Computing entanglement costs of non-local operations on the basis of algebraic geometry

TL;DR

The paper tackles the problem of computing entanglement costs for non-local quantum operations under separable channels by introducing an algebraic-geometric framework built around the minimum finite linear extension ($MFLE$). It proves the existence of the MFLE, derives explicit MFLEs for two-qubit and canonical subspaces, and demonstrates how MFLE constraints can systematically strengthen the DPS hierarchy and be combined with group twirling to yield sharper analytic and numerical bounds for a wide range of tasks, including distillation, non-local unitary implementations, measurements, state verification, and multipartite entanglement distribution. It also shows that, for several deterministic non-local tasks, the resource entanglement must be maximally entangled, generalizing prior results and resolving open questions on local state discrimination. Overall, the MFLE framework provides a unified, scalable method to bound and compute entanglement costs under SEP channels, with broad implications for distributed quantum information processing and quantum network design.

Abstract

In the study of distributed quantum information processing, it is crucial to minimize the entanglement consumption by optimizing local operations. We develop a framework based on algebraic geometry to systematically simplify the optimization over separable (SEP) channels, which serve as widely used models for local operations. We apply this framework to computing one-shot entanglement cost for implementing non-local operations under SEP channels, in both probabilistic and zero-error settings. First, we present a unified generalization of previous analytical results on the entanglement cost. Via the generalization, we resolve an open problem posed by Yu et al. regarding the entanglement cost of local state discrimination. Second, we strengthen the Doherty--Parrilo--Spedalieri hierarchy and determine the trade-off between the entanglement cost and the success probability of implementing various operations -- such as entanglement distillation, non-local unitary channels, measurements, state verification, and multipartite entanglement distribution.

Figures (11)

• Figure 1: Feasible regions for operator $S$ in the range-constrained SEP optimization problem and of its relaxed problem. (a) The set of SEP channels is represented by the red convex cone, with the feasible region being its the intersection with the horizontal plane (range constraint). (b) In the relaxed problem, the PPT cone (blue convex cone) replaces the SEP cone, enlarging the feasible region to a large triangular area on the plane. (c) Due to the difference in the feasible region between the original problem (red triangle) and the relaxed problem (purple triangle), the solution to the relaxed problem is often strictly larger than the original one.
• Figure 2: Examples of MFLE and irreducible components of $\mathbb{E}=\mathbb{S}\left(\mathbb{C}^{2}:\mathbb{C}^{2}\right)\cap\mathcal{V}$. We plot a three-dimensional slice $(1,x,y,z)^T\in\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. In this slice, $\mathbb{S}\left(\mathbb{C}^{2}:\mathbb{C}^{2}\right)$ is represented by the red surface defined by $z=xy$. $\mathbb{E}$ and $\mathcal{V}$ are depicted by black curves (or lines) and brown planes, respectively. Each of $\mathbb{E}$, $\mathbb{S}\left(\mathbb{C}^{2}:\mathbb{C}^{2}\right)$, and $\mathcal{V}$ is closed with respect to the Zariski topology. While $\mathbb{S}\left(\mathbb{C}^{2}:\mathbb{C}^{2}\right)$ and $\mathcal{V}$ are irreducible, the irreducibility of $\mathbb{E}$ differs in the two cases shown. (a) When $\mathcal{V}$ is defined by its normal vector $(1,0,0,-10)^T$, $\mathbb{E}$ is irreducible, as there are no polynomials whose set of zeros defines a proper subset of $\mathbb{E}$. In this case, the MFLE is ${\rm span}\left(\mathbb{E}\right)=\mathcal{V}$. (b) When $\mathcal{V}$ is defined by its normal vector $(0,0,0,1)^T$, $\mathbb{E}=(\mathbb{C}^{2}\otimes| {0} \rangle)\cup(| {0} \rangle\otimes\mathbb{C}^{2})$ is reducible into two subspaces $\mathbb{C}^{2}\otimes| {0} \rangle$ and $| {0} \rangle\otimes\mathbb{C}^{2}$ since each is a proper closed subset of $\mathbb{E}$. In this case, the MFLE is $\mathbb{E}(\subsetneq\mathcal{V})$ itself.
• Figure 3: Hilbert spaces where the canonical subspace is defined. We consider the product vectors between $\hat{\mathcal{A}}$ and $\hat{\mathcal{B}}$.
• Figure 4: General setting of the implementation of a non-local instrument $\{\mathcal{E}_m\}_{m\in\Sigma}$ by using a separable instrument $\{\mathcal{S}_m\}_{m\in\Sigma}\cup\{\mathcal{S}_{\texttt{fail}}\}$ assisted by an entangled state in $\mathcal{H}_{R_A}\otimes\mathcal{H}_{R_B}$. For all $m\in\Sigma\cup\{\texttt{fail}\}$, the Choi operator of $\mathcal{S}_m$ is an element in $\mathbf{SEP}\left(\hat{\mathcal{A}}:\hat{\mathcal{B}}\right)$. We assume that we simulate the non-local instrument without error by post-selecting events corresponding to $m\in\Sigma$.
• Figure 5: Solutions and run time for the relaxed problems of range-constrained SEP optimization problems across (a) $100$ different target states $| {\psi_\theta} \rangle=\cos\theta| {00} \rangle+\sin\theta| {11} \rangle$ or (b-d) $100$ different resource states $| {\tau_\theta} \rangle=\cos\theta| {00} \rangle+\sin\theta| {11} \rangle$. The relaxed problems derived from the DPS hierarchy are denoted by 'PPT' or 'DPS 2nd Lv.'. The relaxed problems obtained from our strengthened DPS hierarchy are denoted by 'PPT+MFLE' or 'PPT*+MFLE'. (a) Success probability of distilling the entangled state $| {\psi_\theta} \rangle$ from a mixed state $\frac{1}{3}\sum_{i=1}^3\tau_i$ using SEP channels, where $| {\tau_1} \rangle=\frac{1}{\sqrt{2}}(| {01} \rangle-| {10} \rangle)$, $| {\tau_2} \rangle=\frac{1}{\sqrt{2}}(| {02} \rangle-| {20} \rangle)$, and $| {\tau_3} \rangle=\frac{1}{\sqrt{2}}(| {12} \rangle-| {21} \rangle)$. The solutions for 'PPT+MFLE' and 'DPS 2nd Lv.' coincide with the analytical lower bound $\min\left\{1,\frac{1}{2\sin2\theta}\right\}$. (b) Success probability of implementing the controlled $T$ gate and (c) success probability of implementing the symmetric joint POVM CZ21MA24PPDG25 using SEP channels assisted by the entangled state $| {\tau_\theta} \rangle$. The true trade-off curve lies within the red-shaded region. The relaxed problem denoted by 'PPT*' partially incorporates constraints from the second level of the DPS hierarchy. (d) Maximum $q$ for deterministically implementing a POVM $\{q\phi,(I-q\phi)\}$, where $| {\phi} \rangle=\frac{\sqrt{3}+1}{2\sqrt{2}}| {00} \rangle-\frac{\sqrt{3}-1}{2\sqrt{2}}| {11} \rangle$. The solution for 'PPT+MFLE' coincides with the lower bound.

Theorems & Definitions (46)

• Definition 1
• Definition 2
• Theorem 1
• Proposition 1
• proof
• proof
• Proposition 2
• proof
• Definition 3
• Theorem 2