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arXiv:2605.08829

Recoverable states on von-Neumann algebras

Saptak Bhattacharya

quant-phmath.FAmath.OASubmitted May 9, 2026• Announced May 12, 2026• 0 citations
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Abstract

Let $(\mathcal{M},τ)$ and $(\mathcal{N},τ^{\prime})$ be tracial von-Neumann algebras and let $φ:\mathcal{M}\to\mathcal{N}$ be a strictly completely positive, trace preserving map. Given a positive, invertible $B\in\mathcal{M}$ with $τ(B)=1$, a state on $\mathcal{M}$ given by a positive $A\in L^1(\mathcal{M}, τ)$ is said to be recoverable if $\mathcal{R}(φ(A))=A$ where $\mathcal{R}$ is the Petz recovery map corresponding to $B$ and $φ$. In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of $\mathcal{R}\circφ$. We show that there exists a completely positive, trace preserving map $ψ:\mathcal{M}\to\mathcal{M}$ such that $ψ(A)$ is recoverable for all $A$ and $(\mathcal{R}\circφ)^n\toψ$ in norm as operators on $L^p(\mathcal{M},τ)$ for all $1\,\textless p\,\textless\infty$, and discuss potential applications to quantum information theory. We also show that this convergence holds strongly in $L^1$. Finally, we prove an interesting decomposition theorem for normal states on $\mathcal{M}$.

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In this paper, we discuss a refinement of quantum data processing inequality for the sandwiched quasi-relative entropy S2\mathcal{S}_2S2​ on a tracial von-Neumann algebra. The main result is a universal recoverability bound with the Petz recovery map, which was previously obtained in the finite dimensional setup.
We prove recovery theorems for ααα-zzz Rényi divergences under normal unital positive maps between von Neumann algebras. Previous results in this setting required 2-positivity, while recent finite dimensional work showed that this assumption can be relaxed to mere positivity. Our proof uses sufficiency for JW*-subalgebras and LpL^pLp-spaces over them to characterize equality in the data processing inequality. As a further application of our methods, we solve a problem discussed by Haagerup and Stormer: Every conditional expectation of a von Neumann algebra onto a JW*-subalgebra factors through a conditional expectation onto the generated von Neumann subalgebra.
Abstract For a state ω\omegaω on a C ∗^{*}∗ -algebra AAA , we characterize all states ρ\rhoρ in the weak* closure of the set of all states of the form ω∘φ\omega \circ \varphiω∘φ , where φ\varphiφ is a map on AAA of the form φ(x)=∑i=1nai∗xai,\varphi (x)=\sum \nolimits _{i=1}^{n}a_i^{*}xa_i,φ(x)=∑i=1n​ai∗​xai​, ∑i=1nai∗ai=1\sum \nolimits _{i=1}^{n}a_i^{*}a_i=1∑i=1n​ai∗​ai​=1 ( ai∈Aa_i\in Aai​∈A , n∈Nn\in \mathbb {N}n∈N ). These are precisely the states ρ\rhoρ that satisfy ∥ρ∣J∥≤∥ω∣J∥\|\rho |J\|\leq \|\omega |J\|∥ρ∣J∥≤∥ω∣J∥ for each ideal JJJ of AAA . The corresponding question for normal states on a von Neumann algebra R\mathcal {R}R (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form ω∘ψ\omega \circ \psiω∘ψ , where ψ\psiψ is a quantum channel on R\mathcal {R}R (that is, a map of the form ψ(x)=∑jaj∗xaj\psi (x)=\sum \nolimits _ja_j^{*}xa_jψ(x)=∑j​aj∗​xaj​ , where aj∈Ra_j\in \mathcal {R}aj​∈R are such that the sum ∑jaj∗aj\sum \nolimits _ja_j^{*}a_j∑j​aj∗​aj​ converge to 111 in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C ∗^{*}∗ -algebra and for properly infinite von Neumann algebras the converse also holds.
We study the interconversion of families of quantum states ("statistical experiments") via positive, trace-preserving (PTP) maps and clarify its mathematical structure in terms of minimal sufficient Jordan algebras, which can be seen to generalize the Koashi-Imoto decomposition to the PTP setting. In particular, we show that Neyman-Pearson tests generate the minimal sufficient Jordan algebra, and hence also the minimal sufficient *-algebra corresponding to the Koashi-Imoto decomposition. As applications, we show that a) equality in the data-processing inequality for the relative entropy or the ααα-zzz quantum Rényi divergence implies the existence of a recovery map also in the PTP case and b) that two dichotomies can be interconverted by PTP maps if and only if they can be interconverted by decomposable, trace-preserving maps. We thoroughly review the necessary mathematical background on Jordan algebras. As a step beyond the finite-dimensional case, we prove Frenkel's formula for approximately finite-dimensional von Neumann algebras.
Abstract We establish an operator algebra generalization of Watrous’ theorem (Watrous 2009 Quantum Inf. Comput. 9 403–413) on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.
Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on M⊗ˉN\mathcal{M}\bar{\otimes}\mathcal{N}M⊗ˉ​N but on the tensor product of M\mathcal{M}M with the \emph{opposite} algebra of N\mathcal{N}N, or equivalently with a commutant; returning to N\mathcal{N}N itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is ∗*∗-anti-isomorphic to itself. Naturality cannot be dropped: whenever M\mathcal{M}M is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.