Recoverable states on von-Neumann algebras

Abstract: Let $(\mathcal{M},τ)$ and $(\mathcal{N},τ<sup>{\prime})$ 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 $(\mathcal{R}\circφ)<sup>n\toψ$ in norm as operators on $L<sup>p(\mathcal{M},τ)$ for all $1\,\textless p\,\textless\infty$, and discuss potential applications to quantum information theory. We also show that this convergence holds strongly in $L<sup>1$. Finally, we prove an interesting decomposition theorem for normal states on $\mathcal{M}$.

• The paper demonstrates that iterated Petz recovery maps converge in L^p norm to a recoverable state projection in infinite-dimensional von Neumann algebras.
• A decomposition theorem uniquely splits any normal state into a recoverable part and a vanishing component, offering insights for quantum error correction.
• Methodologies leveraging spectral analysis, complex interpolation, and ergodic theory underpin the stability results and data processing inequality in non-commutative L^p spaces.

Recoverable States and Iterated Petz Recovery on von Neumann Algebras

Introduction

This paper addresses the structure and approximation of "recoverable" quantum states in the setting of tracial von Neumann algebras, focusing on the action of the Petz recovery map composed with a quantum channel. The context is motivated by quantum information theory, particularly regarding recovery theorems, the quantum data processing inequality (DPI), and the operational significance of relative entropies and recoverability in infinite-dimensional algebras. The core technical results revolve around the dynamics of the composed map $\mathcal{R} \circ \phi$ and its iterates on non-commutative $L^p$ spaces.

Framework and Preliminaries

Let $(\mathcal{M}, \tau)$ and $(\mathcal{N}, \tau')$ be tracial von Neumann algebras, and $\phi: \mathcal{M} \to \mathcal{N}$ a strictly completely positive trace-preserving (CPTP) map. With a reference positive, invertible $B \in \mathcal{M}$ normalized so that $\tau(B) = 1$, consider the Petz recovery map $\mathcal{R}$ defined by:

$$\mathcal{R}(Y) = B^{1/2} \phi^*(\phi(B)^{-1/2} Y \phi(B)^{-1/2}) B^{1/2}.$$

Recoverable states are those $A \in L^1(\mathcal{M}, \tau)$ for which $L^p$0. The work explores how arbitrary states in $L^p$1 can be approximated, in the $L^p$2 norm, by recoverable states via sequences of iterated Petz recovery.

Main Technical Results

Norm Convergence of Iterated Maps

A central result is the construction of a CPTP "projection" channel $L^p$3 satisfying

• $L^p$4 is recoverable: $L^p$5 for all $L^p$6,
• $L^p$7 in the operator norm topology on $L^p$8 for all $L^p$9,
• Convergence holds strongly in $(\mathcal{M}, \tau)$0 but operator norm convergence in $(\mathcal{M}, \tau)$1 remains open.

This result leverages spectral properties of positive contractions on non-commutative $(\mathcal{M}, \tau)$2 spaces and the equivalence of weighted ($(\mathcal{M}, \tau)$3-sandwiched) and standard $(\mathcal{M}, \tau)$4 norms. The proof utilizes spectral projections, complex interpolation, and ergodic theory applied to positive operators on Banach spaces.

Decomposition Theorem

A new structural theorem is established: every normal state $(\mathcal{M}, \tau)$5 can be uniquely decomposed as

$(\mathcal{M}, \tau)$6

where $(\mathcal{M}, \tau)$7 is recoverable and $(\mathcal{M}, \tau)$8 is self-adjoint with $(\mathcal{M}, \tau)$9 in $(\mathcal{N}, \tau')$0. This decomposition is induced by the orthogonal projection relative to the $(\mathcal{N}, \tau')$1-sandwiched inner product and the contractive property of $(\mathcal{N}, \tau')$2.

Stability and Data Processing Inequality

The analysis is grounded on the DPI for sandwiched quasi-relative entropies,

$(\mathcal{N}, \tau')$3

with $(\mathcal{N}, \tau')$4 for $(\mathcal{N}, \tau')$5. The contractivity of $(\mathcal{N}, \tau')$6 in these interpolated norms is crucial for the norm convergence results.

Implications and Discussion

From an operator algebraic and quantum information perspective, the work establishes that:

• Arbitrary states may be approximated to any desired accuracy by recoverable states via iterative application of Petz recovery and the channel, providing a constructive method to mitigate entropy loss in communication scenarios where recoverability is desirable or necessary.
• The dynamics of $(\mathcal{N}, \tau')$7 reveal a form of asymptotic stabilization onto the fixed-point (recoverable) subalgebra.
• These norm convergence results in $(\mathcal{N}, \tau')$8 spaces are notable given that strong operator topology is typically the best possible for infinite-dimensional settings; the norm convergence in all $(\mathcal{N}, \tau')$9 (except $\phi: \mathcal{M} \to \mathcal{N}$0) is an uncommon and robust property.

The decomposition theorem has both conceptual and practical value: it identifies the recoverable content of a state and the "dissipative" part, potentially informing quantum error correction schemes and the design of stable encoding strategies for quantum communication.

Open questions remain regarding the possibility of norm convergence in $\phi: \mathcal{M} \to \mathcal{N}$1, which may depend on the spectral radius of $\phi: \mathcal{M} \to \mathcal{N}$2 restricted to the kernel of $\phi: \mathcal{M} \to \mathcal{N}$3.

Future Directions

Potential directions for further research include:

• Characterizing the spectrum and peripheral eigenstructure of $\phi: \mathcal{M} \to \mathcal{N}$4 in $\phi: \mathcal{M} \to \mathcal{N}$5,
• Extending the analysis to more general quantum channels (e.g., relaxing strict CPTP conditions),
• Exploring operational interpretations of the decomposition, especially in continuous variable quantum information theory,
• Quantitative rates of convergence in specific classes of infinite-dimensional von Neumann algebras.

Conclusion

This paper provides a substantial contribution to the theory of quantum recoverability in infinite-dimensional von Neumann algebras, demonstrating that sequences of iterated Petz recovery and channel composition converge (in various norms) to a CPTP projection onto the recoverable states. The unique decomposition of normal states into recoverable and vanishing parts further enriches the structural understanding of state spaces under quantum operations, with significant implications for quantum error correction and the mathematical foundations of quantum entropy inequalities.