Computable and Faithful Lower Bound on Entanglement Cost

TL;DR

This work develops computable and faithful lower bounds on the entanglement cost under PPT operations by introducing the generalized $k$-negativity divergence and its associated PPT$_k$ hierarchy. The key construct, the logarithmic fidelity of binegativity $E_{N,2}^{1/2}$, is faithful for all NPT states and computable via semidefinite programming, providing a first efficient faithfulness guarantee in NPT entanglement theory. The approach extends to quantum channels and bipartite channels, yielding Choi-state based lower bounds $E_{N,2}^{1/2}(J}$ and $E_C(J)$, and enabling variational SDP methods to tighten channel-cost estimates. The results demonstrate irreversibility of asymptotic entanglement manipulation for full-rank states under PPT operations and outperform prior computable bounds in many cases, offering tools for both static and dynamic entanglement cost analysis with potential applications to channel simulation and quantum networks.

Abstract

Quantifying the minimum entanglement needed to prepare quantum states and implement quantum processes is a key challenge in quantum information theory. In this work, we develop computable and faithful lower bounds on the entanglement cost under quantum operations that completely preserve the positivity of partial transpose (PPT operations), by introducing the generalized divergence of $k$-negativity, a generalization of logarithmic negativity. Our bounds are efficiently computable via semidefinite programming and provide non-trivial values for all states that are non-PPT (NPT), establishing their faithfulness for the resource theory of NPT entanglement. Notably, we find and affirm the irreversibility of asymptotic entanglement manipulation under PPT operations for full-rank entangled states. Furthermore, we extend our methodology to derive lower bounds on the entanglement cost of both point-to-point and bipartite quantum channels. Our bound demonstrates improvements over previously known computable bounds for a wide range of quantum states and channels. These findings push the boundaries of understanding the structure of entanglement and the fundamental limits of entanglement manipulation.

Figures (7)

• Figure 1: General framework of entanglement cost. Given access to copies of ebits, these golden 'coins' are utilized to realize a target quantum state $\rho$ or quantum channel ${\cal N}$ through a process that cannot generate any maximally entangled states (MES) itself. The minimal rate of MES required to achieve this process perfectly is called the entanglement cost of the specific object. This concept plays a crucial role in quantifying entanglement in the studies of resource irreversibility, quantum communication, dynamical entanglement, cryptography, and quantum sensing.
• Figure 2: (a-b) Compare $E_{N,2}^{1/2}$ with $E_{\eta}$ and (c-d) Compare $E_{N,2}^{1/2}$ with $E_{N}^\tau$. Each dot corresponds to one of $500$ randomly generated states according to the Hilbert-Schmidt measure. The red line indicates states $\rho_{AB}$ for which two measures give the same bound value. The dots above the red line (resp. below) indicate the states for which $E_{N,2}^{1/2}$ is tighter (resp. $E_{N,2}^{1/2}$ is looser).
• Figure 3: Comparison between $E_{N,2}^{1/2}$ and the Rains bound for a two-qutrit full-rank state $\widehat{\rho}_v$. The $x$-axis represents the depolarizing noise parameter $p$. The red line represents $E_{N,2}^{1/2}(\widehat{\rho}_v)$, serving as a lower bound on the entanglement cost, while the dashed blue line represents the Rains bound, providing an upper bound on the distillable entanglement. The gap between the two bounds for noise parameter $p \in [0, 0.015]$ indicates the irreversibility of asymptotic entanglement manipulation of $\widehat{\rho}_v$.
• Figure 4: Investigation of the dynamical entanglement during time evolution of a time-independent Hamiltonian. Comparison of the entanglement cost lower bounds $E_{\eta}$Wang2016d, $E^{\tau}_{N}$Lami2023a, and $E^{1/2}_{N,2}$ at each time step of a noisy evolution operator governed by a Heisenberg XXZ Hamiltonian.
• Figure S5: Different bounds on the entanglement cost of noisy Bell states. Panel (a-b) depict different bounds on the entanglement cost of $\rho_{AB} = {\cal A}_{A\rightarrow A'}\otimes {\cal D}_{B\rightarrow B'}(\Phi_{AB}^{+})$. In Panel (a), we set $\gamma=0.1$. The $x$-axis represents the change of the depolarizing noise $p$. In Panel (b), we set $p=0.1$. The $x$-axis represents the change of the amplitude damping noise $\gamma$. The entanglement of formation $E_F(\cdot)$ provides an upper bound on $E_C(\cdot)$. $E_\eta(\cdot)$ and $E_{N}^\tau(\cdot)$ are computable lower bounds given in Ref. Wang2016d and Ref. Lami2023a, respectively. It shows that $E_{N,2}^{1/2}(\cdot)$ outperforms the other two lower bounds, both of which vanish for noisy Bell states having full rank in the case $\gamma >0$.

Theorems & Definitions (18)

• Theorem 1: Lower bound on the static entanglement cost
• Proposition 2
• Proposition 3
• Proposition S4
• Lemma S5
• Lemma S6
• Lemma S7: Faithfulness of $E_{N,k}$
• Proposition S8: Normalization
• Proposition S9: Faithfulness of $E_{N,2}^{1/2}$
• Lemma S10