Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stochastic approximate state conversion for entanglement and general quantum resource theories

Tulja Varun Kondra, Chandan Datta, Alexander Streltsov

TL;DR

The paper addresses the problem of state transformation within general quantum resource theories, focusing on the intermediate regime between probabilistic and approximate conversions. It introduces stochastic-approximate state conversion quantified by $F_p$ (fidelity at a given success probability) and $P_f$ (success probability at a given fidelity) under $\varepsilon$-resource-generating free operations, and derives universal bounds expressed through resource quantifiers like the generalized robustness $R$ and the entropic quantities $E_{\max}$ and $E_{1/2}$. Key contributions include a general single-copy bound valid for all resource theories, an asymptotic-rate bound $r^{M}_{p}(\rho\rightarrow\sigma) \le E_{\max}(\rho)/E_{1/2}(\sigma)$, and complete analytic solutions for stochastic-approximate transformations of pure bipartite states of arbitrary dimension and for two-qubit initial states. The results apply to channels and nonlocality tasks (e.g., PR-box distillation) and provide practical limits for quantum-network tasks, with extensions to non-convex free sets and SDP-representable cases. Overall, the work sharpens the fundamental limits on manipulating quantum resources in realistic, probabilistic settings.

Abstract

Quantum resource theories provide a mathematically rigorous way of understanding the nature of various quantum resources. An important problem in any quantum resource theory is to determine how quantum states can be converted into each other within the physical constraints of the theory. The standard approach to this problem is to study approximate or probabilistic transformations. Here, we investigate the intermediate regime, providing limits on both, the fidelity and the probability of state transformations. We derive limitations on the transformations, which are valid in all quantum resource theories, by providing bounds on the maximal transformation fidelity for a given transformation probability. As an application, we show that these bounds imply an upper bound on the asymptotic rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels, which goes beyond the previously known bounds of channel manipulations. Furthermore, we completely solve the question of stochastic-approximate state conversion via local operations and classical communication in the following two cases: (i) Both initial and target states are pure bipartite entangled states of arbitrary dimensions. (ii) The target state is a two-qubit entangled state and the initial state is a pure bipartite state.

Stochastic approximate state conversion for entanglement and general quantum resource theories

TL;DR

The paper addresses the problem of state transformation within general quantum resource theories, focusing on the intermediate regime between probabilistic and approximate conversions. It introduces stochastic-approximate state conversion quantified by (fidelity at a given success probability) and (success probability at a given fidelity) under -resource-generating free operations, and derives universal bounds expressed through resource quantifiers like the generalized robustness and the entropic quantities and . Key contributions include a general single-copy bound valid for all resource theories, an asymptotic-rate bound , and complete analytic solutions for stochastic-approximate transformations of pure bipartite states of arbitrary dimension and for two-qubit initial states. The results apply to channels and nonlocality tasks (e.g., PR-box distillation) and provide practical limits for quantum-network tasks, with extensions to non-convex free sets and SDP-representable cases. Overall, the work sharpens the fundamental limits on manipulating quantum resources in realistic, probabilistic settings.

Abstract

Quantum resource theories provide a mathematically rigorous way of understanding the nature of various quantum resources. An important problem in any quantum resource theory is to determine how quantum states can be converted into each other within the physical constraints of the theory. The standard approach to this problem is to study approximate or probabilistic transformations. Here, we investigate the intermediate regime, providing limits on both, the fidelity and the probability of state transformations. We derive limitations on the transformations, which are valid in all quantum resource theories, by providing bounds on the maximal transformation fidelity for a given transformation probability. As an application, we show that these bounds imply an upper bound on the asymptotic rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels, which goes beyond the previously known bounds of channel manipulations. Furthermore, we completely solve the question of stochastic-approximate state conversion via local operations and classical communication in the following two cases: (i) Both initial and target states are pure bipartite entangled states of arbitrary dimensions. (ii) The target state is a two-qubit entangled state and the initial state is a pure bipartite state.
Paper Structure (16 sections, 6 theorems, 168 equations, 2 figures)

This paper contains 16 sections, 6 theorems, 168 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stochastic approximate state conversion
  3. Asymptotic and many-copy bounds
  4. Pure entangled state transformations
  5. Entangled states of two qubits
  6. Conclusions
  7. Proof of Theorem \ref{['thm:GeneralBound_resource_generating']}
  8. Asymptotic behaviour of the bound
  9. Deterministic bound for channels
  10. Proof of Eq. (\ref{['eq:distill_singlet_locc']})
  11. Proof of Theorem \ref{['thm:pureQuditConversion']}
  12. Proof for Lemma \ref{['thm:TwoQubitsContinuity']}
  13. Proof of Theorem \ref{['thm:QubitConversion']}
  14. Examples to compare the exact result and the upper bound
  15. Proof of Eq. (\ref{['optprob2']})
  16. ...and 1 more sections

Key Result

Theorem 1

For any quantum resource theory and any two states $\rho$ and $\sigma$ the following inequalities hold: Here, $F^{M, \varepsilon}_{\max}(\sigma)=\max\limits_{\sigma_{\varepsilon}: M(\sigma_{\varepsilon})\leq \varepsilon}F(\sigma,\sigma_{\varepsilon})$.

Figures (2)

  • Figure 1: Optimal fidelity for the transformation $\ket{\psi} \rightarrow \ket{\phi^+}$ is plotted with probability $p$, where we choose $\alpha=0.2$. In the inset, optimal fidelity (solid line) and the upper bound of fidelity (dashed line) are shown with respect to $p$. Note that fidelity is $1$, when $p\leq G(\psi)/G(\phi^+)\approx 0.079$.
  • Figure 2: (a) Optimal fidelity for the transformation $\psi \rightarrow \rho_W$ is plotted with probability $p$, where we choose $\alpha=0.01$ and $r=0.9$. In the inset, optimal fidelity (solid line) and the upper bound of fidelity (dashed line) are shown with respect to $p$. For $p=1$, there is a very small gap of $0.0068$ between the upper bound of fidelity and the exact fidelity. Optimal fidelity is $1$ for $p\leq G(\psi)/G(\rho_W)\approx 0.0004$, which is not visible in the figure. (b) Optimal fidelity is plotted with $r$, where we consider $\alpha=0.01$ and transformation probability to be $0.75$. The vertical dashed line is at $r\approx 0.3487$ and below or equal this $r$ we have unit fidelity.

Theorems & Definitions (6)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Theorem 4