Schrödinger as a Quantum Programmer: Estimating Entanglement via Steering

TL;DR

The paper develops a steering-based framework to test and quantify bipartite entanglement via the fidelity of separability $F_s(\rho_{AB})$. It introduces a quantum interactive proof (QIP) for fidelity of separability and a practical variational quantum steering algorithm (VQSA) that uses mid-circuit steering measurements and parameterized circuits to estimate $F_s$ without full tomography. The theoretical results show the ideal acceptance probability equals $(1+F_s(\rho_{AB}))/2$, and simulations with noisy quantum simulators demonstrate favorable convergence, complemented by SDP benchmarks based on PPT and $k$-extendibility to bound $F_s$. The work extends to multipartite settings and situates the estimation of $F_s$ within the complexity class QIP$_{\text{EB}}(2)$, linking steering, entanglement, algorithms, and quantum computational complexity, while highlighting distributed-VQA opportunities for quantum networks.

Abstract

Quantifying entanglement is an important task by which the resourcefulness of a quantum state can be measured. Here, we develop a quantum algorithm that tests for and quantifies the separability of a general bipartite state by using the quantum steering effect, the latter initially discovered by Schrödinger. Our separability test consists of a distributed quantum computation involving two parties: a computationally limited client, who prepares a purification of the state of interest, and a computationally unbounded server, who tries to steer the reduced systems to a probabilistic ensemble of pure product states. To design a practical algorithm, we replace the role of the server with a combination of parameterized unitary circuits and classical optimization techniques to perform the necessary computation. The result is a variational quantum steering algorithm (VQSA), a modified separability test that is implementable on quantum computers that are available today. We then simulate our VQSA on noisy quantum simulators and find favorable convergence properties on the examples tested. We also develop semidefinite programs, executable on classical computers, that benchmark the results obtained from our VQSA. Thus, our findings provide a meaningful connection between steering, entanglement, quantum algorithms, and quantum computational complexity theory. They also demonstrate the value of a parameterized mid-circuit measurement in a VQSA.

Figures (12)

• Figure 1: Test for separability of mixed states. The verifier uses a unitary circuit $U^\rho$ to produce the state $\psi_{RAB}$, which is a purification of $\rho_{AB}$. The prover (indicated by the dotted box) applies an entanglement-breaking channel $\mathcal{E}_{R\rightarrow A^{\prime}}$ on $R$ by measuring the rank-one POVM $\{\mu^{x}_{R}\}_{x}$ and then, depending on the outcome $x$, prepares a pure state from the set $\{\phi^{x}_{A^{\prime}}\}_{x}$. The final state is sent to the verifier, who performs a swap test. Theorem \ref{['theorem:qip-eb_msf']} states that the maximum acceptance probability of this interactive proof is equal to $\frac{1}{2}(1 + F_{s}(\rho_{AB}))$, i.e., a simple function of the fidelity of separability.
• Figure 2: Quantum part of the VQSA to estimate the fidelity of separability $F_s(\rho_{AB})$. The unitary circuit $U^\rho$ prepares the state $\psi_{RAB}$, which is a purification of $\rho_{AB}$. The parameterized circuit $W_R(\Theta)$ acts on $R$ to evolve $\psi_{RAB}$ to another purification of $\rho_{AB}$. The following measurement, labeled "steering measurement," steers the systems $AB$ to be in a pure state $\psi_{AB}^{x}$ if the measurement outcome $x$ occurs. Conditioned on the outcome $x$, the final parameterized circuit $U^{x}_{A}(\Theta^x)$ and the subsequent measurement accepts with a maximum probability of $F_s(\rho_{AB})$.
• Figure 3: Fidelity of separability calculated for a ($3/4$,$1/4$) classical mixture of $|\Phi^+\rangle$ and $|\Phi^-\rangle$ using our VQSA (blue line). The algorithm converges to 0.93, which agrees with the value obtained using the benchmarks $\widetilde{F}_s^1$ and $\widetilde{F}_s^2$.
• Figure 4: Fidelity of separability calculated for the state $\tilde{\rho}_{AB}$ as specified in \ref{['eqn:rho-tilde']} using our VQSA (blue line) and $\widetilde{F}_s^1$ (orange line).
• Figure 5: Test for separability of multipartite mixed states. The verifier uses the unitary circuit $U^\rho$ to prepare the state $\psi_{RA_1 A_2 A_3 A_4}$, which is a purification of $\rho_{A_1 A_2 A_3 A_4}$. The prover (indicated by the dotted box) applies an entanglement-breaking channel $\mathcal{E}_{R\rightarrow A^{\prime}_1 A^{\prime}_2 A^{\prime}_3}$ on $R$ by measuring the rank-one POVM $\{\mu^{x}_{R}\}_{x\in \mathcal{X}}$ and then, depending on the outcome $x$, prepares a state from the set $\{\phi^{x,1}_{A^{\prime}_1}\otimes\phi^{x,2}_{A^{\prime}_2}\otimes\phi^{x,3}_{A^{\prime}_3}\}_{x\in \mathcal{X}}$. The final state is sent to the verifier, who performs a collective swap test. Theorem \ref{['theorem:qip-eb_msf_multi']} states that the maximum acceptance probability of this interactive proof is equal to $\frac{1}{2}(1 + F_{s}(\rho_{A_1 A_2 A_3 A_4}))$, i.e., a simple function of the fidelity of separability.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: Streltsov2010
• Proposition 5
• Proposition 6
• Remark 7
• Definition 8
• Theorem 9: Streltsov2010
• Theorem 10