Entanglement theory with limited computational resources

TL;DR

This work reframes entanglement theory under computational constraints, showing that the von Neumann entropy loses operational meaning when LOCC must be polynomial-time computable, and that the min-entropy $S_{\min}$ becomes the key determinant of accessible entanglement. It introduces computational distillable entanglement $\hat{E}_{D}^{(k)}$ and computational entanglement cost $\hat{E}_{C}^{(k)}$ for bipartite pure states, and proves a sharp separation from standard information-theoretic rates via state-agnostic and state-aware scenarios. A central technical advance is the construction of pseudoentangled Haar-subsystem states and a Schur-Weyl/Schur-transform based distillation protocol that achieves $\min\{\Theta(S_{\min}),\Theta(\log k)\}$ and remains robust for near-i.i.d. inputs, while showing that entanglement dilution can require $\tilde{\Omega}(n)$ ebits in the worst case, even for tiny $S_{1}$. The authors further develop sample-complexity bounds for entropy estimation and LOCC tomography, including a distillation-through-learning approach that enables LOCC tomography with only linear overhead, thereby linking learning theory to entanglement manipulation under computational limits. Together, these results establish a novel computational resource framework for quantum information, with implications for state compression, tomography, and the fundamental limits of quantum resource theories under realistic computational constraints.

Abstract

The precise quantification of the ultimate efficiency in manipulating quantum resources lies at the core of quantum information theory. However, purely information-theoretic measures fail to capture the actual computational complexity involved in performing certain tasks. In this work, we rigorously address this issue within the realm of entanglement theory, a cornerstone of quantum information science. We consider two key figures of merit: the computational distillable entanglement and the computational entanglement cost, quantifying the optimal rate of entangled bits (ebits) that can be extracted from or used to dilute many identical copies of $n$-qubit bipartite pure states, using computationally efficient local operations and classical communication (LOCC). We demonstrate that computational entanglement measures diverge significantly from their information-theoretic counterparts. While the von Neumann entropy captures information-theoretic rates for pure-state transformations, we show that under computational constraints, the min-entropy instead governs optimal entanglement distillation. Meanwhile, efficient entanglement dilution incurs in a major cost, requiring maximal $(\tildeΩ(n))$ ebits even for nearly unentangled states. Surprisingly, in the worst-case scenario, even if an efficient description of the state exists and is fully known, one gains no advantage over state-agnostic protocols. Our results establish a stark, maximal separation of $\tildeΩ(n)$ vs. $o(1)$ between computational and information-theoretic entanglement measures. Finally, our findings yield new sample-complexity bounds for measuring and testing the von Neumann entropy, fundamental limits on efficient state compression, and efficient LOCC tomography protocols.

Figures (1)

• Figure 1: 1. The information-theoretic distillable entanglement and entanglement cost represent the optimal rates at which one can purify or dilute a collection of identical pure bipartite entangled states $\psi_{A,B}$ within the distant parties paradigm. In this setting, Alice and Bob, can naturally manipulate the quantum states held at their respective laboratories using only LOCC. A single quantity, the von-Neumann entropy of the reduced density matrix $S_1(\psi_A)$ governs both asymptotic rates when computational restrictions are not taken into account. 2. In this work, we consider the computational analogs thereof in which each step is assumed to be sample- and computational-efficient. We find families of states for which the computational distillable entanglement is $S_{\rm min}(\psi_A)$, and other families for which the entanglement cost is (maximally) $\Tilde{\Omega}(n)$. These results apply irrespectively of the actual value of $S_1(\psi_A)$, which therefore looses operational meaning in this computationally limited setting.

Theorems & Definitions (88)

• Definition 1: Computational distillable entanglement. Informal of \ref{['def:distillableentanglement']}
• Definition 2: Computational entanglement cost. Informal of \ref{['def:entanglementcost']}
• Theorem 1: No-go on state-agnostic distillation. Informal version of \ref{['th:upperbounddistillableentanglement']}
• proof : Proof sketch
• Theorem 2: State-agnostic distillation performance. Informal version of \ref{['th:distillation']}
• Theorem 3: Upper bound on the computational distillable entanglement. Informal version of \ref{['appth:upperbounddistillableent']}
• proof : Proof sketch
• Theorem 4: No-go on state-agnostic entanglement dilution. Informal version of \ref{['th:entanglementcostnogo']}
• Theorem 5: Tight lower bound on computational entanglement cost. Informal version of \ref{['appth:lowerboundentcost']}
• Theorem 6: No-go on state-aware entanglement manipulation. Informal version of \ref{['thapp:stateawaredistillation', 'thapp:stateawaredilution']}