ψ a n ∆ { nη,ψ e h { n q n { ψ } , where a n “ T pr Dη B : Dψ B s i { n q ˚ T pr Dη B : Dψ B s i { n q P A ` . (115) Lemma 6.
We have lim n }p e h { n ∆ { nη,ψ a n ∆ { nη,ψ e h { n q n { ψ } “ } η h ` k } . Combining the two lemmas with eq.s (111), (114) gives S A p ψ | η q ´ S B p ψ | η q ě sup h P A s . a . t ω ψ p h q ´ ln ω ψ ˝ T ˝ α η,T p e h qu “ S meas p ω ψ | ω ψ ˝ T ˝ α T,η q , (116)using the variational definition (5.1) of S meas in the last step. Proof of lem. 6:
Since (108) is an absolutely convergent power series in the operatornorm, it follows that a n “ ` n ´ k ` O p n ´ q where O p n α q denotes a family of operators21uch that } O p n α q} ď cn α for all n ą . Since h is bounded, we also have e h { n “ ` n ´ h ` O p n ´ q . Replacing n Ñ n to simplify some expressions we trivially get e h {p n q ∆ {p n q η,ψ a n ∆ {p n q η,ψ e h {p n q “ ∆ { nη,ψ ` n ´ X n ` n ´ Y n (117)where X n , Y n is a finite sum of terms of the form x ∆ s η,ψ x ¨ ¨ ¨ x l ∆ s l η,ψ x l wherein ř s j “ { n, s j ě and } x j } ď c uniformly in n . X n is given explicitly by X n “ h ∆ { nη,ψ ` ∆ { nη,ψ h ` ∆ {p n q η,ψ k ∆ {p n q η,ψ . (118)By [2], II, proof of thm. 3.1, the functions F p z q : “ x ∆ z η,ψ x ¨ ¨ ¨ x j ∆ z j η,ψ x j ` | ψ y , z P ¯ S j { (119)defined for given x j P A are (strongly) analytic in the domain S j { : “ tp z , . . . , z j q P C j : 0 ă Re p z i q , ř Re p z i q ă { u and strongly continuous on the closure. Subharmonicanalysis as in [2], II, proof of thm. 3.1, or [3] furthermore gives the bound } F p z q} ď ź i } x i } , @ z P ¯ S j { . (120)This bound, and the elementary formula p A ` tB q N “ N ÿ j “ t j ÿ m ` ... ` m j “ N ´ j,m j P N A m B ¨ ¨ ¨ A m j ´ BA m j , (121)shows that the difference | ζ n y “ p e h {p n q ∆ {p n q η,ψ a n ∆ {p n q η,ψ e h {p n q q n { | ψ y´ n { ÿ j “ n ´ j ÿ m ` ... ` m j “ n { ´ j,m j P N ∆ m { nη,ψ X n ¨ ¨ ¨ ∆ m j ´ { nη,ψ X n ∆ m j { nη,ψ | ψ y (122)is bounded in norm by } ζ n } ď ` ` n ´ p} h } ` } k }q ` n ´ c ˘ n { ´ ` ` n ´ p} h } ` } k }q ˘ n { (123)for some c ă 8 , hence it tends to zero in norm as n Ñ 8 . Setting now | φ n,j y “ n ´ j ÿ m ` ... ` m j “ n { ´ j,m j P N ∆ m { nη,ψ X n ¨ ¨ ¨ ∆ m j ´ { nη,ψ X n ∆ m j { nη,ψ | ψ y , (124)the strong continuity of the functions F and the usual definition of the Riemann integralimplies | φ j y : “ lim n | φ n,j y“ ż { ds . . . ż s j ´ ds j ∆ s ´ s η,ψ p h ` k q ∆ s ´ s η,ψ p h ` k q . . . ∆ s j ´ ´ s j η,ψ p h ` k q ∆ s j η,ψ | ψ y , (125)22nd the usual perturbation theory by bounded operators as in [4], prop. 16 gives ř j “ | φ j y “ e p ln ∆ η,ψ ` h ` k q{ | ψ y . Hence, lim n p e h {p n q ∆ {p n q η,ψ a n ∆ {p n q η,ψ e h {p n q q n { | ψ y “ e p ln ∆ η,ψ ` h ` k q{ | ψ y (126)strongly, as argued more carefully in [6], proof of lem. 5. We have e p ln ∆ η,ψ ` h ` k q{ | ψ y “ e p ln ∆ η,ψ ` p h ` p k q{ | ψ y (here p “ π A p ψ q P A ). Also, using [28], thm. 12.6., we have ln ∆ η,ψ ` p h ` p k “ ln ∆ η h ` k ,ψ , and this gives | η h ` k y “ J | η h ` k y “ e p ln ∆ η,ψ ` h ` k q{ | ψ y byrelative modular theory. This completes the proof. Proof of lem. 5:
From the definitions, e h { n Γ ψ p { n q “ e h { n ∆ { nη A ,ψ A V ψ ∆ ´ { nη B ,ψ B ∆ { nψ B | ψ B y “ e h { n ∆ { nη A ,ψ A T pr Dη B : Dψ B s i { n q| ψ A y , (127)using the definition of the Connes-cocylce and the fact that r Dη B : Dψ B s i { n P B underour assumption (107), see paper II, proof of lem. 4. In the following, let a “ e h { n , b “ T pr Dη B : Dψ B s i { n q P A and | ψ A y “ | ψ y , | η A y “ | η y etc.By the results of [3] (which hold in the present context since ω ψ is faithful beingdominated by the faithful state ω η ), the vector b ∆ { nη,ψ a | ψ y P L n p A , ψ q has a polar decom-position b ∆ { nη,ψ a | ψ y “ u ∆ { nφ n ,ψ | ψ y , where } b ∆ { nη,ψ aψ } nn,ψ “ } φ n } and where u P A is a partialisometry. To get an expression for | φ n y , we use the formalism of “script” L p -spaces of [3],notation 7.6: As a vector space L ˚ p p A , ψ q , p ě consists of all formal linear combinationsof formal expressions of the form A “ x ∆ z ζ ,ψ x . . . x n ∆ z n ζ n ,ψ x n ` (128)wherein Re p z i q ě , ř i Re p z i q ď ´ { p , x i P A , ζ i P H , the formal adjoint of which isdefined to be A ˚ “ x ˚ n ` ∆ ¯ z n ζ n ,ψ x ˚ n . . . x ˚ ∆ ¯ z ζ ,ψ x ˚ . (129)The notation L ˚ p, p A , ψ q is reserved for formal elements A such that ř i Re p z i q “ ´ { p inaddition to all other conditions. It is then clear that L ˚ p, p A , ψ q L ˚ q, p M , ψ q “ L ˚ r, p M , ψ q as formal products where { r “ { p ` { q with { p “ ´ { p as usual. By [3], lem.7.3, if ď p ď , any element A P L ˚ p p A , ψ q can be viewed as an element of L p p A , ψ q in the sense that | ψ y P D p A q and A | ψ y P L p p A , ψ q . Furthermore, by [3], lem. 7.7 (2),if A , A P L ˚ p p A , ψ q correspond to the same element under this identification, then sodo A ˚ , A ˚ or A B, A B or BA , BA if B P L ˚ q, p A , ψ q (as long as { p ` { q ď { , forexample).We now start with the trivial statement that u ∆ { nφ n ,ψ “ b ∆ { nη,ψ a in the sense that theseelements of L ˚ n , p A , ψ q are identified with the same element of L n p A , ψ q . Then repeatedapplication of [3], lem. 7.7 (2) and the definition of adjoint gives u ∆ { nφ n ,ψ u ˚ “ b ∆ { nη,ψ aa ˚ ∆ { nη,ψ b ˚ in L ˚ n {p n ´ q , p A , ψ q . (130)Forming successively n { products of this equality and applying in each step [3], lem. 7.7(2), we find that u ∆ { φ n ,ψ u ˚ “ p b ∆ { nη,ψ aa ˚ ∆ { nη,ψ b ˚ q n { in L ˚ , p A , ψ q , (131) In fact, } Aψ } p ,ψ ď } x n ` } ś ni “ p} x i }} ζ i } Re p z i q q . H “ L p A , ψ q after we apply them to | ψ y . Thus, }p b ∆ { nη,ψ aa ˚ ∆ { nη,ψ b ˚ q n { ψ } “ } u ∆ { φ n ,ψ u ˚ ψ } “ } uJ uφ n } “ } φ n } (132)using modular theory. Therefore }p b ∆ { nη,ψ aa ˚ ∆ { nη,ψ b ˚ q n { ψ } “ } b ∆ { nη,ψ aψ } nn,ψ , (133)and the proof of the lemma is complete. II) Proof in general case:
We will now remove the majorization condition (107). Thiscondition has been used in an essential way in most of the arguments so far becausewithout it, the operator k in (109) is unbounded and thus not an element of A . Forunbounded operators the Araki-Trotter product formula and the L p -techniques are notavailable and it seems non-trivial extending them to an unbounded framework. We willtherefore proceed in a different way and define a regularization of ω ψ such that themajorization condition (107) holds and such that, at the same time, the desired entropyinequality can be obtained in a limit wherein the regulator is removed. However, it isclear that this regularization must be carefully chosen because the relative entropy isnot continuous but only lower semi-continuous. By itself the latter is insufficient for ourpurposes since the desired inequality (105) has both signs of the relative entropy.Our regularization combines a trick invented in paper I with the convexity of therelative entropy. As in paper I, we consider a function f p t q , t P R with the followingproperties.(A) The Fourier transform of f ˜ f p p q “ ż e ´ itp f p t q dt (134)exists as a real and non-negative Schwarz-space function. This implies that theoriginal function f is Schwarz and has finite L p R q norm, } f } ă 8 .(B) f p t q has an analytic continuation to the upper complex half plane such that the L p R q norm of the shifted function has } f p¨ ` iθ q} ă 8 for ă θ ă 8 .Such functions certainly exist (e.g. Gaussians). We also let f P p t q “ P f p tP q for ourregulator P ą , and we define a regulated version of | ψ y by | ψ P i “ ˜ f P p ln ∆ η,ψ q | ψ i } ˜ f P p ln ∆ η,ψ q ψ } . (135)As shown in paper I, some key properties of the regulated vectors are:(P1) ω ψ P ď c P ω η for some c P ą which may diverge as P Ñ 8 ,(P2) s ´ lim P Ñ8 | ψ P y “ | ψ y (strong convergence),(P3) ´ ´ } f } {} ˜ f } ¯ ` lim sup P Ñ8 S p ψ P | η q ď S p ψ | η q ,24here the first item gives at least “half” of the domination condition (107), the sec-ond states in which sense | ψ P y approximates | ψ y and the third gives us an upper semi-continuity property of the relative entropy opposite to the usual lower semi-continuityproperty which holds for generic approximations. We define for small ε ą : σ p a q “ x η | aη y ρ P,ε p a q “ p ´ ε qx ψ P | aψ P y ` ε x η | aη y . (136)Thus, by P1), the relative majorization condition (107) holds e.g. with c “ max p c P , ε ´ q between ρ P,ε and σ . By P2), lim P Ñ8 lim ε Ñ } ρ ´ ρ P,ε } “ . In P3), we choose a function f such that } f } {} ˜ f } “ (which must be Gaussian). The well-known convexity of therelative entropy gives together with the definition of ρ ε,P that ( ρ P “ x ψ P | . ψ P y ) S p ρ P,ε | σ q ď p ´ ε q S p ρ P , σ q ` ε S p σ | σ q “ p ´ ε q S p ρ P , σ q . (137)Combining this with P3), we get lim sup P Ñ8 lim sup ε Ñ S p ρ P,ε | σ q ď S p ρ | σ q . (138)The norm convergence lim P lim ε ρ P,ε ˝ T “ ρ ˝ T by P2) also gives in combination withthe usual lower semi-continuity of the relative entropy, [2], II thm. 3.7 (2), that lim inf P Ñ8 lim inf ε Ñ S p ρ P,ε ˝ T | σ ˝ T q ě S p ρ ˝ T | σ ˝ T q . (139)Now we combine eq.s (138), (139) with part I of the proof applied to the states ρ P,ε and σ , which obey the relative majorization condition. We get: S p ρ | σ q ´ S p ρ ˝ T | σ ˝ T q ě lim sup P Ñ8 lim sup ε Ñ S meas p ρ P,ε | ρ P,ε ˝ T ˝ α T,σ q . (140)The proof of part II is then finished by proving lower semi-continuity for the measuredrelative entropy: Lemma 7. If µ n , ν n , µ, ν P S p A q are such that lim n µ n “ µ and lim n ν n “ ν in the normsense, then S meas p µ | ν q ď lim inf n S meas p µ n | ν n q .Proof. This is a straightforward consequence of the variational definition (5.1) of S meas ,choosing a near optimal h . Acknowledgements:
SH thanks Tom Faulkner for conversations and the Max-Planck Society for supporting the collaboration between MPI-MiS and Leipzig U., grantProj. Bez. M.FE.A.MATN0003.
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