Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings
UUNIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANNALGEBRAS
MARIUS JUNGE AND NICHOLAS LARACUENTE
Abstract.
Scenarios ranging from quantum error correction to high energy physics use recoverymaps, which try to reverse the effects of generally irreversible quantum channels. The decrease inquantum relative entropy between two states under the same channel quantifies information lost.A small decrease in relative entropy often implies recoverability via a universal map dependingonly the second argument to the relative entropy. We find such a universal recovery map forarbitrary channels on von Neumann algebras, and we generalize to p-fidelity via subharmonicityof a logarithmic p-fidelity of recovery. Furthermore, we prove that non-decrease of relativeentropy is equivalent to the existence of an L -isometry implementing the channel on both inputstates. Our primary technique is a reduction method by Haagerup, approximating a non-tracial,type III von Neumann algebra by a finite algebra. This technique has many potential applicationsin porting results from quantum information theory to high energy settings. Introduction
The quantum channel is a general model of how the state of an open quantum system changeswhen interacting with an initially uncoupled environment. Due to this environmental interaction,the effect of a channel is generally not invertible - it may lose information about the system. Insome special cases, it is nonetheless possible to recover the original input state. For example,quantum error correction defines a ‘code space’ within a larger system, such that perturbations ofstates in the code space are effectively invertible [1, 2]. In the theory of quantum communication[3, 4], one asks how many bits of information one may recover from the output of a quantumchannel with arbitrarily powerful encoding and decoding. Holography in high energy physicsrelies on a reversible map between bulk and boundary theories [5, 6, 7].A key quantity in quantum information is the relative entropy between quantum densitymatrices, denoted D ( ρ | ϕ ) for densities ρ and ϕ . One of the most fundamental inequalities inquantum information theory is the data processing inequality for relative entropy, which statesthat for any quantum channel Φ, D ( ρ | ϕ ) ≥ D (Φ( ρ ) | Φ( ϕ )) . We recall and denote by R ϕ, Φ the Petz recovery map, given by a normalized and re-weightedadjoint of Φ [8, 9]. It is always the case that R ϕ, Φ ◦ Φ( ϕ ) = ϕ . The Petz map for ϕ, Φ sometimes
NL was partially supported by NSF grant DMS 1800872. MJ is partially supported by NSF grants DMS1800872 and Raise-TAQS 1839177. a r X i v : . [ qu a n t - ph ] S e p M. JUNGE AND N. LARACUENTE acts as an inverse on ρ as well. In particular, D ( ρ | ϕ ) = D (Φ( ρ ) | Φ( ϕ )) ⇐⇒ R ϕ, Φ ◦ Φ( ρ ) = Φ( ρ ) . (1)The intuition for data processing is that no stochastic or quantum process may increase the dis-tinction between two probability distributions or densities. Equality of relative entropy faithfullyindicates that Φ also doesn’t destroy any information in ρ relative to ϕ .A natural question is whether a small difference in relative entropy implies approximaterecovery. Holographic theories, for instance, consider approximately invertible maps betweensubsystems of a bulk spacetime and corresponding quantum boundary [6]. Quantum informationapplications such as error correction and communication may work with only approximatelypreserved code spaces, formally outside the strict criteria for perfect recovery via Petz map. Anumber of recent works have begun to quantitatively link relative entropy difference to fidelityof recovered states.A resurgence of activity on approximate recovery started with Fawzi and Renner’s approx-imate Markov chain result [10]. A special form of relative entropy is the conditional mutualinformation on a tripartite system A ⊗ B ⊗ C , given by I ( A : B | C ) ρ = D (cid:16) ρ ABC (cid:13)(cid:13)(cid:13) | A | ⊗ ρ BC (cid:17) − D (cid:16) ρ AC (cid:13)(cid:13)(cid:13) | A | ⊗ ρ C (cid:17) , where ρ BC , ρ AC , and ρ C refer to respective marginals of ρ . Fawzi and Renner show that I ( A : B | C ) ρ ≥ − f ( ρ, R F W ( ρ AC )) , where f ( ρ, ϕ ) = tr ( |√ ρ √ ϕ | ) is the usual fidelity, and for some channel R F W (not necessarily thePetz map). If one can perfectly recover ρ ABC from ρ AC by acting only on C , then the systemis called a quantum Markov chain [11]. In [12], the same inequality is shown for a universalrecovery map, which depends only on ρ AC rather than on ρ ABC . Li and Winter use this form ofrecovery in [13] to show a monogamy of entanglement.Wilde extends approximate recovery to general relative entropy differences in [14], showing D ( ρ | ϕ ) − D (Φ( ρ ) | Φ( ϕ )) ≥ − (cid:16) sup t ∈ R f ( ρ, R tϕ, Φ ( ρ AC )) (cid:17) (2)for a twirled recovery map R tϕ, Φ parameterized by t . In [15], Junge, Renner, Sutter, Wilde, andWinter show that D ( ρ | ϕ ) − D (Φ( ρ ) | Φ( ϕ )) ≥ − (cid:90) R ln f ( ρ, R tϕ, Φ (Φ( ρ ))) dβ ( t ) , (3)where dβ ( t ) = ( π/ πt ) + 1) − dt . Using convexity, one may move the integral inside thelogarithm and fidelity to construct the explicit, universal recovery map given by˜ R ϕ, Φ ( ρ ) = − (cid:90) R R tϕ, Φ ( ρ ) dβ ( t ) . (4) NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 3
Another result by Sutter, Tomamichel and Harrow [16] strengthens the inequality via a pinchedrecovery map. More recently, Carlen and Vershynina show (corollary 1.7 in [17]) that D ( ρ | ϕ ) − D ( E ( ρ ) |E ( ϕ )) ≥ (cid:16) π (cid:17) (cid:107) ∆ ρ,ϕ (cid:107) − (cid:107) R ρ, E ( E ( ϕ )) − ϕ (cid:107) , (5)where ∆ ρ,σ is the relative modular operator, and E is a conditional expectation that restrictsa density to a matrix subalgebra. A recent work by Gily´en, Lloyd, Marvian, Quek, and Wildesuggests a quantum algorithm that implements the Petz recovery map in special cases [18].For recovery’s applications to quantum field theory [19], it would be desirable to extend finite-dimensional results to infinite-dimensional von Neumann algebras, including type III factorsthat lack a finite trace. Applications of recovery appear in finite-dimensional analogs of the theAds/CFT correspondence [6]. Recovery may underpin eventual proofs of ideas related the theRyu-Takayanagi conjecture and analogies to error correction, but field theories are widely believedto be type III, non-tracial algebras, in which much of the finite-dimensional quantum informationmachinery remains conjecture. Two very recent works address the type III extension of recoverymaps. One, by Li and Wilde, extends equations (2) and (5) to the von Neumann algebra setting,also addressing generalizations to optimized f -divergences [20]. Faulkner, Hollands, Swingle, andWang prove an equation in the form of (3) for subalgebraic restriction/inclusion, with applicationsin high energy physics [21].1.1. Primary Contributions.
A motivating result of this work is a universal recovery mapin the style of (3) for channels on all von Neumann algebras. We use a p -generalization of thefidelity similar to that of Liang et al’s in equation (2.14) [22], given by f p ( ρ, ϕ ) = (cid:107)√ ρ √ ϕ (cid:107) p . (6)We denote a twirled recovery map in equivalent form to Wilde’s [14], but parameterized bycomplex z , R zϕ, Φ ( ˆ ρ ) = ϕ ¯ z/ Φ † ( ˆ ϕ − ¯ z/ ˆ ρ ˆ ϕ − z/ ) ϕ z/ , (7)and a logarithmic, twirled p -fidelity of recovery given by F R zϕ, Φ ( ˆ ρ ) = − ln f /Re ( z ) ( ρ Re ( z ) , R zϕ, Φ ( ˆ ρ Re ( z ) )) . (8)For convenience of notation, we may denote R z = R zϕ, Φ when ϕ and Φ are clear from context.Our notion of fidelity of recovery is closely related to that considered earlier in the field [23],though we have included the logarithm in the quantity for convenience. Then we show that: Theorem 1.1.
Let
Φ :
M → N be a normal, completely positive map from von Neumann algebra M to algebra N . Let ρ, σ be densities on M . Then D ( ρ | ϕ ) ≥ D (Φ( ρ ) | Φ( ϕ )) + 2 p (cid:90) R F R (1+ it ) /pϕ, Φ ( ρ ) β ( t ) dt for p ≥ . As with equation (3), we can use convexity of the p -fidelity and negative logarithm to movethe integral inside, constructing an explicit, universal recovery map (see Theorem 5.11). Equation(3) follows as the p = 1 case. Theorem 1.1 follows a more general result for p -fidelity of recovery: M. JUNGE AND N. LARACUENTE
Theorem 1.2.
F R zϕ, Φ is subharmonic. Theorem 1.2 is justified by Remark 5.6 in section 5. Theorem 1.2 converts a mathematicalcomparison from complex interpolation theory into a direct bound on physical quantities.For p = 2 and M ⊂ B ( L ( M )) represented in so-called standard form [24] we may alwaysassume that ρ ( x ) = ( (cid:112) d ρ , x (cid:112) d ρ ) is implemented by its natural ‘purification.’ Then we deduce(see Remark 9.5) that (cid:107) d / ρ − R / ϕ, Φ ( d / ρ ) (cid:107) ≤ D ( ρ | ϕ ) − D (Φ( ρ ) | Φ( ϕ )) . (9)This implies (cid:107) d ρ − R / ϕ, Φ ( d / ρ ) (cid:107) ≤ D ( ρ | ϕ ) − D (Φ( ρ ) | Φ( ϕ ))) . (10)Thus using non-linear recovery maps enables us to obtain a quadratic error formula, whichqualitatively resembles equation (5) and the results in [20].Using the same techniques, we prove a data processing inequality for p -fidelity, that for anyquantum channel Φ and pair of states ρ, ϕ , f p (Φ( ρ ) , Φ( ϕ )) ≥ f p ( ρ, ϕ ) . (11)Finally, we derive a new condition for equality in data processing for states with sharedsupport: Theorem 1.3 (Introduction version of 11.5) . Let ρ, ϕ be states such that ρ ≤ λϕ , and Φ : L ( M ) → L ( ˆ M ) be a quantum channel for von Neumann algebras M, ˆ M . Then the followingare equivalent i) D (Φ( ρ ) | Φ( ϕ )) = D ( ρ | ϕ ) ; ii) There exists a ϕ -conditioned subalgebra M ⊂ M and an completely positive L -isometry u : ˆ M → M such that u ( ϕ ) = Φ( ϕ ) , u ( ρ ) = Φ( ρ ) . Theorem 11.5 is intuitive for finite-dimensional channels with equivalent input and outputspaces, for which perfect recoverability for all states implies unitarity. In the infinite dimensionalsituation and with different input and output spaces Petz’s map gives a precise recovery. However,Theorem 11.5 improves on Petz’s recovery map by providing a local lift from the states spaceof the output to back to the input, motivated by Kirchberg’s work. Assuming equality in anAdS/CFT corresponds, this amount to a an exact lift from boundary to bulk states.The core technique of this paper is a method by Haagerup, Junge & Xu [25]. This techniquerelies on a crossed product construction that embeds a type III von Neumann algebra M within atracial von Neumann algebra ˜ M , and then approximates operators in ˜ M by operators in algebraswith finite trace. The Haagerup approximation method is a general method to transfer resultsfrom tracial to non-tracial settings. We highlight the potential of this technique to yield furtherresults in quantum field theory, where non-tracial algebras are physically relevant.Section 2 reviews relative modular theory formulate the relative entropy. We review themathematical techniques of interpolation theory for Kosaki spaces in section 3 and L p space NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 5 estimates in section 4. In section 5, we prove the necessary properties of twirled recovery, fidelityof recovery, and ultimately Theorem 1.2 for von Neumann algebras with finite trace. Section6 shows some continuity results that become non-trivial in infinite-dimension. In section 8, werecall the Haagerup approximation method and use it to extend the recovery results to generalvon Neumann algebras. Section 10 contains the proof of data processing for p -fidelity. Section11 contains the proof of equivalence between data processing saturation and L -isometries.2. Relative modular operator and entropy
Let M be a von Neumann algebra on Hilbert spaces H and H (cid:48) . Let | ϕ (cid:105) ∈ H and | ρ (cid:105) ∈ H (cid:48) be a pair of normalized vectors for which | ψ (cid:105) is(1) Cyclic, in that { a | ϕ (cid:105) : a ∈ M} is dense in H .(2) Separating, in that if a ∈ M and a | ϕ (cid:105) = 0, then a = 0.The Tomita-Takesaki operator S ϕ,ρ is given by S ϕ,ρ a | ϕ (cid:105) = a † | ρ (cid:105) . S ϕ,ρ has polar decomposition S ϕ,ρ = J ϕ,ρ ∆ / ϕ,ρ , where we call J ϕ,ρ the relative modular conjugation. If | ρ (cid:105) is cyclic and separating as well as | ϕ (cid:105) , then J ϕ,ρ is antiunitary. Regardless, ∆ ϕ,ρ is Hermitian, and it is called the relative modularoperator . Note also that for any density ρ , (cid:107) ρ / (cid:107) = (cid:107) ρ (cid:107) / . Hence we may naturally define∆ ϕ,ρ for a pair of density matrices, and where appropriate will let | ρ (cid:105) = ρ / and | ϕ (cid:105) = ϕ / . Infinite dimension, ∆ ϕ,ρ ( x ) = ρ − xϕ for any x ∈ M . Importantly, for any z ∈ C and x ∈ M , ∆ zϕ,ρ is analytic, and ∆ zϕ,ρ ( x ) ∈ M . The form ∆ itϕ,ρ is analogous to a unitary time-evolution and leadsto the interpretation of ln ∆ ρ,ϕ as a modular Hamiltonian in quantum field theory.We may define the relative entropy as D ( ρ (cid:107) ϕ ) = − ( ρ / , ln(∆ ϕ,ρ ) ρ / ) , taking the inner product on H . In finite dimension, the expression D ( ρ (cid:107) ϕ ) = tr( ρ ln ρ − ρ ln ϕ )also holds and is more common. The definition in terms of modular operators is however preferreddue to its generality to arbitrary von Neumann algebras, holding for instance in non-tracial fieldtheories where von Neumann entropies necessarily diverge. Furthermore, the relative modularoperator was one of the earliest [26] and continues to be [27] a common and intuitive proof ofthe data processing inequality for relative entropy. For more information on modular theory, see[28, 29] and [19] for its field theory applications.3. Preliminary facts on Kosaki L p spaces and interpolation The starting point for Kosaki’s work is normal faithful state ϕ on a von Neumann algebra M . The GNS construction for ϕ allows us to define L ( N, ϕ ) = N (cid:107) (cid:107) M. JUNGE AND N. LARACUENTE as the closure with respect to the Hilbert space norm given by the inner product( a, b ) ϕ = ϕ ( a ∗ b ) . Note that we have a natural inclusion N ⊂ L ( N, ϕ ) and therefore the complex interpolationspace L p ( N, ϕ ) = [
N, L ( N, ϕ )] / p is well-defined and a left N module, i.e. (cid:107) ax (cid:107) p ≤ (cid:107) a (cid:107) N (cid:107) x (cid:107) p . We refer to [30] for general facts on complex interpolation. Indeed, for two Banach spaces A , A ⊂ V embedded in a common topological vector space, the interpolation norm (cid:107) x (cid:107) θ = inf F ( θ )= x (cid:107) F (cid:107) ∞ is obtained by taking continuous functions F : { ≤ (cid:60) ( z ) ≤ } → V such that(1) F is analytic in { z | < (cid:60) ( z ) < } ;(2) F ( i R ) ⊂ A , F (1 + i R ) ⊂ A ;(3) F ( z ) converges to 0 for | z | → ∞ ;(4) F has finite L ∞ -norm (cid:107) F (cid:107) ∞ = max { sup t (cid:107) F ( it ) (cid:107) A , sup t (cid:107) F (1 + it ) (cid:107) A } < ∞ . Although the L ∞ norm is very useful in studying duality for interpolation space, it is possibleto modify the definition without changing the spaces. For this let µ θ be the unique measure onthe boundary of the strip St = { z | ≤ (cid:60) ( z ) ≤ } such that g ( w ) = (cid:90) ∂St g ( z ) dµ w ( z ) . We refer to [30] for the following well-known fact (sometimes called Hirschmann’s Lemma).
Lemma 3.1.
Let w = θ + is (1) Let F be an analytic function vanishing at infinity. Then log (cid:107) a ( w ) (cid:107) [ A ,A ] θ ≤ (cid:90) ∂S log (cid:107) F ( z ) (cid:107) A (cid:60) ( z ) dµ θ ( z ) . (2) µ w ( i R ) = 1 − θ , µ w (1 + i R ) = θ ; (3) µ i R = f w H and µ i R = f w H is absolutely continuous with respect to the -dimensionalHausdorff measure, and moreover f w ( it ) = e π ( s − t ) sin πθ sin ( πθ ) + (cos( πθ ) − e − π ( s − t ) ) and f w (1 + it ) = e π ( s − t ) sin πθ sin ( πθ ) + (cos( πθ ) + e − π ( s − t ) ) . NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 7
Remark 3.2.
In our case the topological vector space is V = L ( A, ϕ ) . In particular, let N ⊂ B ( H ) be a normal representation of N and ξ be a vector representing ϕ . Then L ( N, ϕ ) =
N ξ H = H ϕ ( N ) is an invariant subspace of H . In particular, the space L p ( N, ϕ ) ⊂ H ϕ ( N ) are all continuously embedded in H ϕ ( N ) ⊂ H . Note that our assumption of ϕ being faithful isequivalent if ξ being separating for N .If in addition ξ is cyclic, then H ϕ ( N ) = H . However, in some physical applications given bya quantum filed theory where N is the von Neumann algebra of generator by the observables A O of a certain space (time) region O , it is warranted to consider non-cyclic vectors. One of Kosaki’s main contribution is to identify L p ( N, ϕ ) with the Haagerup L p space L p ( N ),which is defined as follows. Let σ ϕt : N → N the one parameter group of automorphism givenby the modular group of ϕ (for more details see below). Note that (cid:107) σ ϕt ( a ) (cid:107) = (cid:107) a (cid:107) and hence σ t are also unitaries u t = σ t in L ( N, ϕ ). Then we may consider the crossed product M = N (cid:111) R ⊂ B ( L ( R , L ( N, ϕ ))) generated by { u t } and π ( N ) given by the representation π ( f )( h )( t ) = σ − t ( f )( h ( t )) . By Tomita-Takesaki theory (see [31]) it is known that the weight w ( T ) = (cid:90) R ( ξ ϕ , σ t ( T ) ξ ϕ ) dt admits an inner modular group σ wt ( T ) = u ∗ t ( T ) u t for T ∈ M and hence M admits a normalfaithful trace, T r . Moreover, the dual automorphism group θ s ( π ( x )) = π ( x ) , θ s ( u t ) = e ist u t satisfies θ s ( T ) = T ⇔ T ∈ π ( N ) . (12) Theorem 3.3. (Haagerup-Terp) Let < p < ∞ and L p ( N ) be the set of tr measurable operatorssuch that θ s ( a ) = e − s/p a . Then i) L p ( N ) is a linear subspace and an N -bimodule; ii) a ∈ L p ( N ) admits a polar decomposition a = ub /p with b ≥ and b ∈ L ( N ) , u ∈ π ( N ) a partial isometry; iii) There is a 1-1 correspondence between L ( N ) + = { a ∈ L ( N ) | a ≥ } and N + ∗ . iv) T r ( ub ) = ϕ ( b )( u ) extends to a trace on L ( N ) ; iv) (cid:107) a (cid:107) p = T r ( | a | p ) /p defines a norm for ≤ p < ∞ , and a quasi norm for < p < . v) L p ( N ) ∗ = L p (cid:48) ( N ) for ≤ p < ∞ and duality bracket ( a, b ) = T r ( ab ) . M. JUNGE AND N. LARACUENTE
We should mention that Haagerup’s L p spaces carry a natural order structure, and satisfyTakesaki’s paradigm a ∈ L p ( N ) ⇔ a = uϕ /p where ϕ is a positive functional. Here we have to read ϕ /p = d /pϕ , where d ϕ is the uniqueelement such that ϕ ( x ) = T r ( xd ϕ )holds for the Haagerup trace. We will reserve the letters T r for the Haagerup trace.
Remark 3.4.
Let δ > δϕ ≤ ρ ≤ δ − ϕ . The operator d itϕ is a unitary in M , notnecessarily in N . However, g ϕ,ρ ( it ) = d itϕ d − itρ satisfies θ s ( g ϕ,ρ ( it )) = g ϕ,ρ ( it ) and hence does belong to π ( N ) ∼ = N . In fact for z = θ + it , θ ≤ / d θϕ ≤ δ − θ d − θρ that (cid:107) d θρ d − θρ (cid:107) = (cid:107) d − θρ d θϕ d − θρ (cid:107) ≤ δ − θ is bounded. This implies that on { z | < (cid:60) ( z ) < } the function g ϕ,ρ ( z ) = d zϕ d − zρ is well-defined and analytic and, thanks to θ s ( g ϕ,ρ ( z )) = ( e zs d zϕ )( e − zs d zϕ ) = g ϕ,ρ ( z )has values in M . Remark 3.5.
The same argument applies to modular semigroup σ ϕ,ρt ( π ( x )) = d itϕ π ( x ) d − itρ which satisfies θ s ( σ ϕ,ρt ( π ( x )) = σ ϕ,ρt ( π ( x )) and g ϕ,ρ ( it ) = σ ϕ,ρt ( π (1)) ∈ π ( N ) . Moreover, let σ ϕ,ρz be the unique linear extension of the modular group then g ϕ,ρ ( z ) = σ ϕ,ρz (1) ∈ N at least for 0 ≤ (cid:60) ( z ) ≤ / ρ ∈ N ∗ and d ρ ∈ L ( N ) be the corresponding density. Following Kosaki we may considerthe interpolation pair L θp ( N , ρ ) = [ ι θ ( N ) , L ( N )] /p , where the inclusion is given by ι θ ( x ) = d − θρ xd θρ . Note that in full generality this map is not faithful. In [32], Kosaki establishes the link betweenthe interpolation spaces L p ( N, ϕ ) and Haagerup’s L p ( N ) spaces. NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 9
Proposition 3.6.
Let ≤ p ≤ ∞ and d ϕ be the density of a normal faithful state and ξ ϕ theGNS vector representing ϕ . Then i p ( xξ ϕ ) = xd / pϕ extends to isometric isomorphism between [ N, L ( N )] /p and L p ( N ) . Moreover, the map ι η,p ( x ) = d (1 − η ) /pϕ xd η/pϕ extends to an isometry between [ ι η, ( N ) , L ( N )] /p and L p ( N ) . Following this definition we may define the family of norms on N (cid:107) x (cid:107) η,p = (cid:107) d − η/pϕ xd η/pϕ (cid:107) p and observe that they form an interpolation family. The case η = 1 corresponds and 2 ≤ p < ∞ is a special case of this more general construction. Indeed, using (cid:107) xξ ϕ (cid:107) L p := (cid:107) xd /pϕ (cid:107) L p ( N ) , we extend Masuda(?)’s definition of L p norms as norms defined for a dense class of N ξ ϕ ⊂ H ϕ ( N ).For η = 0 and p ≥ a, b ] ϕ = ϕ ( ab ∗ ) and then complexinterpolation (cid:107) x (cid:107) L p = (cid:107) d /pϕ x (cid:107) p = (cid:107) x ∗ d /pϕ (cid:107) p shows that ι p ( x ) = d /p x gives an isometry[ N op , L ( N op )] /p ∼ = L p ( N, ϕ ) ∼ = L p ( N ) . Theorem 3.7 (Kosaki) . For θ = 1 , the map ι p ( x ) = xd p ρ extends to a completely isometricisomorphism between L p ( N , ρ ) and the complemented subspace L p ( N e ) of the Haagerup L p spaces L p ( N ) . Here e is the support of ρ . For our purpose we need a slight extension of Kosaki’s L p spaces for non-faithful states ϕ with support projection e . This can easily be obtained by approximation. Let us assume that N is σ -finite and ψ is a normal faithful state. Then D = d ϕ + (1 − e ) d ψ (1 − e )is a faithful normal density in L ( N ). Note D commutes with e . Corollary 3.8.
The norms (cid:107) x (cid:107) p = (cid:107) xd /pϕ (cid:107) L p ( N ) form an interpolation family on N e for ≤ p ≤ ∞ . Proof.
Recall that (cid:107) x (cid:107) L p ( N,D ) = (cid:107) xD /p (cid:107) L p ( N ) form an interpolation family and the space L p ( N ) e is complemented in the Haagerup L p space.Then we observe that ι D,p ( x ) e = xD /p e = xeD /p = xed /p = ι d,p ( xe ) . This shows that R e ( x ) = xe extends to a contraction from L p ( N, D ) to L p ( N e, d ).4. L p estimates for channels In this section we present a priori estimates on L p spaces which are required to formulate therecovery Theorem in the von Neumann algebra setting. The arguments are very closely relatedto the first author’s lecture notes for proving the data processing inequality for the sandwichedentropy.Our starting point is a completely positive trace preserving map Φ : L ( M ) → L ( ˆ M ). Recallthat the anti-linear duality bracket ( x, d ) = T r ( xd ∗ )allows us to identify ¯ M ∗ with L ( M ) and hence(Φ † ( x ) , d ) = T r ( x Φ( d ) ∗ )defines a normal unital completely positive map Φ † : ˆ M → M . The following fact is well-known.Since it is crucial for all our arguments we indicate a short proof. Lemma 4.1.
Let
Ψ :
M → M be a normal completely positive unital map. Then there exists aHilbert space H normal ∗ -homomorphism π : M → B ( H ) ¯ ⊗ M , and a projection e ∈ B ( H ) suchthat Ψ( x ) = ( e ⊗ π ( x )( e ⊗ . Proof.
We will use the standard GNS construction, see [33, 34]. Let K = M ⊗ Φ M the theHilbert C ∗ -module over M with inner product( a ⊗ x, b ⊗ y ) = x ∗ Ψ( a ∗ b ) y . Let ˜ K be the closure of K in the strong operator topology of the module (see [35]). Then ˜ K admitsa module basis and hence is of the form ˜ K = f ( H ¯ ⊗ M ) for some projection f ∈ B ( H ) ¯ ⊗ M . Thesubspace 1 ⊗ M ⊂ K is an M right module and hence the orthogonal projection q onto (1 ⊗ M )is in in ( M op ) (cid:48) = B ( H ) ¯ ⊗ M . We may define ∗ -representation, see [33] π ( α )( a ⊗ x ) = αa ⊗ x Then we deduce that for e = qf we haveΨ( x ) = eπ ( x ) e . NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 11
It remains to show that π extends to the strong closure of ˜ K , and that π is normal. For simplicitywe assume that ϕ is a normal faithful state and define the Hilbert space L ( K , ϕ ) via the innerproduct ( ξ, η ) ϕ = ϕ (( ξ, η )) . Note that L ( ˜ K, ϕ ) = L ( K , ϕ ) and the inclusion ˜ K ϕ ⊂ L ( ˜ K, ϕ ) is dense, faithful because ϕ isfaithful. Then we see that for all a, b, x, y the function ω a,b,x,y ( α ) = ϕ ( x ∗ Ψ( a ∗ αb ) y )is normal, thanks to Ψ being normal. By norm approximation, we deduce that π extends to anormal representation on L ( K , ϕ ) = L ( ˜ K, ϕ ). Since this is true for all ϕ , we see that π extendsto a representation on the closure ˜ K . Finally, we observe that weak ∗ closure of the adjointablemaps on ˜ K satisfies L w ( ˜ K ) = e ( B ¯ ⊗ M ) e . Since our map π : M → L w ( ˜ K ) is normal, we see that, after identification, that π : M → ( B ¯ ⊗ M )is a normal, not necessarily unital ∗ -homomorphism.In the following, we will fix Φ : L ( M ) → L ( ˆ M ), Ψ = Φ † : ˆ M → M , e ∈ B ( B ( H )) ¯ ⊗ M = ˜ M and the normal ∗ -homomorphism π : ˆ M → ˜ M . Lemma 4.2.
Let Φ( ϕ ) = ˆ ϕ with support s ( ϕ ) , s ( ˆ ϕ ) respectively. Then for all ≤ p ≤ ∞ . (cid:107) π ( y ) es ( ϕ ) (cid:107) L p ( ˜ M,ϕ ) ≤ (cid:107) ys ( ˆ ϕ ) (cid:107) L p ( ˆ M, ˆ ϕ ) . Proof.
Since Φ is trace preserving we note that (cid:107) π ( y ) e (cid:107) L ( ϕ )) = T r ( d ϕ eπ ( y ∗ y ) e ) = T r ( d ϕ Ψ( y ∗ y ))= T r (Φ( d ϕ ) y ∗ y ) = (cid:107) y (cid:107) L ( ˆ ϕ ) . Thus interpolation according to Lemma 3.8 implies the assertion.
Proposition 4.3.
Let d ∈ L ( N ) be the density of a state ϕ and ˆ d = Φ( d ) , with support s = s ( d ) and ˆ s = s ( ˆ d ) . Let ≤ p ≤ ∞ . Then R p ( x ) = d / p Φ † ( ˆ d − / p x ˆ d − / p ) d / p extends to contraction from L p ( ˆ M )) to L p ( M ) .Proof. Let us recall the abstract (Markinciewicz) interpolation theorem: Let ( A , A ) ⊂ V ,( ˆ A , ˆ A ) ⊂ ˆ V be interpolation couples and T : A + A → ˆ A + ˆ A be a linear map such that T ( A ) ⊂ ˆ A and T ( A ) ⊂ ˆ A . Then (cid:107) T : A θ → ˆ A θ (cid:107) ≤ (cid:107) T : A → ˆ A (cid:107) − θ (cid:107) T : A → ˆ A (cid:107) θ . For the proof one considers the analytic function G ( z ) = T ( F ( z )), and then takes the infimumover F such that F ( θ ) = x . In our situation A = ˆ s ˆ M ˆ s and A = ˆ sL ( ˆ M )ˆ s , ˆ A = sM s , ˆ A = sL ( M ) s . The map is given by T ( ˆ d / x ˆ d / ) = d / Φ † ( x ) d / . We also use the map T ∞ ( x ) = s Φ † ( x ) s , and observe the following commuting diagramˆ s ˆ M ˆ s T ∞ → M ↓ ι p, ˆ d ↓ ι p,d ˆ sL p ( ˆ M )ˆ s R p → L p ( M ) ↓ γ p (cid:48) , ˆ d ↓ γ p (cid:48) ,d ˆ sL ( ˆ M )ˆ s T → L ( M )Here γ p,d ( x ) = d / p (cid:48) xd / p (cid:48) is chosen such that γ p,d ι p,d = ι ,d is the symmetric Kosaki embedding.We may think of T ∞ as a densely defined map on ι (ˆ s ˆ M ˆ s ). Thus it remains to show that ι isindeed a contraction. By H¨older’s inequality the map q : L ( ˆ M ) ⊗ L ( ˆ M ) → L ( M ), q ( x ⊗ y ) = xy is a contraction, and indeed a metric surjection, because the adjoint q ∗ : ˆ M → B ( L ( ˆ M )) isisometric. The same is true for ˆ q ( x ⊗ y ) = ˆ sxy ˆ s as a map ˆ q : ˆ sL ( ˆ M ) ⊗ L ( ˆ M )ˆ s → ˆ sL ( ˆ M )ˆ s .Note that ˆ M ˆ d / is dense in L ( M ). This shows that the set D of elementsˆ x ˆ d / xy ˆ d / , (cid:107) ˆ d / x (cid:107) < , (cid:107) y ˆ d / (cid:107) < sL ( ˆ M )ˆ s . Then we recall that (cid:107) π ( y ) ed / (cid:107) = T r ( d Φ † ( y ∗ y )) = T r ( ˆ dy ∗ y ) = (cid:107) y ˆ d / (cid:107) . Taking ∗ ’s we see that similarly (cid:107) d / eπ ( x ) (cid:107) = (cid:107) ˆ d / x (cid:107) . Let u ∈ M be contraction. Then wededuce (where T r is the Haagerup trace) that
T r ( uT ( ˆ d / xy ˆ d / )) = T r ( ud / Φ † ( xy ) d / )= T r ( ud / eπ ( xy ) ed / )= ( π ( x ) ed / , π ( y ) ed / u ) . Thanks to the right module property of L ( M ) we deduce | T r ( uT ( ˆ d / xy ˆ d / ) | ≤ (cid:107) π ( x ) ed / (cid:107) L ( ˜ M ) (cid:107) π ( y ) ed / u (cid:107) L ( ˜ M ) ≤ (cid:107) π ( x ∗ ) ed / (cid:107) L ( ˜ M ) (cid:107) π ( y ∗ ) ed / (cid:107) L ( ˜ M ) (cid:107) u (cid:107) = (cid:107) u (cid:107) (cid:107) ˆ d / x (cid:107) (cid:107) y ˆ d / (cid:107) . Taking the supremum over (cid:107) u (cid:107) ≤
1, we deduce that T ( D ) belongs to the unit ball of L ( M ),and hence T extends to a contraction on ˆ sL ( M )ˆ s . By the abstract Markinkiewic theorem wededuce R p is also a contraction, and the continuous extension of the map R p ( ˆ d / p x ˆ d / p ) = d / p Φ † ( x ) d / p .As an application, we deduce the contraction property of the (twirled) Petz recovery maps,on L p : Lemma 4.4.
Let ϕ be a state and ˆ ϕ = Φ( ϕ ) the image under ϕ with support ˆ e . Then R z (ˆ x ) = ϕ ¯ z/ Φ † ( ˆ ϕ − ¯ z/ ˆ x ˆ ϕ − z/ ) ϕ z/ extends to a (completely) bounded operator on L p ( z ) ( ˆ M ) with values in L p ( z ) ( M ) for p ( z ) = Re ( z ) . Proof.
First, we handle the semifinite case. Let Λ ˆ ϕ,p ( z ) = ˆ ϕ / p ( z ) ˆ M ˆ ϕ / p ( z ) be the image of thesymmetric Kosaki map in L p ( z ) (ˆ e ˆ M ˆ e ). We consider Kosaki’s right-sided interpolation space L p ( z ) = [ ˆ M , L ( ˆ M , ˆ ϕ )] /p ( z ) . For an element ˆ x ∈ L p ( z ) of norm <
1. we can find an analytic function g ( z ) ∈ ˆ M ˆ e such that (cid:107) g ( it ) (cid:107) ∞ ≤ , ˆ ϕ ( g (1 + it ) ∗ g (1 + it )) ≤ t . This allows us to consider G ( z ) = π ( g ( z )) e N ∈ L ( H M )and deduce that (cid:107) G ( z ) (cid:107) L p ( z ) ( L ( H M ) ,ϕ ) ≤ . Indeed, this is obvious for z = it . For z = 1 + it we note that (cid:107) G (1 + it ) (cid:107) L ( L ( H M ) ,ϕ ) = (cid:107) ϕ / G (1 + it ) ∗ G (1 + it ) ϕ / (cid:107) = T r ( ϕ / Φ † ( g ( z + it ) ∗ g (1 + it )) ϕ / )= T r (Φ( ϕ ) g (1 + it ) ∗ g (1 + it )) = (cid:107) g (1 + it ) (cid:107) L ( ˆ M, ˆ ϕ ) ≤ . There we have shown that V z : L p ( z ) ( ˆ M ˆ e ) → L p ( z ) ( L ( H M )), V z (ˆ x ˆ ϕ z/ ) = π (ˆ x ) eϕ z/ extends to a contraction on L p ( z ) ( ˆ M ˆ e ) with values in L p ( z ) ( L ( H M )). Now, we consider anelement ˆ x ∈ Λ p ( z ) , ˆ σ ( ˆ M ). Note that L p ( ˆ M ) = L p ( ˆ M ) L p ( ˆ M ), i.e. we can write ˆ x = ˆ x ˆ x suchthat ˆ e ˆ x = ˆ x and ˆ x ˆ e = ˆ x . By the argument above we know that (cid:107) R z (ˆ x ∗ j ˆ x j ) (cid:107) p ( z ) = (cid:107) ( V z (ˆ x j ) ∗ V z ˆ x j ) (cid:107) p ( z ) ≤ (cid:107) V z (ˆ x j ) (cid:107) p ( z ) ≤ (cid:107) x j (cid:107) p ( z ) holds for j = 1 ,
2. Therefore (cid:107) R z (ˆ x ∗ ˆ x ) (cid:107) p ( z ) ≤ (cid:107) ( V z ˆ x ) ∗ (cid:107) p ( z ) (cid:107) V z (ˆ x ) (cid:107) p ( z ) ≤ (cid:107) ˆ x (cid:107) p ( z ) (cid:107) ˆ x (cid:107) p ( z ) . Taking the infimum over all such decompositions, implies the assertion.In Haagerup spaces, let z = θ + it and p = θ − . Then we have a factorization R z = σ d − t R p σ ˆ dt . Here we use the L p version of the modular group σ dt ( x ) = e − itd xe itd . Note that θ s ( σ dt ( x )) = e − itd e itd e − s/p x = e − s/p x . Thus, by the definition of the Haagerup L p space, σ dt is a contraction with inverse σ d − t . Kosaki L p spaces provide an extremely convenient tool to prove data processing inequalitiesfor the sandwiched relative entropy. Let us briefly sketch this argument. Indeed, let Φ : L ( M ) → L ( ˆ M ) be a completely positive trace preserving map and ϕ a normal faithful sate, which wecall the reference state. Let ˆ ϕ = Φ( ϕ ) be the image with support ˆ e . By continuity Φ( L ( M )) ⊂L (ˆ e ˆ M ˆ e )ˆ e and hence we and will assume ˆ e = 1. We obtain an induced map Φ ∞ : M → ˆ M givenby ˆ ϕ / Φ ∞ ( x ) ˆ ϕ / = Φ( ϕ / xϕ / )More generally, it is easy to show by interpolation that the mapΦ p ( ϕ / p xϕ / p ) = ˆ ϕ / p Φ ∞ ( x ) ˆ ϕ / p is a contraction. Of course interpolation applies exactly because Λ p ( ϕ ) = ϕ / p M ϕ / p is densein the image of the symmetric Kosaki map ι / p : [ ι / ( M ) , L ( M )] /p → L p ( M ).We refer to [36] for the fact that Φ ∞ is indeed a normal completely positive unital map.Therefore Φ ∞ admits a Stinespring dilationΦ ∞ ( x ) = eπ ( x ) e , where π : M → L ( H ˆ M ) is obtained from the W ∗ -module M ⊗ Φ ∞ ˆ M . Lemma 4.5.
Let ≤ p ≤ ∞ and y ∈ M . Then (cid:107) π ( y ) e (cid:107) L p ( L , ˆ ϕ ) ≤ (cid:107) y (cid:107) L p ( M,ϕ ) . Indeed, for p = ∞ this is obvious and for p = 2 we have (cid:107) π ( y ) e (cid:107) = ˆ ϕ ( eπ ( y ∗ y ) e ) = ˆ ϕ (Φ ∞ ( y ∗ y )) = T r ( ϕ / Φ ∞ ( y ∗ y ) ˆ ϕ / )= T r (Φ( ϕ / y ∗ yϕ / ) ≤ T r ( ϕ / y ∗ yϕ / ) . Here we only had to use the trace-reducing property of the original map Φ. In combination withKosaki’s embedding result we deduce that (cid:107) ˆ ϕ − / p (cid:48) Φ( ϕ / y ∗ yϕ / ) ˆ ϕ − / p (cid:48) (cid:107) p = (cid:107) ˆ ϕ − / p (cid:48) ˆ ϕ / Φ ∞ ( y ∗ y ) ˆ ϕ / ˆ ϕ − / p (cid:48) (cid:107) p = (cid:107) ˆ ϕ / p Φ ∞ ( y ∗ y ) ˆ ϕ / p (cid:107) p = (cid:107) π ( y ) e (cid:107) L p ( L , ˆ ϕ ) ≤ (cid:107) y (cid:107) L p ( M,ϕ ) = (cid:107) ϕ / p y ∗ yϕ / p (cid:107) p = (cid:107) ϕ − / p (cid:48) ϕ / y ∗ yϕ / ϕ − / p (cid:48) (cid:107) p . Thus by density we deduce the sandwiched p -Renyi data processing inequality: Theorem 4.6.
Let ϕ be faithful and ≤ p ≤ ∞ . Then (cid:107) Φ( ϕ ) − / p (cid:48) Φ( η )Φ( ϕ ) − / p (cid:48) (cid:107) p ≤ (cid:107) ϕ − / p (cid:48) ηϕ − / p (cid:48) (cid:107) p for all η ∈ L ( M ) . Here (cid:107) · (cid:107) p may refer to Haagerup L p norms and − / p (cid:48) to the pseudo inverseon the support. In terms of sandwiched R´enyi entropy, the inequality is equivalent to D p (Φ( η ) | Φ( ϕ )) ≤ D p ( η | ϕ ) . NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 15 p -fidelities and interpolation A main tool in our analysis of recovery maps will be given by a new definition of p -fidelityfrom [22] F p ( x, y ) = (cid:107)√ y √ x (cid:107) p max (cid:107) x (cid:107) p , (cid:107) y (cid:107) p and for x, y ∈ L p f p ( x, y ) = (cid:107)√ x √ y (cid:107) p . Lemma 5.1.
Let ≤ p ≤ ∞ and ϕ be faithful. Let E : ˜ M → M be a conditional expectationand ˜ ρ = ρ ◦ E , ˜ ϕ = ϕ ◦ E such that ˜ ϕ is also faithful. Then f p ( ˜ ρ /p , ˜ ϕ /p ) = f p ( ρ /p , ϕ /p ) . Proof.
We have to rewrite fidelity by duality as follows f p ( x, y ) = sup (cid:107) z (cid:107) p (cid:48) ≤ T r ( z ∗ x / p y / p )= sup (cid:107) ay /p (cid:48) (cid:107) p (cid:48) ≤ T r ( y / a ∗ x / p y / y − / p )= sup (cid:107) ay /p (cid:48) (cid:107) p (cid:48) ≤ T r ( ay / , ∆ / px,y ( y / )) . According to our assumption M ⊂ ˜ M and also M ( M ) ⊂ M ( ˜ M ). According to Connes’ 2x2matrix trick (see [37]) we know that L ( M ( M )) ⊂ L ( M ( ˜ M )). By approximation we mayassume that ρ and hence ˜ ρ are also faithful. Then ψ ( x ) = ρ ( x )+ ϕ ( x )2 is a faithful state on M ( M ) and ˜ ψ = ψ ◦ E is the corresponding extension. We also have a canonical embedding ι : L ( M ( M )) → L ( M ( ˜ M )) given by ι ( xd / ψ ) = x ˜ d / ψ (see [37]). Moreover, we have thefollowing commutation relation ι ◦ σ ψt = σ ˜ ψt ι , which implies ι ∆ zψ = ∆ z ˜ ψ ι . Let us also recall that for the matrix unit e = | (cid:105)(cid:104) | we have e ⊗ ∆ ρ,ϕ ( ξ ) = ∆ ψ ( e ⊗ ξ ) . In particular, ι ( d / ϕ ) = d / ϕ and∆ / p ˜ ρ, ˜ ϕ ( d / ϕ ) = ∆ / p ˜ ρ, ˜ ϕ ( ι ( d / ϕ )) = ι (∆ / pρ,ϕ ( d / ϕ ) . Now, it is easy to conclude. The map ι p (cid:48) ( ad / ϕ ) = ad / ϕ extends to an isometric embedding of L p (cid:48) ( M ) ⊂ L p (cid:48) ( ˜ M ) and hence f p ( ρ /p , ϕ /p ) ≤ f p ( ˜ ρ /p , ˜ ϕ /p ) . On the other hand for a ∈ ˜ M , we see that for x ∈ M we have( ad / ϕ , xd / ϕ ) = ( E ( a ) d / ϕ , xd / ϕ ) . Since the conditional expectation is extends to a contraction E p (cid:48) ( ad ˜ ϕ /p (cid:48) ) = E ( a ) d /p (cid:48) ϕ , we alsofind the reverse inequality f p ( ˜ ρ /p , ˜ ϕ /p ) ≤ f p ( ρ /p , ϕ /p ).5.1. Interpolation formula for comparable states.
In the following we will assume that ϕ and ρ are densities in L ( M ) such that δϕ ≤ ρ ≤ δ − ϕ . Formally we should probably write d ϕ for the density such that ϕ ( x ) = tr ( xd ϕ ) holds for all x , but we decided to follow Takesaki’s convention. Let Φ : L ( M ) → L ( ˆ M ) be a completelypositive and (sub-)trace preserving map, i.e. the dual map Φ † : ˆ M → M defined by T r (Φ † ( x ∗ ) ϕ ) = T r ( x Φ( ϕ ))is completely positive and (sub-)unital. Let us recall the Stinesping factorizationΦ † ( x ) = eπ ( x ) e for some normal ∗ -homomorphism π : ˆ M → B B ( H ) ¯ ⊗ M and some projection e ∈ M (cid:48) . We will usethe notation ˜ M = e ( B ( H ) ¯ ⊗ M ) e and f for the support of ϕ and ˆ f for the support of ˆ ϕ = Φ( ϕ )or ˆ ρ = ϕ ( ρ ). Indeed, by positivity, δ Φ( ϕ ) ≤ Φ( ρ ) ≤ δ − ϕ ( ϕ )shows that the support projections (both in ˆ M ) coincide. Lemma 5.2.
Let ≤ q , q and q ( θ ) = − θq + θq . Let β θ be the probability density representing θ on the boundary of the strip { ≤ (cid:60) ( z ) ≤ } given by β θ ( t ) = sin ( πθ )2 θ (cosh( πt ) + cos( πθ )) . Then G ( z ) = π ( ˆ ρ z/ ˆ ϕ − z/ ˆ f ) ef ϕ z/ ρ − z/ is analytic in ˜ M and i) For all θ in the complex strip, ln (cid:107) G ( θ ) (cid:107) L q ( θ ) ( ˜ M,ρ ) ≤ (1 − θ ) (cid:90) ln (cid:107) G ( it ) (cid:107) L q ( ˜ M,ρ ) β − θ ( t ) dt + θ (cid:90) ln (cid:107) G (1 + it ) (cid:107) L q ( ˜ M,ρ ) β θ ( t ) dt ;ii) (cid:82) − ln (cid:107) G (1 + it ) (cid:107) q β θ ( t ) dt ≤ − ln (cid:107) G ( θ ) (cid:107) q ( θ ) θ ; iii) (cid:82) − ln (cid:107) G (1 + it ) (cid:107) q β ( t ) dt ≤ lim inf θ → − ln (cid:107) G ( θ ) (cid:107) q ( θ ) θ . NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 17
Proof.
Let us recall that µ θ is the unique measure such that f ( θ ) = (1 − θ ) (cid:90) f ( it ) dµ − θ ( t ) + θ (cid:90) f (1 + it ) dµ θ ( t ) . (13)Therefore i) is a reformulation of Lemma 3.1 so that dµ θ (1 + it ) = 1 θ β θ ( t ) dt, dµ − θ ( t ) = 11 − θ β − θ ( t ) dt . The analyticity of G follows from Remark 3.4 and (cid:60) ( z ) ≤
1. For z = it the element ˆ ρ it ˆ ϕ − it is inˆ M and a partial isometry, the same applies to ϕ it ˆ ρ − it and hence (cid:107) G ( it ) (cid:107) L q ( ˜ M ) ,ρ ) ≤ T r ( ρ ) ≤ . Thus ln (cid:107) G ( it ) (cid:107) q ≤
0. Dividing by − θ yields ii). The function h ( t ) = − ln (cid:107) G (1 + it ) (cid:107) q is continuous, lim θ → πθ ) θ converges to 1 /π and the measures β θ are uniformly bounded by Ce −| t | . Thus the dominated convergence theorem implies the assertion (see [15] for calculationof β ).Let us fix 0 < q < q and 1 q θ = 1 − θq + θq . Then Hirschmann’s improvement (see [30, page 93]) tells us thatln (cid:107) G ( θ ) (cid:107) q ( θ ) ≤ (1 − θ ) (cid:90) ln (cid:107) G ( it ) (cid:107) q β − θ ( t ) dt + θ (cid:90) ln (cid:107) G (1 + it ) (cid:107) q β θ ( t ) dt (14)holds for certain probability measures β − θ ( t ) dt and β θ ( t ) dt on the real line. We note that (cid:107) G ( it ) (cid:107) L q ( ρ ) = (cid:107) π ( g it ˆ ρ, ˆ ϕ ) eg itϕ,ρ ρ /q (cid:107) q ≤ . Hence (cid:90) − ln (cid:107) G (1 + it ) (cid:107) q β θ ( t ) dt ≤ ln (cid:107) G ( θ ) (cid:107) q ( θ ) θ . Our abstract recovery formula is summarized in the equation: − (cid:90) ln (cid:107) G (1 + it ) (cid:107) q β ( t ) dt ≤ lim inf θ → − ln (cid:107) G ( θ ) (cid:107) q ( θ ) θ . Here β ( t ) dt is obtained as the pointwise limit of the measures β θ ( t ) dt . Before we launch intomore fidelity estimates, we need a few L p norm inequalities. These will allow us to more formallystate and prove the result. Remark 5.3. a) For semifinite von Neumann algebras the L p continuity of R z (ˆ x ) = ϕ z/ Φ † ( ˆ ϕ − z/ ˆ x ˆ ϕ − z/ ) ϕ z/ is an immediate application of Stein’s analytic family interpolation theorem. However, for non-semifinite von Neumann algebras this map is not necessarily well-defined.b) We have R z (Φ( ϕ ) Re ( z ) ) = ϕ Re ( z )8 M. JUNGE AND N. LARACUENTE for all z in the strip { z | ≤ Re ( z ) ≤ } .c) For z = θ + it we see that R z = σ ϕt/ R θ σ ˆ ϕ − t/ is indeed a rotated, generalized Petz recovery map. Lemma 5.4.
Let z = θ + it . Then the twirled Petz map (with respect to ϕ ) satisfies (cid:107) G ( z ) (cid:107) L /θ ( ˜ M,ρ ) = f /θ ( ρ θ , R z (Φ( ρ ) θ )) . Proof.
Let p = 1 /θ . Using the calculation in the Haagerup L p spaces we deduce from thedefinition of R z that (cid:107) G ( z ) (cid:107) L p = (cid:107) π ( ˆ ρ z/ ˆ ϕ − z/ ) eϕ z/ ρ − z/ ρ /p (cid:107) L p ( ˜ M ) = (cid:107) ρ /p G ( z ) ∗ G ( z ) ρ /p (cid:107) p/ = (cid:107) ρ / p − θ ρ − it/ ϕ − it/ ϕ +1 / p Φ † ( ˆ ϕ it/ ϕ − θ/ ˆ ρ θ ϕ − θ/ ˆ ϕ − it/ ) ϕ +1 / p ϕ − it/ ρ + it/ ρ / p − θ (cid:107) p/ = (cid:107) ρ / p ϕ / p σ ϕt/ Φ † ( σ ˆ ϕ − t/ ( ˆ ϕ − θ/ ρ θ ˆ ϕ − θ/ )) ϕ / p ρ / p (cid:107) p/ = f p ( R z ( ˆ ρ /p ) , ϕ /p ) . Corollary 5.5.
Let z = θ + it . Then f /θ ( ϕ θ , R z (Φ( ρ ) θ )) ≤ . Proof.
By H¨older’s inequality, (cid:107) ρ / p ϕ / p σ ϕt/ Φ † ( σ ˆ ϕ − t/ ( ˆ ϕ − θ/ ρ θ ˆ ϕ − θ/ ) ϕ / p ρ / p (cid:107) p/ ≤ (cid:107) ρ / p (cid:107) p (cid:107) σ t/ R p ( σ ˆ ϕ − t/ ( ˆ ρ /p )) (cid:107) p ≤ (cid:107) R p ( σ ˆ ϕ − t/ ( ˆ ρ /p )) (cid:107) p ≤ (cid:107) σ ˆ ϕ − t/ ( ˆ ρ /p ) (cid:107) p ≤ (cid:107) ρ /p (cid:107) p . We use that tr ( ρ ) = 1, the modular group extends to an isometry on L p , and Proposition 4.3.The analyticity of G allows us to reformulate the interpolation formula for G as an interpo-lation of complex families of fidelities. Remark 5.6.
Theorem 1.2 then follows from Lemma 5.4 and Lemma 4.4. We use Equation (13) as a reformulation of Lemma 3.1 based on Lemma 5.2, after applying the re-iteration Theorem(see [30] for more information), which allows us to replace the boundaries of the complex strip i R and i R by p + i R and p + i R . Remark 5.7.
For any p , ∆ z/ ϕ,ρ ( ρ /p ) = ρ /p − z/ ϕ z/ = ρ − z/ ρ /p ϕ z/ , NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 19 z Figure 1.
Using complex interpolation and the re-iteration theorem, we estimatethe value of an analytic function at point z ∈ { ≤ Re ( z ) ≤ } by the nearestpoints along the lines p + i R and p + i R . and for any ω and p , (cid:107) ( ˆ ρ z/ ˆ ϕ − z/ ⊗ E ) ω (cid:107) p = (cid:107) ( ˆ ϕ − z/ ⊗ E ) ω ( ˆ ρ z/ ⊗ E ) (cid:107) p = (cid:107) (∆ z/ ρ, ˆ ϕ ⊗ E ) ω (cid:107) p . Hence (cid:107) G ( z ) (cid:107) ρ,p = (cid:107) (∆ z/ ρ, ˆ ϕ ⊗ E ) U ∆ − z/ ρ,ϕ (cid:107) ρ,p , where U is the finite-dimensional Stinespring isometry with environment E . This is not clear intype III, where we lack the tracial property. G ( z ) is a more useful form in type III, due to resultswe leverage from operator algebras. In particular, we have G ( z ) = π ( g z/ ρ, ˆ ϕ ) eg z/ ϕ,ρ , and we use in proving Lemma 5.2 that g itϕ,ρ and g it ˆ ρ, ˆ ϕ are respectively in M and ˆ M . As notedin Remarks 3.4 and 3.5, g ϕ,ρ has good analytic and algebraic properties that work well with theinterpolation methods we require. The correspondence between G ( z ) and its finite-dimensionalequivalent in terms of modular operators may nonetheless merit future investigation. Differentiation.
For the twirled recovery map we have to use a suitable differentiationresult, first under the additional assumption of regularity δϕ ≤ ρ ≤ δ − ϕ . More generally, wedifferentiate Kosaki norms for smooth functions with values in the underlying von Neumannalgebra. Lemma 5.8.
Let ( M, τ ) be a finite von Neumann algebra with trace τ . Let h : I → M be adifferentiable function such that h (0) = . Let ϕ be a faithful state. Let p be a differentiablefunction and p (0) > . Then i) ddθ (cid:107) ϕ / p ( θ ) h ( θ ) ϕ / p ( θ ) (cid:107) p ( θ ) (cid:12)(cid:12)(cid:12) θ =0 = lim θ → θ − ( (cid:107) ϕ / p ( θ ) h ( θ ) ϕ / p ( θ ) (cid:107) p ( θ ) −
1) = − ϕ ( h (cid:48) (0)) p (0) ; ii) lim θ → − ln (cid:107) ϕ / p ( θ ) h ( θ ) ϕ / p ( θ ) (cid:107) p ( θ ) θ = ϕ ( h (cid:48) (0)) .Proof. We consider g ( θ ) = (cid:107) ϕ / p ( θ ) h ( θ ) ϕ / p ( θ ) (cid:107) p ( θ ) p ( θ ) and assume first that p ( θ ) >
1. We mayassume by continuity that h ( θ ) > θ = 0. Let H ( t ) = ϕ / p ( θ ) h ( tθ ) ϕ / p ( θ ) .Using the differentiation formula for p -norms and convexity, we get for fixed p = p ( θ ) that g ( θ ) − (cid:107) H (1) (cid:107) pp − (cid:107) H (0) (cid:107) pp = p (cid:90) τ ( H ( t ) p − H (cid:48) ( t )) dt = pθ (cid:90) τ ( H ( t ) p − ϕ / p h (cid:48) ( tθ ) ϕ / p ) dt = pθ (cid:90) τ (( H ( t ) p − − H (0) p − ) ϕ / p h (cid:48) ( tθ ) ϕ / p ) dt + pθ (cid:90) τ ( ϕ p − p ϕ / p h (cid:48) ( tθ ) ϕ / p ) dt . For the second term we observe that τ ( ϕ p − p ϕ / p h (cid:48) ( tθ ) ϕ / p ) = τ ( ϕh (cid:48) ( tθ ))and hence pθ (cid:90) τ ( ϕ p − p ϕ / p h (cid:48) ( tθ ) ϕ / p ) dt = pτ ( ϕ ( h ( θ ) − h (0))) . As for the error (first) term, we observe that | τ (( H ( t ) p − − H (0) p − ) ϕ / p h (cid:48) ( tθ ) ϕ / p ) | ≤ (cid:107) ( H ( t ) p − − H (0) p − ) (cid:107) p (cid:48) (cid:107) ϕ / p h (cid:48) ( tθ ) ϕ / p (cid:107) p by H¨older’s inequality. Now, we may use the continuity of the Mazur map, see [38, Cor 2.3] for α = p − p (cid:48) = pp − and deduce that (cid:107) ( H ( t ) p − − H (0) p − ) (cid:107) p (cid:48) ≤ p − (cid:107) H ( t ) − H (0) (cid:107) p max {(cid:107) H ( t ) (cid:107) p , (cid:107) H (0) (cid:107) p } p − ≤ p − (cid:107) h ( tθ ) − h (0) (cid:107) ∞ max {(cid:107) H ( t ) (cid:107) p , (cid:107) H (0) (cid:107) p } p − ≤ p − (cid:107) h (cid:48) (cid:107) ∞ tθ max {(cid:107) H ( t ) (cid:107) p , (cid:107) H (0) (cid:107) p } p − . We deduce that p (cid:90) τ (( H ( t ) p − − H (0) p − ) ϕ / p h (cid:48) ( tθ ) ϕ / p ) dt ≤ (cid:107) h (cid:48) (cid:107) ∞ p ( p − (cid:90) max {(cid:107) H ( t ) (cid:107) p , (cid:107) H (0) (cid:107) p } p − (cid:107) ϕ / p h (cid:48) ( tθ ) ϕ / p (cid:107) p tθdt ≤ (cid:107) h (cid:48) (cid:107) ∞ (cid:107) ϕ / p h (cid:48) ϕ / p (cid:107) ∞ p ( p − θ (cid:90) max {(cid:107) H ( t ) (cid:107) p , (cid:107) H (0) (cid:107) p } p − tdt . The faithfulness of ϕ and fact that h (0) = 1 imply that (cid:107) H (0) (cid:107) p > p , so the integral onthe right hand side remains finite. As θ →
0, this term becomes 0. Thus for p (0) >
1, we canfind θ such that p ( θ ) − > δ for θ ≤ θ and hencelim θ → g ( θ ) − θ = p (0) τ ( ϕh (cid:48) (0)) . Let us now define the function F ( θ, p ) = g ( θ ) /p in two parameters. We find that ddθ F = − p g ( θ ) /p − g (cid:48) ( θ ) and dFdp = − p g ( θ ) /p ln g ( θ ). As ϕ is faithful, g ( θ ) is non-zero when h ( θ ) isalways positive and not equal to zero. Hence dF/dp is continuous and differentiable. To showthat dF ( p, θ ( p )) /dθ is continuous and differentiable, we must also check the dF/dθ part, whichinvolves g (cid:48) ( θ ). We again apply separation of variables. First, ddθ (cid:107) ϕ / p h ( θ ) ϕ / p (cid:107) pp = (cid:107) ϕ / p h ( θ ) ϕ / p (cid:107) pp (cid:18) ddθ ln (cid:107) ϕ / p h ( θ ) ϕ / p (cid:107) p (cid:19) NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 21
The prefactor is continuous by the continuity of g ( θ ) for p >
1. We now use a fact of Banachspaces, that for any continuous, differentiable function H ( θ ) and p fixed, ddθ (cid:107) H ( θ ) (cid:107) p = (cid:28)(cid:16) H ( θ ) (cid:107) H ( θ ) (cid:107) p (cid:17) p/p (cid:48) , ddθ H ( θ ) (cid:29) . Letting H ( θ ) = (cid:107) ϕ / p ( θ ) h ( θ ) ϕ / p ( θ ) (cid:107) p ( θ ) , left side of the braket is again the Mazur map andtherefore continuous. For the right side, ddθ ( ϕ / p ( θ ) h ( θ ) ϕ / p ( θ ) ) = ϕ / p h (cid:48) ( θ ) ϕ / p . We again see continuity of this expression. Finally, positivity of θ and the chain rule for thenatural logarithm give us continuity of the entire expression. We still however must contendwith the p derivative. Here we apply separation of variables yet another time, writing ddp (cid:107) ϕ / p h ( θ ) ϕ / p (cid:107) pp = ddp (cid:107) ϕ / q hϕ / q (cid:107) pp + ddq (cid:107) ϕ / q hϕ / q (cid:107) pp (cid:12)(cid:12)(cid:12) p = q . First, we deal with the p -derivative, noting that the quantity inside of the norm is assumed p -independent. We obtain ddp (cid:107) ϕ / q h ( θ ) ϕ / q (cid:107) pp = ddp tr(( ϕ / q hϕ / q ) p ) = tr(( ϕ / q hϕ / q ) p ln( ϕ / q hϕ / q )) . This is finite whenever ϕ / q hϕ / q >
0, so this derivative is continuous. For the q derivative, ddq (cid:107) ϕ / q hϕ / q (cid:107) pp = ddq tr(( ϕ / q hϕ / q ) p ) = p ( ϕ / q hϕ / q ) p − ddq ( ϕ / q hϕ / q ) . Since we only care about continuity and will not rely here on explicitly evaluating this derivative,we merely note that the product rule allows us to differentiate the remaining factor, and that ϕ / q − is finite by the positivity of ϕ . This term is therefore continuous.Hence F is differentiable, and ddθ F ( θ, p ( θ )) = − p ( θ ) g ( θ ) /p ( θ ) − g (cid:48) ( θ ) − p ( θ ) g ( θ ) /p ( θ ) ln g ( θ ) dp ( θ ) dθ . For θ = 0, we deduce from g (0) = 1 that ddθ F ( θ, p ( θ )) | θ =0 = − p (0) ϕ ( h (cid:48) (0)) . This concludes the proof of i) in this case. For ii) we note thatln (cid:107) ϕ / p ( θ ) h ( θ ) ϕ / p ( θ ) (cid:107) p ( θ ) θ = 1 p ( θ ) ln g ( θ ) θ Using ddθ ln g ( θ ) | θ =0 = g (cid:48) (0) g (0) we deduce indeed ii). Theorem 5.9.
Let δϕ ≤ ρ ≤ δ − ρ and ≤ p < ∞ . Then (cid:90) R ( − ln f p ( ρ /p , R itp (Φ( ρ ) /p ))) β ( t ) dt ≤ D ( ρ | ϕ ) − D (Φ( ρ ) | Φ( ϕ ))2 p . Proof.
Let q ≥ q >
2. We define q ( θ ) = − θq + θq . Then we may apply Lemma 5.2 for G q ( z ) = G ( z/q ) = π ( ˆ ρ z/ q ˆ ϕ − z/ q ˆ f ) ef ϕ z/ q ρ − z/ q which remains analytic as long as q ≥
1. Using (cid:107) G q ( it ) (cid:107) q ≤
1, we deduce as in Lemma 5.2 thatlim θ → ln (cid:107) G q ( θ ) (cid:107) L q ( θ )1 θ ≤ (cid:90) ln (cid:107) G q (1 + it ) (cid:107) L q β ( t ) dt . Let us recall that, according to Lemma 5.4 we have (cid:107) G q (1 + it ) (cid:107) L q = f q ( ρ /q , R itq (Φ( ρ ) /q )) . However, we have used the dominated convergence theorem to interchange integral and limit,which is possible thanks to the continuity interpolated fidelity, proved in the next section. Weare left to calculate the limit. We may introduce p ( θ ) = q ( θ )2 so that p (0) >
1. Then we see that (cid:107) G q ( θ ) (cid:107) L q ( θ )1 = (cid:107) ρ /q ( θ ) ρ − / q ( θ ) ϕ / q ( θ ) Φ † ( ˆ ϕ − / q ( θ ) ˆ ρ /q ( θ ) ˆ ϕ − / q ( θ ) ) ϕ / q ( θ ) ρ − / q ( θ ) ρ /q ( θ ) (cid:107) p ( θ ) = (cid:107) ρ / p ( θ ) h q ( θ ) ρ / p ( θ ) (cid:107) p ( θ ) holds for h q ( θ ) = ρ − / q ( θ ) ϕ / q ( θ ) Φ † ( ˆ ϕ − / q ( θ ) ˆ ρ /q ( θ ) ˆ ϕ − / q ( θ ) ) ϕ / q ( θ ) ρ − / q ( θ ) = h (cid:16) θq (cid:17) . For q = 1, our derivative of h ( θ ) = ρ − θ/ ϕ θ/ Φ † ( ˆ ϕ − θ/ ˆ ρ θ/ ˆ ϕ − θ/ ) ϕ θ/ ρ − θ/ satisfies h (cid:48) (0) = − ln ρ + ln ϕ + Φ † (ln ˆ ρ ) − Φ † (ln ˆ ϕ ) . This implies tr ( ρh (cid:48) (0)) = − tr ( ρ ln ρ ) + tr ( ρ ln( ϕ )) + tr (Φ( ρ ) ln Φ( ρ ) − ln Φ( ϕ )) = − D ( ρ | ϕ ) + D (Φ( ρ ) | Φ( ϕ )) . Using the chain rule, we get − qtr ( ρh (cid:48) q (0)) = D ( ρ | ϕ ) − D (Φ( ρ ) | Φ( ϕ )) . Remark 5.10.
In a type
III situation is is better to write h ( θ ) = ∆ θ/ ρ,ϕ Φ † ((∆ θ/ ρ, ˆ ϕ ) ∗ ∆ θ/ ρ, ˆ ϕ )∆ θ/ ρ,ϕ and hence h (cid:48) (0) = − ln ∆ ρ,ϕ + Φ † (ln ∆ ˆ ρ, ˆ ϕ ) . This implies again tr ( ρh (cid:48) (0)) = − ( ρ / , ln ∆ ρ,ϕ ρ / ) + tr (Φ( ρ ) / , ∆ Φ( ρ ) , Φ( ϕ ) Φ( ρ ) / )= − D ( ρ | ϕ ) + D (Φ( ρ ) | Φ( ϕ )) . NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 23
Corollary 5.11.
Let ≤ p ≤ ∞ . Then (cid:90) R ( − ln f p ( ρ /p , R itp (Φ( ρ ) /p ))) β ( t ) dt ≤ D ( ρ | ϕ ) − D (Φ( ρ ) | Φ( ϕ ))2 p . Furthermore, the (generally non-linear) universal recovery map ˜ R p ( x ) = (cid:16) (cid:90) R p,t ( x /p ) dµ ( t ) (cid:17) p satisfies − ln f p ( ρ, ˜ R p (Φ( ρ ))) ≤ p [ D ( ρ | ϕ ) − D (Φ( ρ ) , Φ( ϕ ))] . The same holds for the general von Neumann algebra version in Section ?? .Proof. We refer to Sections 6 and 8 for the discussion that assuming ρ ≤ λϕ is enough and tojustify the differentiation Lemma. For the ‘moreover’ part, we recall that ln is concave and f p isjointly concave, and hence (cid:90) ln f p ( ρ, R p,t ( ˆ ρ /p ) p ) dµ ( t ) ≤ ln (cid:90) f p ( ρ, R p,t ( ˆ ρ /p ) p ) dµ ( t ) ≤ ln f p (cid:16) (cid:90) ρdµ ( t ) , (cid:90) R p,t ( ˆ ρ /p ) p dµ ( t ) (cid:17) = ln f p ( ρ, ˜ R p ( ˆ ρ )) . Theorem 5.9 essentially finishes the finite-dimensional case. Though we have yet to removethe restriction that δϕ ≤ ρ ≤ δ − ρ for some δ , we will leave this to the general continuity resultsin Section 6, and the Haagerup approximation results in Section 8. Infinite dimensions introduceadditional subtleties with the continuity arguments, and it is not so simple to show that we candrop the restriction that δϕ ≤ ρ ≤ δ − ρ . Section 6 resolves these issues, extending recoveryto type II. The obvious barrier in type III is the lack of a trace. Were this the only barrier,the Haagerup L p spaces and corresponding trace would suffice. The deeper problem is that thedifferentiability of h ( θ ) as used in Lemma 5.8, and the continuity of the trace of the operatorlogarithm are not clear without a finite trace. Hence we must approximate the crossed productby finite von Neumann algebras in Section 8, our main use of the techniques of [25].6. Continuity for fidelity of Recovery
In this section, we show some continuity results for the fidelity of recovery, which are notimmediate in infinite dimension. We continue to use our standard assumptions on ϕ , ρ and Φ. Lemma 6.1.
Let A be an (possibly unbounded) positive operator on a Hilbert space H , ξ in thedomain of A / and f n : R → R be sequence of functions such that | f n ( x ) | ≤ C (1 + | x | / ) and lim n f n ( x ) = f ( x ) for all x . Then lim n (cid:107) ( f n ( A ) − f ( A ))( ξ ) (cid:107) H = 0 , where f n extends to operators by elementary functional calculus.Proof. Let dµ ξ ( x ) be the spectral measure of A , i.e.( ξ, f ( A ) ξ ) = (cid:90) f ( x ) dµ ξ ( x )for all measurable f . Then we observe by the triangle inequality that | f n ( x ) − f ( x ) | ≤ C (1+ | x | ) holds for all n ∈ N and moreover, (cid:107) A / ξ (cid:107) H = ( A / ξ, A / ξ ) = (cid:90) | x | dµ ξ ( x )Since ξ has finite norm, we deduce that x (cid:55)→ (1 + | x | ) is in L ( µ ξ ). By the dominated convergencetheorem, we deduce thatlim n (cid:107) ( f n ( A ) − f ( A )) ξ (cid:107) H = lim n (cid:90) | f n ( x ) − f ( x ) | dµ ξ ( x ) = 0 . Proposition 6.2.
Let δϕ ≤ ρ ≤ δ − ρ . Then the function F ( z ) = f Re ( z ) ( ρ Re ( z ) , R z (Φ( ρ ) Re ( z ) )) is continuous in z on { z | < Re ( z ) ≤ } .Proof. Here we recall Lemma 6.1 as a general fact.Let ρ and ϕ be states and ψ ( x ) = ϕ ( x )+ ρ ( x )2 the corresponding positive functional on M ( M ) considered by Connes [31]. Then ϕ / = ∆ ϕ,ρ ( ρ / ) = ∆ ψ (cid:18) (cid:18) ρ / (cid:19) (cid:19) belongs to the domain of ∆ / ψ . And hencelim z → w (cid:107) ∆ zψ − ∆ ψ ( | (cid:105)(cid:104) | ⊗ ρ / ) (cid:107) = 0as long as (cid:60) ( z ) , (cid:60) ( w ) ≤ /
2. Note that thanks to the calculation in the core M (cid:111) R we knowthat ϕ z/ ρ − z/ ρ / = ϕ z/ ρ / ρ − z/ = ∆ z/ ϕ,ρ ( ρ / ) ∼ = ∆ ψ ( | (cid:105)(cid:104) | ⊗ ρ / ) . This means we have L convergence in z for 0 ≤ (cid:60) ( z ) ≤
1. Using Kosaki’s interpolation resultwe deduce that (cid:107) ( ϕ z/ ρ − z/ − ϕ w/ ρ − w/ ) ρ / p (cid:107) p ≤ | ( ϕ z/ ρ − z/ − ϕ w/ ρ − w/ ) (cid:107) − /p ∞ (cid:107) ( ϕ z/ ρ − z/ − ϕ w/ ρ − w/ ) ρ / (cid:107) /p . Therefore, we see deduce that 0 ≤ (cid:60) ( z ) , (cid:60) ( w ) ≤ z → w (cid:107) ( ϕ z/ ρ − z/ − ϕ w/ ρ − w/ ) ρ / p (cid:107) p = 0holds uniformly on compact sets. NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 25
Now it is time we address the fidelity. We will use functional calculus and observe that ϕ z/ ρ − z/ − ϕ w/ ρ − w/ = ϕ z/ (1 − ϕ w − z/ ρ ( z − w ) /z ) ρ − z/ . Let us define the ∗ homomorphism π : C ( R ) → B ( L ( M )) given by π ( F ⊗ F ) = L F ( ρ ) R F ( ϕ ) .Using | e a − | ≤ ae | a | , we observe that | ( x/y ) w − ( x/y ) z | = | ( e ln x − ln y ( w − z ) − x/y ) z | ≤ | w − z || ln( x/y ) || ( x/y ) z | . Let δ ≤ D ≤ δ − be a bounded operator. Using | e x − | ≤ xe | x | and functional calculus wededuce that (cid:107) D w − D z (cid:107) = (cid:107) ( e (ln D ) w − z − D z (cid:107) ≤ | w − z || ln δ | e | ln δ || w − z | δ −| Re ( z ) | = | w − z | ( δ − ) | w − z | + | z | . This allows us to estimate (cid:107) G ( z ) − G ( w ) (cid:107) = (cid:107) π (∆ z/ ρ, ˆ ϕ )∆ z/ ϕ,ρ − π (∆ w/ ρ, ˆ ϕ )∆ w/ ϕ,ρ (cid:107)≤ δ − ) | w − z | + | z | | w − z | . Let us know consider the case p ≤ p where p = Re ( w ), Re ( z ) = p . Then we find that (cid:107) G ( w ) (cid:107) L p ( L ( H M ) ,ρ ) ≤ (cid:107) G ( w ) − G ( z ) (cid:107) L p ( L ( H M ) ,ρ ) + (cid:107) G ( z ) (cid:107) L p ( L ( H M ) ,ρ ) ≤ C ( δ, w, z ) | w − z | + (cid:107) G ( z ) (cid:107) L p ( L ( H M ) ,ρ ) . Since C ( δ, w, z ) is bounded in bounded regions of C , we deduce continuity for Re ( w ) ≥ Re ( z ).More precisely, we have continuity for fixed Re ( z ), and moreover, F ( w ) ≤ lim inf z → w,Re ( w ) ≥ Re ( z ) F ( z ) ≤ lim sup z → w,Re ( w ) ≥ Re ( z ) F ( z ) , (15)lim sup z → w,Re ( z ) ≥ Re ( w ) F ( z ) ≤ F ( w ) . (16)To prove the missing inequality in (15), we may assume Im ( z ) = Im ( w ) = 0. Let us now assumethat Re ( w ) = p > Re ( z ) = p , i.e. p > p for fixed p . Let p ≥
1, Then we can find η such that p = − ηp + ηp . We use that standard interpolation estimate and deduce from (cid:107) G (1 /p ) (cid:107) ≤ (cid:107) G (1 /p ) (cid:107) p ≤ (cid:16) (cid:90) R f p ( η ) ( ρ /p , R itp (Φ( ρ ) /p ( η ) )) β η ( t ) dt (cid:17) − η . Here q = p − p . We may now send η →
0. Thanks to the continuity in the imaginary part andthe explicit form of the measure (see [30, p=93]) dµ η ( t ) = h η ( t ) dt , h η ( t ) = e − πt sin πη (1 − η )(sin πη + (cos πη − e − πt ) )we deduce thatlim sup η → (cid:107) G (1 /p ( η )) (cid:107) p ( η ) ≤ lim sup η → (cid:16) (cid:90) R f p ( ρ /p , R itp (Φ( ρ ) /p )) β η ( t ) dt (cid:17) − η = f p ( ρ /p , R itp (Φ( ρ ) /p )) . This shows that lim sup z → w,Re ( z ) >Re ( w ) F ( z ) ≤ F ( w ) . Similarly, we prove the missing inequality F ( w ) ≤ lim inf z → w,Re ( z ) >Re ( w ) F ( z ) . in (16) using uniform continuity in the imaginary axes. All four inequalities together then yieldcontinuity. Lemma 6.3.
Let ≤ p < ∞ . The function h ( z ) = g ϕ,ρ ( z ) is continuous in L p ( M, ρ ) .Proof. We will first prove the assertion for p = 2. Following Connes we consider M ( N ) and thestate ψ ( x ) = ( ϕ ( x ) + ρ ( x )). Let e i,j = | i (cid:105)(cid:104) j | be the matrix units in M . Then we see that∆ ψ ( e ⊗ ξ ) = e ⊗ ϕξρ − = e ⊗ ∆ ϕ,ρ ( ξ ) . Moreover, ∆ / ( ρ / ) = ϕ / shows that e ⊗ ρ / belongs to the domain. Note however, that,thanks to calculation in the core M (cid:111) R we have g ϕ,ρ ( z ) ρ / = ϕ z/ ρ − z/ ρ / = ∆ z/ ϕ,ρ ( ρ / ) . Let lim n z n = z such that 0 ≤ (cid:60) ( z n ) ≤
1. Then f n ( x ) = x z n / and f ( x ) = x z/ satisfy theassumption of Lemma 6.1, and hence we have convergence. For 2 < p < ∞ we deduce fromKosaki’s interpolation theorem that also have (cid:107) a (cid:107) L p ≤ (cid:107) a (cid:107) − θL (cid:107) a (cid:107) θ ∞ provided a is bounded and p = − θ . We apply this to a = g ϕ,ρ ( z n ) − g ϕ,ρ ( z ) which is uniformlybounded, see Remark 3.4. Therefore, convergence in L implies convergence for all 2 ≤ p < ∞ . Lemma 6.4.
Let a ∈ M . Then h ( p ) = (cid:107) a (cid:107) L p ( ρ ) is continuous.Proof. Let p ≤ q ≤ p and θ ( q ) such that1 q = 1 − θp + θp Then we deduce from Kosaki’s interpolation theorem that (cid:107) a (cid:107) p ≤ (cid:107) a (cid:107) q ≤ (cid:107) a (cid:107) − θ ( q ) p (cid:107) a (cid:107) θ ( q ) p . Note that q converges to p iff θ ( q ) converges to 0. This implies the assertion. NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 27
Proof. (6.2) Let us consider G ( z ) = π ( g ˆ ρ, ˆ ϕ ( z/ e and G ( z ) = g ϕ,ρ ( z/
2) such that G ( z ) = G ( z ) G ( z ) . Let us the notation p ( z ) = (cid:60) ( z ). From the triangle inequality we deduce that |(cid:107) G ( z ) (cid:107) p ( z ) − (cid:107) G ( w ) (cid:107) p ( w ) | ≤ |(cid:107) G ( z ) (cid:107) p ( z ) − (cid:107) G ( w ) (cid:107) p ( z ) | + |(cid:107) G ( w ) (cid:107) p ( z ) − (cid:107) G ( w ) (cid:107) p ( w ) |≤ (cid:107) G ( z ) − G ( w ) (cid:107) p ( z ) + |(cid:107) G ( w ) (cid:107) p ( z ) − (cid:107) G ( w ) (cid:107) p ( w ) | A glance at (the proof of Lemma (6.4)) show that because (cid:107) G ( w ) (cid:107) ≤ M uniformly for Re ( w ) ≤ w → z |(cid:107) G ( w ) (cid:107) p ( z ) − (cid:107) G ( w ) (cid:107) p ( w ) | = 0 . For the first part we use Kosaki’s interpolation result and get (cid:107) G ( z ) − G ( w ) (cid:107) p ( z ) ≤ (cid:107) G ( z ) − G ( w ) (cid:107) − Re ( z )2 . Thus for Re ( z ) >
0, it suffices to show that L estimate. Then we observe that (cid:107) G ( z ) − G ( w ) (cid:107) = (cid:107) G ( z ) G ( z ) − G ( w ) G ( w ) (cid:107) ≤ (cid:107) G ( z )( G ( z ) − G ( w )) (cid:107) + (cid:107) ( G ( z ) − G ( w )) G ( w ) (cid:107) ≤ (cid:107) G ( z ) (cid:107) ∞ (cid:107) g ϕ,ρ ( z/ ρ / − g ϕ,ρ ( w/ ρ / (cid:107) + (cid:107) ( G ( z ) − G ( w )) G ( w ) (cid:107) . Thanks to Remark 3.4, we deduce convergence for the first of the two terms from Lemma 6.3.Let us consider the remaining term and w = 1 /q + it . Then we deduce from H¨older’s inequalityand interpolation that (cid:107) aG ( w ) (cid:107) = (cid:107) aϕ w/ ρ / − w/ (cid:107) = (cid:107) aϕ / q ϕ it/ ρ − it/ ρ / − /q (cid:107) ≤ (cid:107) aϕ / q (cid:107) q ≤ (cid:107) a (cid:107) − /q ∞ (cid:107) aϕ / (cid:107) /q . Therefore we are left with an L -norm estimate. In our case a = π ( G ( z ) − G ( w )) e and hencefor b = G ( z ) − G ( w ) we find that (cid:107) aϕ / (cid:107) = T r ( ϕ / Φ † ( b ∗ b ) ϕ / ) = T r ( ˆ ϕ ( b ∗ b ))= (cid:107) ˆ ρ z/ ˆ ϕ − z/ ˆ ϕ / − ˆ ρ w/ ˆ ϕ − w/ ˆ ϕ / (cid:107) . Therefore Lemma 6.3 concludes the proof.7.
Approximation of relative entropy
In this section we will work with Lindblad’s definition of relative entropy D Lin ( ρ | ϕ ) = ( √ ρ, log ∆ ρ,ϕ ( √ ρ )) + ϕ (1) − ρ (1)Indeed, D Lin is the unique homogeneous joint extension of the relative D entropy, i.e.i) D Lin ( tρ | ϕ ) = tD Lin ( ρ | ϕ );ii) D Lin ( ρ | ϕ ) = D ( ρ | ϕ ) if ρ (1) = ϕ (1) = 1. Finite von Neumann algebras.Proposition 7.1.
Let ( N, τ ) be a finite von Neumann algebra and a ≤ d ϕ ≤ a − . Let d ψ be adensity of a state ψ . Then d M,δ = 1 [0 ,M ] ( d ψ ) d ψ + δd ϕ satisfies δd ϕ ≤ d M,δ ≤ ( a + δ ) d ϕ ; i) lim M →∞ lim δ → (cid:107) d M,δ − d M (cid:107) = 0 ; ii) lim M →∞ lim δ → D Lin ( d M,δ | d ϕ ) = D ( d | d ϕ ) .Proof. In the tracial setting, we have (see [19]) that D ( ψ | ϕ ) = D ( d ψ | d ϕ ) = τ ( d ψ ln d ψ ) − τ ( d ψ ln d ϕ ) = D Lin ( d ψ | d ϕ ) . For fixed M , we denote by d M = 1 [0 ,M ] ( d ψ ) d ψ the density obtained by functional calculus. Then d M,δ = d M + δd ϕ converges in operator norm, and L norm to d M . Therefore, the continuity of f ( x ) = x ln x implies thatlim δ → τ ( d M + δd ϕ ln d M + δd ϕ ) − τ (( d M + δd ϕ ) ln d ϕ ) + τ ( d ϕ ) − τ ( d M + δd ϕ ) = D Lin ( d M | d ϕ ) . Here we use that d ϕ is bounded below and above and hence ln d ϕ is in L ∞ ( N ). Using this factagain, we deduce from Fatou’s lemma τ ( d ψ ln d ψ ) − τ ( d ψ ln d ϕ ) + τ ( d ϕ ) − τ ( d ψ ) = lim M →∞ τ ( d M ln d M ) − τ ( d M ln d ϕ ) + τ ( d ϕ ) − τ ( d M ) . Note here that D ( d ψ | d ϕ ) is finite iff τ ( d ψ ln d ψ ) is finite.For the convenience of the reader let us briefly review how to transition from trace freedefinition to the one using trace. Indeed, in L ( N , τ ) the vector (cid:112) d ϕ , the purification of thestate ϕ , implements the GNS representation with respect to the usual left-regular representation π ( x ) (cid:112) d ϕ = x (cid:112) d ϕ for x ∈ N . We will use π again in the Haagerup construction, section 8.2Moreover, using Connes’ 2 × ξ ∈ L ( N , τ ) that∆ ϕ,ψ ( ξ ) = d ϕ ξd − ψ and hence ∆ itψ,ϕ ( x ) = d itψ xd − itϕ . This implies ln ∆ ψ,ϕ ( d / ψ ) = ln d ψ d / ψ − d / ψ ln d ϕ . Taking the inner product, we find( d / ψ , ln ∆ ψ,ϕ ( d / ψ )) = τ ( d ψ ln d ψ ) − τ ( d ψ ln d ϕ ) = D τ ( d ψ | d ϕ ) . NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 29
Haagerup construction.
Haagerup’s construction for type
III algebras provides a con-venient tool to deduce properties of type
III algebras from finite von Neumann algebras. Thisconstruction is a discrete version of the usual continuous core. The starting point is a normalfaithful state ϕ with modular group ( σ ϕt ) t ∈ R . Instead of working with R , we use the groupdiscrete G = (cid:83) n − n Z ⊂ R and the crossed product˜ M = M (cid:111) σ ϕ G .
The advantage here is that we have conditional expectation E : ˜ M → M given by E ( (cid:88) g x g λ ( g )) = x . (17)Let us state the main facts (see [25]:)Hi) E and ˜ ϕ = ϕ ◦ E are faithful.Hii) There exists an increasing family of subalgebras ˜ M k and normal conditional expectation F k : ˜ M → ˜ M k such that ˜ ϕF k = ˜Φ;Hiii) lim k (cid:107) F k ( ψ ) − ψ (cid:107) ˜ M ∗ = 0 for every normal state ψ ∈ ˜ M ;Hiv) For every k there exists a normal faithful trace trace τ k ( x ) = ˜ ϕ ( d k ( x )) such that d k ∈ ˜ M (cid:48) k and a k ≤ d k ≤ a − k for some scalars a k ∈ R + .Thanks to the conditional expectation, we have a canonical map E ∗ : M ∗ → ˜ M ∗ given by E ∗ ( ρ ) = ρ ◦ E . We will use the notation ˜ ρ = E ∗ ( ρ ). Remark 7.2.
Let us recall two possible ways to represent the crossed product M (cid:111) G for anaction α of a discrete group on Hilbert space. We may assume that M ⊂ B ( H ) and consider (cid:96) ( G, H ). Then M (cid:111) G = (cid:104) λ H ( G ) , π ( M ) (cid:105) is generated by a copy of λ ( G ), the left regularrepresentation of G , and π ( M ). Here we may assume π ( x ) = (cid:88) g | g (cid:105)(cid:104) g | ⊗ α g − ( x )is given by a twisted diagonal representation and λ H ( g ) = λ ( g ) ⊗ H . Alternatively, we maychoose ˆ π ( x ) = 1 ⊗ x and ˆ λ H ( g ) = λ ( g ) ⊗ u g such that u ∗ g xu g = α g − ( x ). Both of theserepresentations are used in the literature, and their equivalence is used in the proof of Takai’stheorem. For the equivalence we note that λ H ( g ) − π ( x ) λ H ( g ) = π ( α g − ( x )) . Similarly, λ ( g ) − ⊗ u − g (1 ⊗ x ) λ ( g ) ⊗ u g = 1 ⊗ α − g ( x ). This shows that the algebraic relations ofthese two representations coincide. Using a GNS construction this extends to the generated vonNeumann algebras. Lemma 7.3.
Let ρ, ϕ be states on the von Neumann algebra M with corresponding ˜ ρ, ˜ ϕ in ˜ M ∗ .Then D ( ˜ ρ | ˜ ϕ ) = D ( ρ | ϕ ) .Proof. We consider the Hilbert space H = (cid:96) ( G, L ( M )) and still use the symbol λ ( g ) insteadof λ L ( M ) ( g ). Our first goal is to calculate the modular operator for an analytic state ϕ withdensity d in L ( M ), and ˜ ϕ = ϕ ◦ E , E : M (cid:111) G → M the canonical conditional expectation. Then ξ = | (cid:105) ⊗ d / implements the state ˜ ϕ on the crossed product. In order to calculate themodular operator ∆ = S ∗ S , we recall that( yξ, ∆( xξ )) = ( x ∗ ξ, y ∗ ξ ) . We start with finitely supported y = (cid:80) g λ ( g ) π ( y g ) , z = (cid:80) g λ ( g ) π ( z g ) and observe that( yξ, zξ ) = ( (cid:88) g | g (cid:105) y g d / , (cid:88) g | g (cid:105) z g d / ) = (cid:88) g ϕ ( y ∗ g x g ) . On the other hand, we find( x ∗ ξ, y ∗ ξ ) = ( (cid:88) g | g − (cid:105) α g ( x ∗ g ) d / , (cid:88) g | g − (cid:105) α g ( y ∗ g ) d / ) = (cid:88) g ϕ ( α g ( x g y ∗ g )) . Let d g − = α − g ( d ). Then we see that ϕ ( α g ( x g y ∗ g )) = tr ( d g − x g y ∗ g ) = tr ( d / y ∗ g d g − x g d − d / ) = ( y g d / , d g − x g d − d / ) . This means that the diagonal operator ∆ g ( ξ g ) = ∆ d g − ,d is a good candidate for the modularoperator, and is indeed well-defined for finitely supported sequences of σ tα − g ( ϕ ) ,ϕ -analytic ele-ments, which are dense. Now, it is easy to identify the polar composition using the isometry J ( (cid:80) g | g (cid:105) ξ g ) = (cid:80) g | g − (cid:105) α g ( ξ ∗ g ) on (cid:96) ( G, L ( M )), because α g extends to an isometry on L ( M ).This formula S = J ∆ / follows by calculation. Finally, we use Connes’ 2 × ϕ, ψ and the diagonal state ˆ ϕ ( x ab ) = ϕ ( x )+ ψ ( x ). Note that M ( M ) (cid:111) G = M ( M (cid:111) G )and hence ∆ ˜ ϕ, ˜ ψ is the 1 , G -diagonal operator ∆ α − g ( ϕ ) ,ψ . This implies D ( ˜ ϕ | ˜ ψ ) = ( ξ ψ , log ∆ ˜ ϕ, ˜ ψ ( ξ ψ )) = ( d / ψ , ∆ α − ( ϕ ) ,ψ ( d / ψ ))= ( d / ψ , log ∆ ϕ,ψ ( d / ψ )) = D ( ϕ | ψ ) . Here we use that the relative entropy can be calculated on any representing Hilbert space.However, the representation of M (cid:111) G is in standard form, which may be used as a definition ofthe relative entropy.A similar result holds for the fidelity. Theorem 7.4.
Let ϕ be a faithful state. Then there exists a sequence of states ρ α such that i) δ α ϕ ≤ ρ α ≤ δ − α for some δ α > ; ii) lim α ρ α = ρ ; iii) D ( ρ | ϕ ) = lim α D ( ρ α | ϕ ) .Proof. Let us define ψ k = F k ( ˜ ρ ). Thanks to the Haagerup construction we know that lim k ψ k = ˜ ρ .We may apply Proposition 8.1 and find d k,m,δ = α k,m,δ (1 [0 ,m ] ( d ψ k ) d ψ k + δd ϕ k ), where α k,m,δ ischosen such that d k,m,δ has trace 1. Denote by ψ k,m,δ the corresponding state on ˜ M k and ψ k,m,δ = ψ k,m,δ ◦ F k . Let ρ k,m,δ be the restriction to M . Certainly, we find condition i). Moreover,by the data processing inequality (see Witten’s notes [19]) D ( ρ k,m,δ | ϕ ) ≤ D ( ψ k,m,δ | ϕ ) NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 31 and hence lim sup k →∞ ,m →∞ ,δ → D ( ρ k,m,δ | ϕ ) ≤ lim sup k D ( ψ k | ˜ ϕ ) ≤ D ( ˜ ρ | ˜ ϕ ) = D ( ρ | ϕ ) . However, we deduce from
Hiii ) and Proposition 8.1 thatlim k lim m lim δ ψ k,m,δ = ˜ ρ . Taking the conditional expectation E by restriction these state to M preserves this property.Thus by the semicontinuity of D Lin , we deduce that D ( ρ | ϕ ) ≤ lim inf k,m,δ D ( ρ k,m,δ | ϕ ) ≤ lim sup k,m,δ D ( ψ k,m,δ | ˜ ϕ ) ≤ D ( ρ | ϕ ) . This allows us to find a suitable convergent subsequence.8.
Approximation of relative entropy
In this section we will work with Lindblad’s definition of relative entropy D Lin ( ρ | ϕ ) = ( √ ρ, log ∆ ρ,ϕ ( √ ρ )) + ϕ (1) − ρ (1)Indeed, D Lin is the unique homogeneous joint extension of the relative D entropy, i.e.i) D Lin ( tρ | ϕ ) = tD Lin ( ρ | ϕ );ii) D Lin ( ρ | ϕ ) = D ( ρ | ϕ ) if ρ (1) = ϕ (1) = 1.8.1. Finite von Neumann algebras.Proposition 8.1.
Let ( N, τ ) be a finite von Neumann algebra and a ≤ d ϕ ≤ a − . Let d ψ be adensity of a state ψ . Then d M,δ = 1 [0 ,M ] ( d ψ ) d ψ + δd ϕ satisfies δd ϕ ≤ d M,δ ≤ ( a + δ ) d ϕ ; i) lim M →∞ lim δ → (cid:107) d M,δ − d M (cid:107) = 0 ; ii) lim M →∞ lim δ → D Lin ( d M,δ | d ϕ ) = D ( d | d ϕ ) .Proof. In the tracial setting, we have (see [19]) that D ( ψ | ϕ ) = D ( d ψ | d ϕ ) = τ ( d ψ ln d ψ ) − τ ( d ψ ln d ϕ ) = D Lin ( d ψ | d ϕ ) . For fixed M , we denote by d M = 1 [0 ,M ] ( d ψ ) d ψ the density obtained by functional calculus. Then d M,δ = d M + δd ϕ converges in operator norm, and L norm to d M . Therefore, the continuity of f ( x ) = x ln x implies thatlim δ → τ ( d M + δd ϕ ln d M + δd ϕ ) − τ (( d M + δd ϕ ) ln d ϕ ) + τ ( d ϕ ) − τ ( d M + δd ϕ ) = D Lin ( d M | d ϕ ) . Here we use that d ϕ is bounded below and above and hence ln d ϕ is in L ∞ ( N ). Using this factagain, we deduce from Fatou’s lemma τ ( d ψ ln d ψ ) − τ ( d ψ ln d ϕ ) + τ ( d ϕ ) − τ ( d ψ ) = lim M →∞ τ ( d M ln d M ) − τ ( d M ln d ϕ ) + τ ( d ϕ ) − τ ( d M ) . Note here that D ( d ψ | d ϕ ) is finite iff τ ( d ψ ln d ψ ) is finite.For the convenience of the reader let us briefly review how to transition from trace freedefinition to the one using trace. Indeed, in L ( N , τ ) the vector (cid:112) d ϕ , the purification of thestate ϕ , implements the GNS representation with respect to the usual left-regular representation π ( x ) (cid:112) d ϕ = x (cid:112) d ϕ for x ∈ N . We will use π again in the Haagerup construction, section 8.2Moreover, using Connes’ 2 × ξ ∈ L ( N , τ ) that∆ ϕ,ψ ( ξ ) = d ϕ ξd − ψ and hence ∆ itψ,ϕ ( x ) = d itψ xd − itϕ . This implies ln ∆ ψ,ϕ ( d / ψ ) = ln d ψ d / ψ − d / ψ ln d ϕ . Taking the inner product, we find( d / ψ , ln ∆ ψ,ϕ ( d / ψ )) = τ ( d ψ ln d ψ ) − τ ( d ψ ln d ϕ ) = D τ ( d ψ | d ϕ ) . Haagerup construction.
Haagerup’s construction for type
III algebras provides a con-venient tool to deduce properties of type
III algebras from finite von Neumann algebras. Thisconstruction is a discrete version of the usual continuous core. The starting point is a normalfaithful state ϕ with modular group ( σ ϕt ) t ∈ R . Instead of working with R , we use the groupdiscrete G = (cid:83) n − n Z ⊂ R and the crossed product˜ M = M (cid:111) σ ϕ G .
The advantage here is that we have conditional expectation E : ˜ M → M given by E ( (cid:88) g x g λ ( g )) = x . (18)Let us state the main facts (see [25]:)Hi) E and ˜ ϕ = ϕ ◦ E are faithful.Hii) There exists an increasing family of subalgebras ˜ M k and normal conditional expectation F k : ˜ M → ˜ M k such that ˜ ϕF k = ˜Φ;Hiii) lim k (cid:107) F k ( ψ ) − ψ (cid:107) ˜ M ∗ = 0 for every normal state ψ ∈ ˜ M ;Hiv) For every k there exists a normal faithful trace trace τ k ( x ) = ˜ ϕ ( d k ( x )) such that d k ∈ ˜ M (cid:48) k and a k ≤ d k ≤ a − k for some scalars a k ∈ R + .Thanks to the conditional expectation, we have a canonical map E ∗ : M ∗ → ˜ M ∗ given by E ∗ ( ρ ) = ρ ◦ E . We will use the notation ˜ ρ = E ∗ ( ρ ). NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 33
Remark 8.2.
Let us recall two possible ways to represent the crossed product M (cid:111) G for anaction α of a discrete group on Hilbert space. We may assume that M ⊂ B ( H ) and consider (cid:96) ( G, H ). Then M (cid:111) G = (cid:104) λ H ( G ) , π ( M ) (cid:105) is generated by a copy of λ ( G ), the left regularrepresentation of G , and π ( M ). Here we may assume π ( x ) = (cid:88) g | g (cid:105)(cid:104) g | ⊗ α g − ( x )is given by a twisted diagonal representation and λ H ( g ) = λ ( g ) ⊗ H . Alternatively, we maychoose ˆ π ( x ) = 1 ⊗ x and ˆ λ H ( g ) = λ ( g ) ⊗ u g such that u ∗ g xu g = α g − ( x ). Both of theserepresentations are used in the literature, and their equivalence is used in the proof of Takai’stheorem. For the equivalence we note that λ H ( g ) − π ( x ) λ H ( g ) = π ( α g − ( x )) . Similarly, λ ( g ) − ⊗ u − g (1 ⊗ x ) λ ( g ) ⊗ u g = 1 ⊗ α − g ( x ). This shows that the algebraic relations ofthese two representations coincide. Using a GNS construction this extends to the generated vonNeumann algebras. Lemma 8.3.
Let ρ, ϕ be states on the von Neumann algebra M with corresponding ˜ ρ, ˜ ϕ in ˜ M ∗ .Then D ( ˜ ρ | ˜ ϕ ) = D ( ρ | ϕ ) .Proof. We consider the Hilbert space H = (cid:96) ( G, L ( M )) and still use the symbol λ ( g ) insteadof λ L ( M ) ( g ). Our first goal is to calculate the modular operator for an analytic state ϕ withdensity d in L ( M ), and ˜ ϕ = ϕ ◦ E , E : M (cid:111) G → M the canonical conditional expectation.Then ξ = | (cid:105) ⊗ d / implements the state ˜ ϕ on the crossed product. In order to calculate themodular operator ∆ = S ∗ S , we recall that( yξ, ∆( xξ )) = ( x ∗ ξ, y ∗ ξ ) . We start with finitely supported y = (cid:80) g λ ( g ) π ( y g ) , z = (cid:80) g λ ( g ) π ( z g ) and observe that( yξ, zξ ) = ( (cid:88) g | g (cid:105) y g d / , (cid:88) g | g (cid:105) z g d / ) = (cid:88) g ϕ ( y ∗ g x g ) . On the other hand, we find( x ∗ ξ, y ∗ ξ ) = ( (cid:88) g | g − (cid:105) α g ( x ∗ g ) d / , (cid:88) g | g − (cid:105) α g ( y ∗ g ) d / ) = (cid:88) g ϕ ( α g ( x g y ∗ g )) . Let d g − = α − g ( d ). Then we see that ϕ ( α g ( x g y ∗ g )) = tr ( d g − x g y ∗ g ) = tr ( d / y ∗ g d g − x g d − d / ) = ( y g d / , d g − x g d − d / ) . This means that the diagonal operator ∆ g ( ξ g ) = ∆ d g − ,d is a good candidate for the modularoperator, and is indeed well-defined for finitely supported sequences of σ tα − g ( ϕ ) ,ϕ -analytic ele-ments, which are dense. Now, it is easy to identify the polar composition using the isometry J ( (cid:80) g | g (cid:105) ξ g ) = (cid:80) g | g − (cid:105) α g ( ξ ∗ g ) on (cid:96) ( G, L ( M )), because α g extends to an isometry on L ( M ).This formula S = J ∆ / follows by calculation. Finally, we use Connes’ 2 × states ϕ, ψ and the diagonal state ˆ ϕ ( x ab ) = ϕ ( x )+ ψ ( x ). Note that M ( M ) (cid:111) G = M ( M (cid:111) G )and hence ∆ ˜ ϕ, ˜ ψ is the 1 , G -diagonal operator ∆ α − g ( ϕ ) ,ψ . This implies D ( ˜ ϕ | ˜ ψ ) = ( ξ ψ , log ∆ ˜ ϕ, ˜ ψ ( ξ ψ )) = ( d / ψ , ∆ α − ( ϕ ) ,ψ ( d / ψ ))= ( d / ψ , log ∆ ϕ,ψ ( d / ψ )) = D ( ϕ | ψ ) . Here we use that the relative entropy can be calculated on any representing Hilbert space.However, the representation of M (cid:111) G is in standard form, which may be used as a definition ofthe relative entropy.A similar result holds for the fidelity. Theorem 8.4.
Let ϕ be a faithful state. Then there exists a sequence of states ρ α such that i) δ α ϕ ≤ ρ α ≤ δ − α for some δ α > ; ii) lim α ρ α = ρ ; iii) D ( ρ | ϕ ) = lim α D ( ρ α | ϕ ) .Proof. Let us define ψ k = F k ( ˜ ρ ). Thanks to the Haagerup construction we know that lim k ψ k = ˜ ρ .We may apply Proposition 8.1 and find d k,m,δ = α k,m,δ (1 [0 ,m ] ( d ψ k ) d ψ k + δd ϕ k ), where α k,m,δ ischosen such that d k,m,δ has trace 1. Denote by ψ k,m,δ the corresponding state on ˜ M k and ψ k,m,δ = ψ k,m,δ ◦ F k . Let ρ k,m,δ be the restriction to M . Certainly, we find condition i). Moreover,by the data processing inequality (see Witten’s notes [19]) D ( ρ k,m,δ | ϕ ) ≤ D ( ψ k,m,δ | ϕ )and hence lim sup k →∞ ,m →∞ ,δ → D ( ρ k,m,δ | ϕ ) ≤ lim sup k D ( ψ k | ˜ ϕ ) ≤ D ( ˜ ρ | ˜ ϕ ) = D ( ρ | ϕ ) . However, we deduce from
Hiii ) and Proposition 8.1 thatlim k lim m lim δ ψ k,m,δ = ˜ ρ . Taking the conditional expectation E by restriction these state to M preserves this property.Thus by the semicontinuity of D Lin , we deduce that D ( ρ | ϕ ) ≤ lim inf k,m,δ D ( ρ k,m,δ | ϕ ) ≤ lim sup k,m,δ D ( ψ k,m,δ | ˜ ϕ ) ≤ D ( ρ | ϕ ) . This allows us to find a suitable convergent subsequence.
NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 35 Recovery of positive vectors
In this section, we explain how to recover certain vectors in a Hilbert space from a Petzrecovery map. Our starting point is representation of a von Neumann algebra M ⊂ B ( H ) and aseparating vector h ∈ M , i.e. the map x (cid:55)→ xh is injective. This implies that the correspondingnormal state ϕ ( x ) = ( h, xh ) has full support in M ∗ . Then we may apply the GNS constructionand a partial isometry U : M h → L ( M ) via U ( xh ) = xϕ / . Indeed, ( U ( xh ) , U ( yh )) = T r ( ϕ / x ∗ yϕ / ) = ϕ ( x ∗ y ) = ( xh, yh )shows that U extends to an isometry between M h and L ( M ). Recall that the inclusion M ⊂ B ( L ( M )) is in standard position. This means there is a real subspace L ( M ) + ⊂ L ( M ) andpartial isometry J such that J | L ( M ) + = id . In fact, all these objects can be constructed byTomita-Takesaki theory and J ϕ = U ∗ J U is indeed the anti-linear part of S = J ∆ / in the polardecomposition of S ( xh ) = x ∗ h . Of particular importance here is the real subspace H + = U ∗ ( L ( M ) + ) . The space of positive vectors is the range of Mazur map. Let us be more precise. For every normone vector k ∈ H we may consider the state ω k ( x ) = ( k, xk )which admits a density d k ∈ L ( M ) such that ω k ( x ) = T r ( d k x ) . Thanks to Størmer’s inequality the map d k (cid:55)→ d / k is continuous and hence | k | = U ∗ d / k ∈ H + . This allows us to reformulate the usual polar decomposition theorem.
Proposition 9.1.
Let h be a separating vector and H h = M h . Then every element k ∈ M h admits a polar decomposition k = v | k | where v ∈ M is a partial isometry, uniquely determined by v ∗ v = supp( ω k ) . Remark 9.2.
Since U ∗ : L ( M ) → M h we can also work with polar decomposition for theadjoint U ( k ) = | U ( k ) ∗ | w = R w ( | U ( k ) ∗ )where w belongs to the M , R w is the right multiplication and hence k = U ∗ R w U U ∗ ( | U ( k ) ∗ ) ∈ M (cid:48) H +6 M. JUNGE AND N. LARACUENTE admits a polar decomposition with respect to the commutant. In this form the theorem extendsto all of H . Indeed, let H = (cid:88) i M h i be a direct sum of irreducible subspaces with projections e i H = M h i in M (cid:48) . Then M h i ∼ = L ( M ) f i for some projection f i corresponding to the support of h i . Using an isomorphism V between H and ⊕ i L ( M ) f i we see that M (cid:48) ( M h ) = M (cid:48) h is dense in H . Using this isomorphism,we now deduce that k = wV ∗ ( | V ( k ) ∗ | )admits a polar decomposition with a partial isometry w ∈ M (cid:48) and V ∗ ( | V ( k ) ∗ | ) ∈ H + .For 1 ≤ p ≤ ∞ we may now consider the Kosaki interpolation space L p ( M, ω h ) as embeddedin H . Indeed, we have already the inclusion L ∞ ( M, ω h ) ∼ = M h ⊂ H ∼ = L ( M, ω h )and by interpolation we find an injective map U ∗ p : L p ( M, ω h ) → H .
This allows us to define the corresponding p -norm (cid:107) k (cid:107) p = sup {| ( ah, h ) | | (cid:107) aω /p (cid:48) h (cid:107) p < ∞} for 1 ≤ p ≤ ∞ . For 1 ≤ p ≤ H p = { k | (cid:107) k (cid:107) p < ∞} is dense in H and isomorphic L p ( M ). Therefore we find natural cones H p + = H p ∩ H + as the range of U ∗ ( L p ( M ) + ). Let us explain how these cones appear naturally in the context ofPetz maps. We will assume that Φ : L ( M ) → L ( ˆ M ) is a completely positive trace preservingmap and, for simplicity, that ϕ and ˆ ϕ = Φ( ϕ ) have full support. Then the Petz map R /p : L p ( ˆ M ) → L p ( M ) , R /p ( ˆ ϕ / p x ˆ ϕ / p ) = ϕ / p Φ † ( x ) ϕ / p is a contraction and sends L p ( ˆ M ) + to L p ( ˆ M ). Therefore we also find a contraction R /p : ˆ H p + → H p + . Let us describe this map more explicitly, by assuming that ω k ≤ Cω h and hence, as above, a ( z ) = ω z/ k ω − z/ h , ˆ a ( z ) = ˆ ω z/ k ˆ ω − z/ h are well defined. Then we find that R /p (ˆ ω /pk ) = ω / ph Φ † (ˆ a (1 / p ) ∗ ˆ a (1 / p )) ω / ph = ∆ / pω h (Φ † (ˆ a (1 / p ) ∗ ˆ a (1 / p )) ω /ph ) . NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 37
If we define b = Φ † (ˆ a (1 / p ) ∗ ˆ a (1 / p )) we see that R /p (ˆ ω /pk ) = ∆ / pω h ( bh ) ∈ H p + . On the other hand we see that k ∈ ˆ H p + is represented ˆ U ( k ) = ˆ ω /pk ˆ ω − /ph ˆ ω /ph . This impliesˆ a (1 / p ) ∗ ˆ a (1 / p ) = ˆ ω − / ph ˆ ω /pk ˆ ω − / ph = ∆ − / p ˆ ω h (ˆ ω /pk ˆ ω − /ph ) . Let us recall the map Φ † p ( b ˆ ω /ph ) = Φ † ( b ) ω /ph which we extend to a densely map on H p as followsΦ † p ( b ˆ h ) = Φ † ( b ) h . Then we can combine the calculations above and find that R /p = ∆ / pω h Φ † p ∆ − / p ˆ ω h . (19)Our fidelity result can be formulated as follows: Corollary 9.3.
Let h be a separating vector for M with associated vector state ω h , and let Φ † : ˆ M → M be a normal, unital completely positive map and ˆ ω h = ω h ◦ Φ † the associated vectorstate. Then map R /p : ˆ H p → H p R /p = ∆ / pω h Φ † p ∆ − / p ˆ ω h extends to a contraction and satisfies − ln f p ( k, R /p (ˆ k )) ≤ p ( D ( ω k | ω h ) − D (ˆ ω k | ˆ ω h )) . for every k ∈ H p + . Our next application tells us that if we use the standard form of representing a states on vonNeumann algebras, then we may recover the implementing vector:
Corollary 9.4.
Let H = L ( M ) . Then implementing vectors ξ ρ for ρ and ξ ˆ ρ satisfy (cid:107) ξ ρ − R / ( ξ ˆ ρ ) (cid:107) ≤ D ( ρ | ϕ ) − D (Φ( ρ ) | Φ( ϕ )) . . Proof.
Let us first consider a, b ∈ L ( M ) + of norm 1 and h = b − a . Then0 = (cid:107) b (cid:107) − (cid:107) a (cid:107) = (cid:107) a + h (cid:107) − (cid:107) a (cid:107) = 2( a, h ) + (cid:107) h (cid:107) . On the other hand1 − f ( a, b ) = (cid:107) a (cid:107) − (cid:107) a / b / (cid:107) = tr ( a ) − tr ( ab ) = tr ( a ( a − b ))= − ( a, h ) = (cid:107) h (cid:107) . Then ln(1 + x ) ≤ x implies for a = ρ / and b = R / ( ˆ ρ / ) that − ln f ( a, b ) = − ln(1 − (1 − f ( a, b ) )) ≥ (1 − f ( a, b ) ) ≥ (cid:107) a − b (cid:107) . The assertion then follows from Theorem 5.9.
Remark 9.5.
The proof of equations (9) and (10) in the introduction follows via the triangleand Cauchy-Schwarz inequalities.
As an illustration we will now assume that ˆ M ⊂ M is a subalgebra and that there exists anormal conditional expectation E : M → ˆ M such that ω h = ω h | ˆ M ◦ E .
In this case Φ † = ι is just the inclusion map ˆ M ⊂ M and moreover, Φ † commutes with themodular group (see [37]). Then E extends to map E : L ( M ) + → L ( ˆ M ) + via E ( xω / h ) = E ( x )ˆ ω / h . Under these additional assumptions, we see that R /p : ˆ H p → H p is simply the inclusion map.In his particular case the fidelity can also be expressed easily. Indeed, according to the proof ofLemma 5.1 we know that f p ( k (cid:48) , k ) = sup (cid:107) ak (cid:107) p (cid:48) ≤ | ( ak, ∆ / pk (cid:48) ,k ( k )) | . The case p = 2 is particularly interesting and gives the self-polar form f ( x, y ) = (cid:107) x / y / (cid:107) = T r ( x / y / ) . For elements k, k (cid:48) ∈ H + we may assume k = aω / h and k (cid:48) = bω / h , and x / = U ( aω / h ), y / = U ( bω / h ). This means f ( x, y ) = T r ( ω h b ∗ a ) = ( h, b ∗ ah ) = ( bh, ah ) = ( k (cid:48) , k ) . Corollary 9.6.
In addition to the assumption of 9.3 assume that ω h = ˆ ω h ◦ E holds for a normalconditional expectation. For k ∈ H + − ln( k, E ( k )) ≤ D ( ω h | ω k ) − D (ˆ ω h | ˆ ω k ) . Remark 9.7.
Without assuming the existence of E , we can still describe the Petz map for L inthis special case. Indeed, let us assume that ˆ M ⊂ M and denote by ˆ ι : ˆ M h → M h the canonicalinclusion map. We will assume that k ∈ H + ( ˆ M ) and ω k ≤ Cω h (which implies ˆ ω k ≤ C ˆ ω h .Then ˆ ω / k = ˆ ω / k ˆ ω − / h ˆ ω / h implies k = ˆ ω / k ˆ ω − / h h and ˆ∆ − / ( k ) = ˆ∆ − / (ˆ ω / k ˆ ω − / h ) h . Thanks to (19) this implies ξ = R / ( k ) = ∆ / (ˆ ι (( ˆ∆ − / ( k ))) . NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 39
Let P / be the orthogonal projection onto the rotated space ˆ H / = ∆ / ( ˆ M h ). Then ξ ∈ ˆ H / implies ( | k | , ξ ) = ( P / | k | , ξ ) = (cid:107) P / | k |(cid:107)(cid:107) ξ (cid:107) ≤ (cid:107) P / | k |(cid:107) . Therefore we deduce that − ln (cid:107) P / | k |(cid:107) ≤ D ( ω h | ω k ) − D (ˆ ω h | ˆ ω k ) . In particular, if the relative entropy difference is small, then P / | k | ≈ | k | implies that U ( | k | )almost commutes with ω h .10. Data processing inequality for p -fidelity Theorem 10.1.
Let
Φ : L ( M ) → L ( ˆ M ) be a channel. Then f p (Φ( ρ ) , Φ( σ )) ≥ f p ( ρ, σ ) . We need the following L p norm inequality Proposition 10.2.
Let Φ † be a normal, unital, completely, positive adjoint map of a channel Φ ,and ϕ be a normal state on M such that Φ( ϕ ) = ˆ ϕ . Then Φ p : L p ( ˆ M ) → L p ( M ) given by Φ p ( x ) = ϕ / p Φ † ( ˆ ϕ / p x ˆ ϕ / p ) ϕ / p is a completely positive contraction.Proof. We may assume that the density ϕ of a given state has full support, let ˆ e be the supportof ˆ ϕ , so that we may assume that Φ p is defined on ˆ eL p ( ˆ M ))ˆ e . This allows us to use the Kosakiisomorphism L p ( ˆ M ) = L p ( ˆ M , ˆ ϕ ). With the help of this automorphism, we consider the denselydefined map T ( ˆ ϕx ˆ ϕ ) = ϕ / Φ( x ) ϕ / . Since Φ † : ˆ M → M is contraction, we see that (cid:107) T ( x ) (cid:107) ∞ ≤ (cid:107) x (cid:107) ∞ . On the other hand let us assume that x = ab . Then we see deduce from the Cauchy-Schwarzinequality for completely positive maps that (cid:107) T ( ˆ ϕ / ab ˆ ϕ / ) (cid:107) = (cid:107) ϕ / Φ † ( ab ) ϕ / (cid:107) ≤ (cid:107) ϕ Φ † ( aa ∗ ) ϕ (cid:107) / (cid:107) ϕ Φ † ( b ∗ b ) ϕ (cid:107) / = tr ( ϕ Φ † ( aa ∗ )) / tr (Φ † ( b ∗ b ) ϕ ) / = tr ( ˆ ϕ ( aa ∗ )) / tr ( ˆ ϕb ∗ b ) / = (cid:107) ˆ ϕa (cid:107) (cid:107) b ˆ ϕ (cid:107) . By density of ˆ M ˆ ϕ / in L ( ˆ M )ˆ e , we deduce that (cid:107) T ( ξη ) (cid:107) ≤ (cid:107) ξ (cid:107) (cid:107) η (cid:107) for any ξ and η . Thus T extends to a completely positive contraction on ˆ eL ( ˆ M )ˆ e . By the generalRiesz-Thorin theorem (see [30]), we deduce that T : L p ( ˆ M , ˆ ϕ ) → L p ( M, ϕ ) is a contraction. ByKosaki’s theorem, this completes the proof.
Corollary 10.3.
Let ϕ, ρ be two densities of states. Then T ϕ,ρp ( x ) = ϕ / p Φ † ( ˆ ϕ − / p x ˆ ρ − / p ) ρ / p ) extends to a contraction from L p ( ˆ M ) to L p ( M ) .Proof. We use Connes’ matrix trick and consider σ = (cid:18) ρ ϕ (cid:19) on M ( M ) for Φ = id M ⊗ Φ.The assertion follows from applying Proposition 10.2 to y = (cid:18) x (cid:19) . Proof of 10.1.
Let x = ˆ ϕ / p ˆ ρ / p . Then we deduce that T ϕ,ρp ( ˆ ϕ / p ˆ ρ / p ) = ϕ / p ρ / p . Since T ϕ,ρp is a contraction, we deduce that f p ( ϕ, ρ ) = (cid:107) ϕ / p ρ / p (cid:107) p ≤ (cid:107) ˆ ϕ / p ˆ ρ / p (cid:107) p = f p ( ˆ ϕ, ˆ ρ ) . Corollary 10.4. If D ( ρ | ϕ ) = D (Φ( ρ ) | Φ( ϕ )) for a channel Φ : L ( M ) → L ( ˆ M ) , then T ϕ,ϕp ( σ ˆ ϕs ) = σ ϕs ( ρ /p ) holds for all ≤ p ≤ ∞ and s ∈ R . Moreover, there exists a modular group intertwining channel Ψ : L ( M ) → L ( ˆ M ) such that ˆ T p ( x ) = σ / p Ψ † (ˆ σ − / p x ˆ σ − / p ) σ / p satisfies ˆ T p ( ˆ ρ /p ) = ρ /p and Ψ( ρ ) = Φ( ρ ) . Proof.
In this case − ln f p ( ρ, R p,t ( ˆ ρ /p ) p ) = 0holds µ almost everywhere. By continuity this holds for all t . In other words, thanks to theMazur map, we get ρ / p σ − it/ p Φ † (ˆ σ − (1 − it/ p ˆ ρ /p ˆ σ − (1+ it ) / p ) σ (1+ it ) / p ρ / p = ρ /p for all t . This implies T ϕp ( σ ˆ ϕ ( s )( ˆ ρ /p )) = σ ϕ ( s ) ρ /p for all s . For the moreover part we consider the family R p ( x ) = ˆ σ − / p (cid:48) Φ( σ / p (cid:48) xσ / p (cid:48) )ˆ σ / p .Thanks to data processing inequality for sandwiched relative entropy, this map is contraction,and hence Ψ ( x ) = lim T, U (cid:90) T − T σ ˆ ϕ ( s )Φ ( σ ϕ ( − s )( x )) ds T NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 41 exists as a bounded operator on L . By density of L in L we deduce thatΨ ( ϕ / xϕ / ) = ˆ ϕ / Ψ ( x ) ˆ ϕ / = lim T, U (cid:90) T − T σ ˆ ϕ ( s )Φ ( σ ϕ ( − s )( x )) ds T is a completely positive map on L ( M ). Its adjoint Ψ † is normal, unital completely positivemap, defined as a point weak ∗ limit of averages. Hence our assumption shows that ˆ T p ( x ) = ϕ / p Ψ † ( ˆ ϕ − / p x ˆ ϕ − / p ) ϕ / p also satisfiesˆ T p ( ˆ ρ /p ) = ρ /p for 1 ≤ p ≤ ∞ . For the final assertion, we have to establish a simple duality relation. UsingKosaki L p spaces, we see that the family of mapsΦ p ∼ = Φ | ι p ( L p ( M,ϕ )) is really the same map, via the topological embedding ι p ( x ) = ϕ / p xϕ / p . Similarly, ϕ / p (cid:48) T p ( ˆ ϕ / p xϕ / p ) ϕ / p (cid:48) = T ( ˆ ϕ / x ˆ ϕ / )show that T p = T | ι p ( L p ) is also the same map. Moreover, T r (Φ( ϕ / xϕ / y )) = tr ( ϕ / xϕ / Φ † ( y )) = tr ( xT (ˆ σ / y ˆ σ / ))shows that T p = Φ † p (cid:48) , by density. The same holds for ˆ T p = Ψ † p (cid:48) . Now, it is easy to conclude.Our assumption implies 1 = T r ( ρ /p ρ /p (cid:48) ) = T r ( ρ /p ˆ T p (cid:48) ( ˆ ρ /p (cid:48) ))= ( ι p ( ρ /p ) , ˆ T ( ι p (cid:48) ( ˆ ρ /p (cid:48) )))= (Ψ( ι p ( ρ /p ) , ι p (cid:48) ( ˆ ρ /p (cid:48) )))= T r (Ψ p ( ρ /p ) ρ /p (cid:48) ) . By uniform convexity of L p we deduce thatΨ p ( ρ /p ) = ˆ ρ /p = Φ( ρ ) /p . For p →
1, we deduce the assertion. 11. L isometries In the theory of von Neumann algebras completely isometric embeddings of L ( N ) into L ( M ) are completely characterized (see [39] for more information on the crucial work by Kirch-berg). Indeed, a map u : L ( N ) → L ( M ) is complete isometry iff there exists a normalconditional expectation E : M → N ⊂ N , a ∗ -homomorphism π : M → N and J ∈ N (cid:48) suchthat u ( ϕ / xϕ / ) = ˆ ϕπ ( x ) J ˆ ϕ . Such a map is completely positive if J is completely positive. Moreover, the inverse u − extendsto L ( M ). Let us formulate a simple consequences of the the data processing inequalities. Lemma 11.1.
Let u be a completely positive complete isometry u : L ( N ) → L ( ˜ N ) . Then D ( u ( ϕ ) | u ( ρ )) = D ( ϕ | ρ ) provided they are finite. Moreover, f p ( u ( ρ ) , u ( ϕ )) = f p ( ρ, ϕ ) . Lemma 11.2.
Let ˆ M and ˆ N be semifinite and Φ : L ( M ) → L ( ˆ M ) , ρ ≤ Cϕ such that D ( ρ | ϕ ) = D (Φ( ρ ) | Φ( ϕ )) . Then there exists completely positive L -isometry u such that ˆ ϕ = u ( ϕ ) and ˆ ρ = u ( ρ ) .Proof. Let Ψ † : ˆ M → ˆ N the averaged map. Then we see thatΨ † ( ˆ ρ / ) = ρ / and hence ρ = Ψ † ( ˆ ρ / )Ψ † ( ˆ ρ / ) ≤ Ψ † ( ˆ ρ ) = ρ . Thus we equality in Kadison’s inequality, and ˆ ρ belongs to the (extended) multiplicative domain m ⊂ ˆ M . Since Ψ is normal and invariant under σ ˆ ϕ , we see that the multiplicative domain m admits a ϕ -invariant conditional expectation E : ˆ M → m such that ˆ ϕE = ˆ ϕ , see e.g. [31] andalso [37]. In particular we have completely isometric, completely positive inclusion ι : L ( m ) → L ( ˆ M ) such that ι ( ˆ ϕ / x ˆ ϕ / ) = ˆ ϕ / x ˆ ϕ / . Let us denote by ˆ M ( ˆ ρ, ˆ ϕ ) ⊂ m be the smallest von Neumann algebra generated by C ∗ ( ˆ ρ ) and σ ˆ ϕt , which remains ˆ ϕ -complemented. Let f : R → R be a bounded function. Then we deducethat Ψ † ( f ( ρ )) = f ( ρ ) , Ψ † ( σ ˆ ϕ ( t )( f ( ρ ))) = σ t f ( ρ ) . This means that Ψ † extends to a natural isomorphism between ˆ M ( ˆ ρ, ˆ ϕ ) and M ( ρ, ϕ ) such that tr ( ϕ Ψ † ( x )) = tr (Ψ( ϕ ) x ) = tr ( ˆΦ( x )) . The adjoint of u = (Ψ † | ˆ M (ˆ ρ, ˆ ϕ ) ) † satisfies u ( ϕ ) = ˆ ψ and tr ( u ( ρ ) x ) = tr ( ρ Ψ † ( x )) = tr ( ˆ ρ ( x )) . Since M ( ρ, ϕ ) is also ϕ -conditioned, we deduce the assertion. Remark 11.3.
It follows easily that u ( ϕ ) /p = ˆ ϕ /p and u ( ρ ) /p = ˆ ρ /p holds for all 1 ≤ p ≤ ∞ , under the assumptions above. NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 43
We want to extend this result to type
III von Neumann algebras. For this we need thenotion of the multiplicative domain. For a completely positive unital map Φ : M → N withStinespring dilation Φ( x ) = V ∗ π ( x ) V , we recall that x belongs to the right domain ifΦ( x ) ∗ Φ( x ) = Φ( x ∗ x ) (20)or equivalently V ∗ π ( x )(1 − V V ∗ ) π ( x ) V = 0. If x and x ∗ satisfy (20), then [ V, π ( x )] = 0 holdsfor a minimal Stinespring dilation. The setmdom(Φ) = { x | [ V, π ( x )] = 0 } = { x | Φ( x ∗ )Φ( x ) = Φ( x ∗ x ) and Φ( x )Φ( x ∗ ) = Φ( xx ∗ ) } is a sub- C ∗ -algebra of M and for normal Φ, hence normal π , see [36, 35], this is even a sub-vonNeumann algebra. Lemma 11.4.
Let Φ n : ˆ M → M be a sequence of normal completely positive maps such that i) The weak ∗ limit Φ ∞ ( x ) = lim n Φ n ( x ) ;ii) Φ † n ( σ ) = ˆ σ for normal faithful states σ and ˆ σ ; iii) ( σ / Φ n ( x ) , Φ m ( y ) σ / ) = ( σ / Φ min( n,m ) ( x ) , Φ min( n,m ) ( y ) σ / ) .Let ( a n ) be a bounded sequence in the multiplicative domain of Φ n , converging strongly to a .Then a belongs to the multiplicative domain of Φ .Proof. We follow Kirchberg and use the C ∗ -algebra C ( ˆ M ) of all bounded sequences ( a n ) suchthat a n converges in the strong and strong ∗ -algebra. Similarly, we consider C ( ˆ M ) and thecorresponding quotient maps ˆ q and q : C ( M ) → M given by q (( a n )) = w ∗ lim n a n . We claimthat Φ • C ( ˆ M ) ⊂ C ( M ). Indeed, assume that lim n a n − a converges to 0 strongly. Then a n − a ˆ σ converges to 0 in L ( ˆ M ). Let us fix n ≤ mm . We find that (cid:107) (Φ n ( a n ) − Φ m ( a m )) σ / (cid:107) = T r ( σ / Φ n ( a ∗ n a n )ˆ σ / ) + T r ( σ / Φ m ( a ∗ m a m )ˆ σ / ) − T r ( σ / Φ n ( a ∗ n )Φ m ( a m ) σ / ) − T r ( σ / Φ m ( a m ) ∗ Φ n ( a n ) σ / )= T r (Φ ∗ n ( σ )( a ∗ n a n )) + T r (Φ ∗ m ( σ )( a ∗ m a m )) − T r (Φ ∗ n ( σ )( a ∗ n a m )) − T r (Φ ∗ n ( σ )( a ∗ m a n ))= T r (ˆ σ ( a ∗ n a n + a ∗ m a m − a ∗ n a m − a ∗ m a n ))= (cid:107) ( a n − a m )ˆ σ (cid:107) . Since σ is faithful and (Φ n ( a n )) bounded, we deduce that Φ n ( a n ) is also strongly convergent.Let ˆ M n ⊂ M the multiplicative domain of A = { ( x n ) | x n ∈ ˆ M n } the corresponding subalgebraof (cid:96) ∞ ( ˆ M ). Then Φ • : A → (cid:96) ∞ ( M ) is a ∗ -homomorphism, and we may define A = C ( ˆ M ) ∩ A .Then Φ ∞ | A : A → C ( M )is a C ∗ -homomorphism. Let ˆ J ⊂ C ( ˆ M ) be the kernel of the quotient map ˆ q . Since Φ ∞ preservesstrong convergence, we deduce that Φ ∞ ( ˆ J ) ⊂ J , J the kernel q . We deduce that there exists a ∗ -homomorphism π : ˆ q ( A ) ⊂ C ( ˆ M ) / ˆ J = ˆ M to M = C ( M ) /J such that q Φ ∞ ( a n ) = σ ( q ( a n )) . Note that σ is the restriction of the completely positive map ˜Φ : C ( ˆ M ) / ˆ J → C ( M ) /J . Byapplying this map to the constant sequence ( b n ) = b , we deduce that ˜Φ = Φ ∞ . Thus for everystrongly convergent sequence in A , we deduce that a = lim n a n belongs to the multiplicativedomain of Φ ∞ because σ ( a ∗ a ) = σ ( a ) ∗ σ ( a ) and σ ( a ) ∗ σ ( a ) = σ ( aa ∗ ). Theorem 11.5.
Let ρ ≤ λϕ , and Φ : L ( M ) → L ( ˆ M ) . Then the following are equivalent i) D (Φ( ρ ) | Φ( ϕ )) = D ( ρ | ϕ ) ; ii) There exists a ϕ -conditioned subalgebra M ⊂ M and an completely positive L -isometry u such that u ( ϕ ) = Φ( ϕ ) , u ( ρ ) = Φ( ρ ) . Proof.
Thanks to Lemma 11.1 we only have to prove i ) ⇒ ii ). In view of Corollary 10.4, wemay assume that Φ = Ψ intertwines σ ϕ and σ ˆ ϕ . Let G = (cid:83) k − k Z . Since Ψ is σ -invariantwe know that Ψ G = Ψ (cid:111) G extends to the cross product. Recall that ϕ G = ϕ ◦ E G , and ρ G = ρ ◦ E G naturally extend to the discrete crossed product. Let us recall that Ψ G extends toa map T G : L ( ˆ M G ) → L ( M G ) via T G ( ˆ ϕ / G x ˆ ϕ / G ) = ϕ / G Φ † G ϕ / G . Since D ( ρ G | ϕ G ) = D ( ρ | ϕ ) and D (Ψ G ( ρ ) | Ψ G ( ϕ G )) = D (Ψ( ρ ) | Ψ( ϕ )), we deduce that T G ( ˆ ρ G ) = ρ G . Let E n be the conditional expectation given by the Haagerup construction. Note that T G E n = E n T G follows from the fact that Ψ commutes with the modular group. Thus for every n ∈ N ,we may apply Lemma 11.2 and find A n = ˆ M n ( E n ( ρ G )) , E n ( ϕ G )) in the multiplicative domainwhich is modular group invariant.Let us now assume that ρ = ϕ / hϕ / for a bounded h and hence (using the map Ψ insteadof Φ) that ˆ ρ = ˆ ϕ / ˆ h ˆ ϕ , ˆ ρ G = ˆ ϕ G ˆ h ˆ ϕ G . Let d n and ˆ d n the densities of ˆ ϕ G | ˆ M n and ϕ G | M ( n ) , respectively. Recall that ˆ d n , and d n belongto the center of ˆ M ( n ) and M ( n ). Then E n ( ˆ ρ G ) = ˆ d / n E n (ˆ h ) ˆ d / n implies that ˆ h n = E n (ˆ h ) also belongs to the multiplicative domain of Ψ † n = Ψ † E n . In order toapply Lemma, we recall that ϕ G and ˆ ϕ G are E n invariant. Since ˆ M n are increasing, we deducethat for n ≤ mT r ( ϕ / G E n Ψ † ( a ) E m ( b ) ϕ / G ) = T r ( ϕ / G Ψ † ( E n ( a ) E m ( b )) ϕ / G )= T r (Ψ( ϕ G ) E m ( E n ( a ) b )) = T r ( ˆ ϕ G ( E n ( a ) b ))= T r ( ˆ ϕ G ( E n ( a ) E n ( b ))) = T r ( ϕ / G E n Ψ † ( a ) E n ( b ) ϕ / G ) . NIVERSAL RECOVERY AND P-FIDELITY IN VON NEUMANN ALGEBRAS 45
Note that for Φ n = Ψ † G E n we have Φ ∞ = Ψ † and hence ˆ k = lim n ˆ k n belongs to the multiplicativedomain of Ψ † G , and hence to multiplicative domain of Ψ. Indeed, we may consider a n ˆ k / n . Then a ∗ n a n converges weakly to ˆ k if a n − ˆ k / converges strongly to 0. Using E n ( ˆ ρ G ) = ˆ ϕ / G E n (ˆ k ) ˆ ϕ / G we deduce that weak-convergence from the crucial inequalitylim n (cid:107) E n ( ˆ ρ G ) − ˆ ρ G (cid:107) in the Haagerup construction. Note also that (cid:113) ˆ d / n ˆ k n ˆ d / n = ˆ d / n ˆ k / n ˆ d / n because ˆ d n belongs to the center of ˆ M ( n ), which allows us to use Størmer’s inequality. Sincethe multiplicative domain of Φ † is invariant under the modular group of ˆ ϕ and ˆ k belongs tothe smallest modular group invariant von Neumann subalgebra ˆ M which is mapped to M thesmallest modular group invariant generated by h , we can now conclude as in Lemma 11.2.12. Acknowledgements
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Appendix A. Extending the Differentiation Lemma
Lemma A.1.
Let h : I → M be a differential function such that h (0) = 1 and ϕ be a faithfulstate. Let p be a differentiable function and p (0) > . If in addition h ( θ ) = v ( θ ) ∗ v ( θ ) and v ( θ ) is invertible, then part ii) of Lemma 5.8 also holds for p (0) > / .Proof. Let g ( θ ) = (cid:107) ϕ / p ( θ ) h ( θ ) ϕ / p ( θ ) (cid:107) p ( θ ) p ( θ ) . Let H ( t ) = ϕ / p ( θ ) h ( tθ ) ϕ / p ( θ ) . We recall theproof of Lemma 5.8. For the case p (0) = 1, we consider q ( θ ) = 2 p ( θ ) and H ( t ) = H ( t, θ ) = (cid:18) v ( tθ ) ρ /q ( θ ) ρ /q ( θ ) v ( tθ ) ∗ (cid:19) We still have (for q fixed) (cid:107) ρ /q h ( tθ ) ρ /q (cid:107) pp − (cid:107) H ( t ) (cid:107) qq − (cid:107) H (0) (cid:107) qq q (cid:90) tr ( H ( t ) q − H (cid:48) ( t )) dt . We may write H ( t ) q − = H ( t ) H ( t ) q − . Thanks to the off diagonal structure, we see that only theeven part of H ( t ) q − can contribute to the trace. Let H ( t ) = W t | H ( t ) | be the polar decompositionand note that the even part of H ( t ) q − equals | H ( t ) | q − . Thus we get tr ( H ( t ) q − H (cid:48) ( t )) = tr ( H ( t ) | H ( t ) | q − H (cid:48) ( t )) = tr ( W t | H ( t ) | q − H (cid:48) ( t )) . Since H ( t ) is invertible (as unbounded operator) the same us true for | H ( t ) | and hence W t is aunitary. Therefore we deduce that W t = (cid:18) (cid:19) U t , U t = (cid:18) u ( t, θ ) 00 u ( t, θ ) (cid:19) , and tr ( H ( t ) q − H (cid:48) ( t )) = tr ( H ( t ) | H ( t ) | q − H (cid:48) ( t )) = tr ( | H ( t ) | q − H (cid:48) ( t ) W t )= tr (( | H ( t ) | q − − | H (0) | q − ) H (cid:48) ( t ) W t ) + tr ( | H (0) | q − H (cid:48) ( t ) W t ) Since now q ( θ ) converges to 2, we can use the same H¨older type estimate and continuity ofthe Mazur map to control the first error term. For the second term we observe that H (0) = (cid:18) ρ /q ρ /q (cid:19) and hence | H (0) | = (cid:18) ρ /q ρ /q (cid:19) . This implies | H (0) | q − = (cid:18) ρ − /q ρ − /q (cid:19) and hence tr ( | H (0) | q − H (cid:48) ( t ) W t ) = θ ( tr ( ρ − /q h (cid:48) ( tθ ) ρ /q u ( t, θ )) + tr ( ρ − /q ρ /q h (cid:48) ( tθ ) ∗ u ( t, θ ))) . By the dominated convergent theorem we deduce thatlim θ → (cid:107) ρ /q ( θ ) h ( tθ ) ρ /q ( θ ) (cid:107) p ( θ ) p ( θ ) − θ = q (0)2 lim θ → (cid:90) ( tr ( ρ − /q h (cid:48) (0) ρ /q u ( t, θ )) + tr ( ρ − /q ρ /q h (cid:48) (0) ∗ u ( t, θ ))) dt . The family of operators H ( t, q ( θ ) converges in the measure topology to H (0 , q (0)) and, thanks toinvertibility, we deduce that lim θ → u j ( t, θ ) = 1. Thus our function g ( θ ) = (cid:107) ρ /q ( θ ) h ( tθ ) ρ /q ( θ ) (cid:107) p ( θ ) p ( θ ) satisfies g (cid:48) (0) = p (0) tr ( ρv (cid:48) (0)) + tr ( ρv (cid:48) (0) ∗ ) . Note that h ( t ) = v ( t ) ∗ v ( t ) satisfies h (cid:48) (0) = v (cid:48) (0) ∗ v (0) + v (0) ∗ v (cid:48) (0) = v (cid:48) (0) ∗ + v (cid:48) (0) and hence thisis exactly the same formula as for p (0) > Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
E-mail address , Marius Junge: [email protected]
Department of Physics, University of Illinois, Urbana, IL 61801, USA
E-mail address , Nick LaRacuente:, Nick LaRacuente: