Trace- and improved data processing inequalities for von Neumann algebras

We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The inequality, which generalizes previous work by Sutter et al. on finite dimensional density matrices, yields a bound how well a quantum state can be recovered after it has been passed through a channel. The natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III. Along the way we generalize various multi-trace inequalities to general von Neumann algebras.


Introduction
The relative entropy Spρ|σq " Trpρ ln ρ ´ρ ln σq is an important operationally defined measure for the distinguishability of two statistical operators ρ, σ.A fundamental property of S is that Spρ|σq ´SpT pρq|T pσqq ě 0 for a quantum channel T , i.e. completely positive linear trace preserving map 1 .The above difference represents the loss of distinguishability between σ, ρ if these are passed through the channel T .
An important general question that can be abstracted from concrete settings such as quantum communication or quantum error correction is to what extent the action of a quantum channel can be reversed, i.e. to what extent it may be possible to recover ρ from T pρq.It was understood already a long time ago by Petz that the question of recoverability is intimately linked to the case of saturation of the data processing inequality (DPI) (1), see e.g.[28].As was understood by [17] -and has subsequently been generalized in various was by [25,35,17,7,10,23,34,39] -explicit lower bounds in the DPI or related information theoretic inequalities can provide information how well a channel may be reversed if the inequality is e.g.nearly saturated.
The current best result in this direction appears to be that by Sutter, Berta, and Tomamichel [35].It provides an explicit recovery channel, such that the recovered state is close to the original state ρ in a suitable information theoretic measure provided the difference in the DPI is also small.The recovery channel α σ,T is called "explicit" because it 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 pT pσqq " σ.In fact, it is closely related -though not precisely equal -to the channel originally proposed by Petz [29,30,31,28].
The above mentioned works (though not [29,30,31,28]) establish their results only for type I von Neumann algebras -in particular [35] assumes a finite-dimensional Hilbert space.While this is well-motivated by applications in quantum computing, there are cases of interest when the algebras are not of this type.A notable example of this are quantum field theoretic applications related to the "quantum null energy condition" (see e.g.[12]) where the algebras are of type III [9,19].With this application in mind we proved in [15] a generalization of [25] in the case when the channel T corresponds to an inclusion of general von Neumann algebras.This result has been generalized to arbitrary 2-positive channels T in [16], where the following improved DPI has been demonstrated: Here, s P r1{2, 1q and D s are the so-called "sandwiched Renyi entropies" [27,40], which for s " 1{2 become the negative log squared fidelity.β 0 ptqdt is a certain explicit probability density and α t η,T is an explicit 1-parameter family of recovery channels that is a disintegration of α η,T in the sense ş dt β 0 ptq α t η,T " α η,T .Using convexity of D s and Jensen's inequality, the bound implies Spρ|σq ´SpT pρq|T pσqq ě 1 ´s s D s pα σ,T pT pρqq|ρq.
A qualitatively similar result has been proved for general von Neumann algebras by Junge and LaRacuente [26].In their result, the sandwiched Renyi entropies are now replaced by some other information theoretic quantity with an operational meaning.Both [16,26] lead to the same inequality for s " 1{2.For type I algebras and s " 1{2 (2) is the result by [25], but the relation for general s is unclear to the author.We also mention recent results by Gao and Wilde [18] of a roughly similar flavor but different emphasis, which apply 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 yet another information theoretic quantity on the right side, namely (thm. 1) Spρ|σq ´SpT pρq|T pσqq ě S meas pα σ,T pT pρqq|ρq.
Here, S meas is the "measured relative entropy", defined as the maximum possible value of the relative entropy restricted to a commutative subalgebra.We show below (prop.1) that for s " 1{2, 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 the commutative case, as noted already by [35].So in this respect (4) is sharp unlike its predecessors.
Our proof technique is similar in several respects to that in [35] and related antecedents such as [25] in that we also use interpolation arguments for L p -spaces.However, there are also 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 a trace.As in our previous papers [15,16] -referred to as papers I,II -our solution to the first problem is to work entirely with Araki's relative modular operator, the log of which can 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 entropies2 .For these norms, we require a complex interpolation theory, see lem. 1, which generalizes a result in [15].This result is then applied to a specially constructed analytic family of vectors and combined with certain cutoff-techiques for appropriately extended domains of analyticity in a similar way as in [15].However, in [15,16], such cutoff techniques were needed to control the limit of the Araki-Masuda norms as p Ñ 2, whereas in the present paper, it is the limit p Ñ 8 which is relevant.The regularization is necessary here to apply the powerful technique of bounded perturbations of normal states of a von Neumann algebra, and a (somewhat modified) version of the Lie-Trotter product formula for 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 from the theory of von Neumann algebras.In sec. 3 we establish an interpolation theorem for the Araki-Masuda L p -norms, which we apply in sec.4 to obtain generalizations of various known mutli-trace inequalities to von Neumann algebras.In sec.5 we establish our main result, thm. 1.The definition of the L p -norms is relegated to the appendix.

Von Neumann algebras and modular theory
Let A " M n pCq.The fundamental representation of this algebra is on C n , but one can also work in the "standard" Hilbert space (H where the functional and the state is denoted by the same symbol.These linear functionals are alternatively characterized by the property ωpa ˚aq ě 0, ωp1q " 1. A (σ-finite) von Neumann algebra in standard form M is an ultra-weakly closed linear subspace of the bounded operators on a Hilbert space H . M should contain 1, be closed under products and the ˚-operation should have a cyclic and separating vector |ψy P H . Cyclic and separating means that M|ψy is dense in H and m|ψy " 0 implies m " 0. In the matrix example, ψ should therefore be invertible.The set of ultra-weakly continuous positive linear functionals (thus satisfying ωpa ˚aq ě 0, ωp1q " 1) is called S pMq.For a detailed account of von Neumann algebras see [36].
Associated with a von Neumann algebra in standard form3 is a convex cone P 7 M and an anti-linear involution J, called "modular conjugation" leaving this cone invariant.A possible choice of this non-unique "natural cone" for A " M n pCq is the subset of positive semi-definite matrices in H , and in this case, J ˇˇζ " ˇˇζ ˚ .A general property of J which is easily verified in this example is that JMJ " M 1 , the latter meaning the commutant of M on H .Given vectors |ψy, |ηy, |ζy 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|ψy `p1 ´πM 1 pψqq|ζy ¯" π M pψqm ˚|ηy.(7) Here π M pψq P M is the orthogonal projection onto the closure of the subspace M 1 |ψy and π M 1 pψq P M 1 that onto the closure of M|ψy.The definition is consistent because mπ M pψq " 0 if m|ψy " 0. One shows that S η,ψ is a closable operator and that if One calls the self-adjoint, non-negative operator ∆ η,ψ the "relative modular operator".Its support is π M pηqπ M 1 pψq and complex powers ∆ z η,ψ are understood as 0 on the orthogonal complement of the support.The modular conjugation and relative modular operators of A " M n pCq with the above choice of natural cone are: where we invert the density matrix ω ψ on the range of π M 1 pψq which in the case at hand is 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 pMq corresponds to one and only one vector |ξ ω y in the natural cone P 6 M such that ωpaq " xξ ω |aξ ω y.Vice versa, any vector |ψy (in the natural cone or not) gives rise to a linear functional ω ψ paq " xψ|aψy, for all a P A.
For A " M n pCq, this linear functional is identified with the density matrix ω ψ " ψψ ånd the natural cone vectors correspond to the unique positive square root of the corresponding density matrix, now thought of as pure states in the standard Hilbert space.So the vector representative of a density matrix ω in the natural cone is |ξ ω y " |ω 1{2 y.An important fact used implicitly in several places below is that if two linear functionals are close in norm, then the vectors in the natural cone are as well, and vice versa: where the norm of a linear functional is }ω} " supt|ωpmq| : m P M, }m} " 1u.In the case A " M n pCq, the latter norm is }ω} " Tr |ω|, so the above relation expresses the Powers-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 algebras, always assumed σ-finite.Calligraphic letters H , K , . . .denote complex Hilbert spaces, always assumed to be separable.S pMq denotes the set of all ultra-weakly continuous, positive, normalized linear functionals on M ("states"), which are in one-to-one correspondence with density matrices if A " M n pCq.M `is the subset of all non-negative self-adjoint operators in M and M s.a. the subset of all self-adjoint elements of the von Neumann algebra M. We use the physicist's "ket"-notation |ψy for vectors in a Hilbert space.The scalar product is written as and is anti-linear in the first entry.The norm of a vector is written simply as }|ψy} ": }ψ}.The action of a linear operator T on a ket is sometimes written as T |φy " |T φy.
In this spirit, the norm of a bounded linear operator T on H is written as }T } " sup |ψy:}ψ}"1 }T ψ}.

Interpolation of non-commutative L p norms
For the algebra A " M n pCq the standard Hilbert space H -M n pCq on which A acts by left multiplication can be equipped with various norms.We have already mentioned that the 2-norm (or Hilbert-Schmidt norm) actually defines the Hilbert space norm on H (so the subscript "2" is generally omitted).For p ą 0, one can generalize this to Given a faithful vector |ψy P H with associated linear functional ω ψ paq " xψ|aψy " Trpaω ψ q (Hilbert Schmidt inner product), one can also define the yet more general norms: The faithful condition is relevant for p ą 2 as it ensures that ω ψ is invertible.The generalized L p -norms }ζ} p,ψ evidently reduce to usual L p -norms if ω ψ paq " Trpaq{n is the tracial state.A general von Neumann algebra M in standard form need not have such a tracial state, but Araki and Masuda [3] have shown that one can still define the above "non-commuting L p -norms" for p ě 1 using the relative modular operators based on |ψy, see also [23,24,8].Their basic definitions are recalled for convenience in the appendix.The following interpolation result for the Araki-Masuda L p -norms is one of the main workhorses of this article.
Lemma 1.Let |Gpzqy be a H -valued holomorphic function on the strip S 1{2 " t0 ă Rez ă 1{2u that is uniformly bounded in the closure, and let |ψy P H a state of a σ-finite von Neumann algebra M in standard form acting on H .For 0 ă θ ă 1{2, p 0 , p 1 P r1, 2s or p 0 , p 1 P r2, 8s, let Then Proof.We may assume |ψy P P 6 M by invariance of the L p -norms.In parts (a1), (a2) of this proof we first apply that |ψy is faithful in order to apply the results by [3].
(a1) Assume that p 0 , p 1 P r1, 2s.This part of the proof is taken from paper I and only included for convenience.Denote the dual of a Hölder index p by p 1 , defined so that 1{p `1{p 1 " 1. [3] have shown that the non-commutative L p pM, ψq-norm of a vector |ζy relative to |ψy can be characterized by (dropping the superscript on the norm) }ζ} p,ψ " supt|xζ|ζ 1 y| : }ζ 1 } p 1 ,ψ ď 1u. ( They have furthermore shown ( [3], thm. 3) that when p 1 ě 2, any vector |ζ 1 y P L p 1 pM, ψq has a unique generalized polar decomposition, i.e. can be written in the form |ζ 1 y " u∆ 1{p 1 φ,ψ |ψy, where u is a unitary or partial isometry from M. Furthermore, they show that }ζ 1 } p 1 ,ψ " }φ} p 1 .We may thus choose a u and a normalized |φy, so that perhaps up to a small error which we can let go zero in the end.Now we define p θ as in the statement, so that and we define an auxiliary function f pzq by noting that by construction.By Tomita-Takesaki-theory, f pzq is holomorphic in S 1{2 .For the values at the boundary of the strip S 1{2 , we estimate |f pitq| " |xu∆ Here we used the version of Hölder's inequality proved by [3], we used }a ˚ζ } p 1 0 ,ψ ď }a}}ζ} p 1 0 ,ψ for any a P A, see [3], lem.4.4, and we used }∆ which we prove momentarily.A similar chain of inequalities also gives To prove the remaining claim, let |ζ 1 y " ∆ z φ,ψ |ψy and z " 1{p 1 `2it.Then we have, using the variational characterization by [3] of the L p 1 pM, ψq-norm when p 1 ě 2: Using [3], lem.4.4, we continue this estimation as which gives the desired result.Next, we use the Hirschman improvement of the Hadamard three lines theorem [21].
Applying this to g " f gives the statement of the theorem.
(a2) Now we assume that p 0 , p 1 P r2, 8s.[3] have shown that for any 4 ζ 1 `P L p1 pM, ψq :" L p 1 -closure of P 1{p2p 1 q M , 1 ď p 1 ď 2 there is φ P H such that for all ζ P L p pM, ψq we have 4 The cone P 1{p2p 1 q M is defined as the closure of ∆ 1{p2p 1 q ψ M `|ψy and its properties are discussed in [3].
(b) In the remaining part of the proof, we remove the faithful condition on the state |ψy.Suppose that ω ψ is non-faithful.For σ-finite M, there exists some cyclic and separating vector |ηy for M and we put ω ψε " p1 ´εq ω ψ `ε ω η (34) so that |ψ ε y P P 6 M is now faithful for M (and M 1 ).The proof is then completed by the following lemma because we can apply part (a1),(a2) to the faithful state |ψ ε y and obtain b) by taking the limit ε Ñ 0 and using the dominated convergence theorem under the integral.
To deal with the powers, we employ the standard formula for α P p0, 1q, X ě 0. We use this with X " ∆ 1{2 and " ∆ 1{2 n giving us that In the rest of the proof we denote by c any constant depending only on α, C. We split the integration domain into three parts: p0, δq, pδ, Lq, pL, 8q.
(ii) Range pδ, Lq: and the convergence is uniform for λ in the compact set rδ, Ls.
Combining ( 48) with (49) implies the statement of lem. 3 in the case This completes the proof of lem. 1.

Multi-trace inequalities for von Neumann algebras
As applications of lem. 2 we now prove various inequalities that reduce to "multi-trace inequalities" 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 pm ˚mq " 0 implies m " 0 for all m P M.
Proof.We choose p 1 " p, p 0 " 8, θ " r{2 and in lem. 2. Then }Gpzq} is uniformly bounded on S 1{2 and p θ " p{r.At the lower boundary of the strip: because a 2it k are unitary operators (using the isomeric identification of L 8 pM, ψq Q a|ψy Þ Ñ a P M proven in [3].)Thus the term from the lower boundary does not contribute and we obtain the statement.
Another corollary of a similar nature is: Proof.A proof for this has already been given in [8], thm.12, so the only point is to show an alternative proof.We may assume that }∆ r{4 ζ,ψ ψ} ă 8, otherwise the statement is trivial.Also, we may assume without loss of generality that |ζy is in the natural cone.In lem.2, we take Gpzq " ∆ rz{2 ζ,ψ ψ, p 1 " 2, p 0 " 8, θ " 1{r, so p θ " r.Then Gpzq is holomorphic and uniformly bounded in S 1{2 , see e.g.lem. 3 of [6].
On the left side of lem. 2 we obtain ln }∆ 1{2 ζ,ψ ψ} r r,ψ " ln }ζ} r r,ψ .We compute at the lower boundary of the strip: Here uptq " ∆ it ζ,ψ ∆ ´it ψ,ψ is the Connes cocycle which is a unitary from M and we used again the isomeric identification of L 8 pM, ψq Q a|ψy Þ Ñ a P M proven in [3].Thus the term from the lower boundary does not contribute.At the upper boundary of the strip: which no longer depends upon t, using that the L 2 norm is equal to the Hilbert space norm [3] and that ∆ it ζ,ψ is a unitary operator.Since ş dtβ θ ptq " 1 we obtain the statement.
Let h be a self-adjoint element of M and |ψy P H a normalized state vector.Following Araki [4], the non-normalized perturbed state |ψ h y is defined by the absolutely convergent series which can also be written as e pln ∆ ψ `hq{2 |ψy [6].This technique of perturbations has been 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 framework generalized to so-called "extended-valued upper bounded self-adjoint operators affiliated with 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 pMq Q σ Þ Ñ σphq P R Y t8u, and (ii) the set tσphq : σ P S pMqu 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 pMq , is given by the unique extremizer of the convex variational problem cpσ, hq " suptρphq ´Spρ|σq : ρ P S pMqu (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 the von Neumann algebra M, and in this case the above "thermodynamic" definition of the perturbed state is up to normalizations equivalent to Araki's "perturbative" definition (56): wherein |ηy 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) Here, q P M is a projector onto the subspace where h is ´8, and the measure E h pdλq takes values in the projections in p1 ´qqMp1 ´qq, so it commutes with q.
Corollary 3. (Generalized Golden-Thomson inequality) For h i P M ext , |ψy P H , }ψ} " 1 there holds Proof.Case I).First we assume each h j P M s.a., i.e. it is bounded.We let By standard results of Tomita-Takesaki theory, this family of vectors is analytic on S 1{2 and uniformly bounded in the norm of H on the closure, for instance by the maximum of 1 and ś k i"1 }e h i } using a standard Phragmen-Lindelöf type argument.In lem.2, we use this with p 1 " 2, p 0 " 8, θ " 1{n, n P 2N, so p θ " n.At the lower boundary of S 1{2 , we get }Gpitq} 2,ψ " 1 -the L 2 -norm is the Hilbert space norm -so this does not contribute.Keeping therefore only the term from the upper boundary, we have Now we consider the left side, putting a n " e h 1 {n ¨¨¨e h k {n .By [3] (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 1 , . . ., h k , using that e h 1 {n ¨¨¨e h k {n " 1 `n´1 ph 1 hk q`Opn ´2q where }Opn ´2q} ď Cn ´2 for all n ą 0) see [6], rem.s 1 and 2, establishes that s ´lim n p∆ 1{p2nq ψ e h 1 {n ¨¨¨e h k {n ∆ 1{p2nq ψ q n{4 |ψy " |ψ h 1 `¨¨¨`h k y " e pln ∆ ψ `h1 `¨¨¨`h k q{2 |ψy, (66) 5 The cone P 1{p2nq M is defined as the closure of ∆ 1{p2nq ψ M `|ψy in H .
Case II).The proof can be generalized to the case when h j P M ext by reducing to the case I) via an approximation argument: Elements k P M ext can be approximated by bounded self-adjoint elements k n P M s.a. by introducing a cutoff in the spectral decomposition (69), as in ´n λE k pdλq ´n ¨q ; (69) in fact one shows that |ψ kn y Ñ |ψ k y strongly, see [14], prop.3.15.We perform this cutoff 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 p1{2`itqh j,n Ñ e p1{2`itqh j as n Ñ 8 strongly and uniformly in t (as can be seen by decomposing H " q j H `p1 ´qj qH ).
Examples: 1) In the previous corollary we take k " 1, h 1 " h.Then the norm in the integrand no longer depends upon t and we can use that ş dtβ 0 ptq " 1 to get: as shown previously by [6].
2) Finite-dimensional type I algebras.Let A " M n pCq.We will work in the standard Hilbert space (H » M n pCq » C n˚b C n ) and identify state functionals such as ω ψ with density matrices via ω ψ paq " Trpaω ψ q.Vectors |ζy in H are thus identified with matrices ζ P M n pCq.We have already mentioned that the L p pA, ψq-norms can be computed using the well known correspondence between these norms and the sandwiched relative entropy discussed in [8]: where |ζy P H is identified with a matrix ζ P M n pCq as described.Let a i be non-negative matrices.The multi-matrix inequality in cor. 1 then reads, when ω ψ is the normalized tracial state ω ψ paq " Trpaq{n, which generalzes the Araki-Lieb-Thirring inequality (corresponding to k " 2).This has been derived previously in [41, ?], so our result can be seen as a generalization of these results to arbitrary von Neumann algebras.Cor. for any hermitian matrices h i .This reduces to the Golden-Thomson inequality for k " 2, Tr e h 1 `h2 ď Trpe h 1 e h 2 q, (74) using that the trace in the integrand no longer depends on t and ş dtβ 0 ptq " 1.For arbitrary number of matrices this is due to [35], who also explain the relation with Lieb's triple matrix inequality (for k " 3).
5 Improved DPI and recovery channels

Relative entropy and measured relative entropy
For the von Neumann algebra A " M n pCq, the relative entropy between two states (density matrices) ω ψ , ω η is defined by: Spω ψ |ω η q " Trpω ψ ln ω ψ ´ωψ 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 for general von Neumann algebras.The main technical difference in the general case is that the individual terms in the above expression such as the von Neumann entropy ´Trpω ψ ln ω ψ q are usually infinite.Thus form a mathematical viewpoint, the relativeand not the absolute entropy is the primary concept.Let pM, J, P6 M , H q be a von Neumann algebra in standard form acting on a Hilbert space H , with natural cone P 7 M and modular conjugation J.According to [1,2], if π M pηq ě π M pψq, the relative entropy may be defined in terms of them by 6   Spψ|ηq " ´lim otherwise, it is by definition infinite.Here, |ξ ψ y denotes the unique representer of a vector |ψy in the natural cone.The relative entropy only depends on the functionals ω ψ , ω η on M, but not the choice of vectors |ψy, |ηy that define these functionals.We will therefore use interchangeably the notations Spψ|ηq " Spω ψ |ω η q.Araki's definition of Spω ψ |ω η q still satisfies the data processing inequality (1) [37] along with many other properties, see e.g.[28].For t P R, the Connes-cocycle pDψ : Dηq t is the isometric operator from M satisfying pDψ : It only depends on the state functionals ω ψ , ω η .In terms of the Connes-cocycle, the relative entropy (76) may also be defined as The last expression has the advantage that it does not require one to know the vector representative of |ψy in the natural cone; in particular it shows that S only depends on the state functionals. 7ater we will use the following variational expression for the relative entropy [33] we can write the measured relative entropy in the following way: S meas pω ψ |ω η q " suptSpµ h,ψ |µ h,η q : h P M s.a.u " suptSpω ψ|C |ω η|C q : C Ă M a commutative von Neumann subalgebrau. (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 counting measures on the finite set t1, . . ., nu.

Petz recovery map
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 map between two von Neumann algebras A, B in standard form acting on Hilbert spaces 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 pAq, T : B Ñ A gives a corresponding map T : S pAq Ñ S pBq by ω Þ Ñ T pωq :" ω ˝T .For finite dimensional von Neumann algebras A, B where state functionals are identified with density matrices through ωpaq " Trpωaq, we can think of T as the linear operator on density matrices defined by Tr ωT pbq " Tr T pωqb @b P B.
This operator T is completely positive and trace-preserving.The quantum data processing inequality (DPI) [37] states that where the right side could also be written as Sp T pω ψ q| T pω η qq.
We recall the definition of the Petz-map.Let |η A y be a cyclic and separating vector in the natural cone of a von Neumann algebra A in standard form.Then the KMS scalar product on A is defined as 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 faithful9 has a vector representative |η B y P K in the natural cone.So: |η A y resp.|η B y give KMS scalar products for A resp.B, which we can use to define the adjoint T `: A Ñ B (depending on the choices of these vectors) of the normal, unital and 2-positive 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 wherein σ A is the density matrix of |η A y and σ B " T pσ A q for |η B y.The rotated Petz map, which we call α t η,T : A Ñ B, is defined by conjugating this with the respective modular flows, i.e.
where ς t η,A " Ad∆ it η,A is the modular flow for A, |η A y etc.For finite dimensional matrix algebras, α t η,T gives by duality a linear operator αt η,T acting on density matrices ρ for B, which is αt η,T pρq " An equivalent definition of the rotated Petz map is: 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 t η,T : A Ñ B is defined implicitly by the identity: for all a P A, b P B.
Closely related to the Petz map is the linear map V ψ : K Ñ H defined 10 ω ψ by [32,30] V ψ b|ξ B ψ y :" T pbq|ξ A ψ y pb P Bq.
As in paper II, we introduce a vector valued function the existence and properties of which are established in lem.s 3, 4 in paper II.In particular, |Γ ψ pzqy is holomorphic inside the strip S 1{2 and bounded in the closure S 1{2 in norm by 1. Furthermore, the representation (24) of paper I shows in conjunction with Stone's theorem that this function is strongly continuous on the boundaries of the strip S 1{2 , i.e. for Repzq " 0 or Repzq " 1{2, which is used implicitly below e.g. when we consider integrals involving this quantity along these boundaries.The relation to the Petz map is as follows, paper II, lem.2: 10 As it stands, the definition is actually consistent only when |ξ B ψ y is cyclic and separating.In the general case, one can define [32] instead V ψ pb|ξ B ψ y `|ζyq :" T pbq|ξ B ψ y pb P B, π B 1 pψq|ζy " 0q. (101)

Improved DPI
Our main theorem is: Theorem 1.Let T : B Ñ A be a two-positive, unital (in the sense (92)) linear map between two von Neumann algebras, and let ω ψ , ω η be normal states on A, with ω η faithful.Then Spω ψ |ω η q ´Spω ψ ˝T |ω η ˝T q ě S meas pω ψ |ω ψ ˝T ˝αT,η q. (105) with the recovery channel Remarks: 1) The theorem should generalize to non-faithful ω η by applying appropriate support 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 due to [35].The recovery channel is given explicitly by (99) in this case as an operator on density 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 case where there exists 8 ą c ě 1 such that Note that this implies c ´1ω 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 |ψy " |η h y{}η h } such that }h} ď ln c, and vice versa.As is well known, this furthermore implies that the Connes cocycle rDη B : Dψ B s iz is holomorphic in the two-sided strip tz P C : |Repzq| ă 1{2u and bounded in norm (by c Repzq ) 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 series expansion with bounded operators k l P B such that }k l } ď C l .We set Using [28], cor.12.8, and the definition of the relative entropy in terms of the Connes cocycle, which is one side of the inequality that we would like to prove.The variational expression (116) then gives: where we used |pη k q h y " |η k`h y see [28], thm.12.10.To get the desired DPI we will establish an upper bound on ln }η h`k } 2 .In lem. 1, we take |Gpzqy " e zh |Γ ψ pzqy, p 0 " 8, p 1 " 2 where θ " 1{n with n P 4N and h " h ˚P A. At the lower boundary we have with u B ptq :" rDη B : Dψ B s t P B, u A ptq :" rDη A : Dψ A s t P A the unitary Connes cocycles, where we used }ς t η rT pbqs} " }T pbq} ď }b} (from the positivity of T and ς t η " Ad∆ it η A ) as well as the isomeric identification of L 8 pA, ψq Q a|ψy Þ Ñ a P A proven in [3].Since p θ " n and ln }Gpitq} p 0 ,ψ ď 0 as just shown, we get from lem. 1 using (104) in the third line and Jensen's inequality in the second (noting that the integrand is continuous and uniformly bounded).Taking the lim-sup n Ñ 8, we get using the definition of the recovery channel α T,η : The next lemmas give an expression for the lim-sup: Lemma 5. We have }e h{n Γ ψ p1{nq} n ψ,n " }pe h{n ∆ 1{n η,ψ a n ∆ 1{n η,ψ e h{n q n{4 ψ} 2 , where a n " T prDη B : Dψ B s i{n q ˚T prDη B : Dψ B s i{n q P A `.
Proof of lem.5: From the definitions, ) using the definition of the Connes-cocylce and the fact that rDη B : Dψ B s i{n P B under our assumption (107), see paper II, proof of lem. 4. In the following, let a " e h{n , b " T prDη B : Dψ B s i{n q P A and |ψ A y " |ψy, |η A y " |ηy etc.
By the results of [3] (which hold in the present context since ω ψ is faithful being dominated by the faithful state ω η ), the vector b∆ 1{n η,ψ a|ψy P L n pA, ψq has a polar decomposition b∆ 1{n η,ψ a|ψy " u∆ 1{n φn,ψ |ψy, where }b∆ 1{n η,ψ aψ} n n,ψ " }φ n } 2 and where u P A is a partial isometry.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 pA, ψq, p ě 1 consists of all formal linear combinations of formal expressions of the form wherein Repz i q ě 0, ř i Repz i q ď 1 ´1{p, x i P A, ζ i P H , the formal adjoint of which is defined to be The notation L p,0 pA, ψq is reserved for formal elements A such that ř i Repz i q " 1´1{p in addition to all other conditions.It is then clear that L p,0 pA, ψqL q,0 pM, ψq " L r,0 pM, ψq as formal products where 1{r 1 " 1{p 1 `1{q 1 with 1{p 1 " 1 ´1{p as usual.By [3], lem.7.3, if 1 ď p ď 2, any element A P L p pA, ψq can be viewed as an element of L p 1 pA, ψq in the sense that |ψy P DpAq and A|ψy P L p 1 pA, ψq.11Furthermore, by [3], lem.7.7 (2), if A 1 , A 2 P L p pA, ψq correspond to the same element under this identification, then so do A 1 , A 2 or A 1 B, A 2 B or BA 1 , BA 2 if B P L q,0 pA, ψq (as long as 1{p 1 `1{q 1 ď 1{2, for example).
We now start with the trivial statement that u∆ 1{n φn,ψ " b∆ 1{n η,ψ a in the sense that these elements of L n1 ,0 pA, ψq are identified with the same element of L n pA, ψq.Then repeated application of [3], lem. 7.
Forming successively n{4 products of this equality and applying in each step [3] and the proof of the lemma is complete.
II) Proof in general case: We will now remove the majorization condition (107).This condition has been used in an essential way in most of the arguments so far because without it, the operator k in (109) is unbounded and thus not an element of A. For unbounded operators the Araki-Trotter product formula and the L p -techniques are not available and it seems non-trivial extending them to an unbounded framework.We will therefore proceed in a different way and define a regularization of ω ψ such that the majorization condition (107) holds and such that, at the same time, the desired entropy inequality can be obtained in a limit wherein the regulator is removed.However, it is clear that this regularization must be carefully chosen because the relative entropy is not continuous but only lower semi-continuous.By itself the latter is insufficient for our purposes 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 the relative entropy.As in paper I, we consider a function f ptq, t P R with the following properties.(B) f ptq has an analytic continuation to the upper complex half plane such that the L 1 pRq norm of the shifted function has }f p¨`iθq} 1 ă 8 for 0 ă θ ă 8.
Such functions certainly exist (e.g.Gaussians).We also let f P ptq " P f ptP q for our regulator P ą 0, and we define a regulated version of |ψy by where the first item gives at least "half" of the domination condition (107), the second states in which sense |ψ P y approximates |ψy and the third gives us an upper semicontinuity property of the relative entropy opposite to the usual lower semi-continuity property which holds for generic approximations.We define for small ε ą 0: σpaq " xη|aηy ρ P,ε paq " p1 ´εqxψ P |aψ P y `εxη|aηy.
Proof.This is a straightforward consequence of the variational definition (5.1) of S meas , choosing a near optimal h.
Here, the norm is }ζ} p,ψ " sup In [3], it is assumed for most results that |ψy is cyclic and separating.When using such results in the main text, we will be in that situation.A somewhat different approach replacing the relative modular operator by the Connes spatial derivative and containing also many new results is laid out some detail in [8].

(A) The Fourier transform of f f ppq " ż 8 ´8
e ´itp f ptqdt (134) exists as a real and non-negative Schwarz-space function.This implies that the original function f is Schwarz and has finite L 1 pRq norm, }f } 1 ă 8.
Vectors |ζy in H are thus identified with matrices ζ P M n pCq.H » M n pCq is both a left and right module and the inner product on H is the Hilbert-Schmidt inner product xζ 1 |ζ 2 y " Trpζ 1 ζ 2 q.A mixed state, represented by a density matrix ω, gives rise to a linear functional on A by ωpaq " Trpωaq, 2 is another generalization of this inequality which gives nothing new in the present case.Cor. 3 gives the following inequality.Under the above identification of vectors |ψy P H and matrices, the perturbed vector is |ψ h y " |e ln ψ`h{2 y (72) (assuming |ψy to be in the natural cone, i.e. self-adjoint and non-negative), and then choosing |ψ " 1 n { ?ny as the vector representing the tracial state on A, we have ln Tr e h 1 `¨¨¨`h k ď ż R dt β 0 ptq ln Tr |e p1{2qh 1 e p1{2`itqh 2 ¨¨¨e p1{2`itqh k´1 e p1{2qh k | 2 , (73) The relative entropy between these measures is defined (but can be `8) if suppµ η,a Ă suppµ ψ,a , wherein dµ ψ,a {dµ η,a means the Radon-Nikodym derivative between the measures.We may perform the maximization in over f phq with 8 f P L 8 pR; Rq and h P M s.a. because f phq P M s.a..Maximizing first for fixed h over f and using (" eq.(75) in the commutative case) tω ψ phq ´ln }e h{2 η} 2 u.(80)From the Golden-Thomson inequality (70) we findS meas pψ|ηq ď Spψ|ηq.(81)Smeascanalsobe written in terms of the classical relative entropy Spµ|νq (Kullback-Leibler divergence) of two probability measures as follows.Let a P M s.a.be a self-adjoint element of M. Then it has a spectral decomposition a "ż λE a pdλq(83)with an M-valued projection measure E a pdλq.Given |ψy, |ηy P H , we get Borel measures dµ ψ,a " xψ|E a pdλqψy, and likewise for |ηy.Physically, these correspond to the probability distributions for measument outcomes of a in the states |ψy resp.|ηy.