Estimating Entanglement via Quantum Steering: A Variational Quantum Algorithm for Quantifying Separability

Core Concepts

The core message of this paper is to develop a quantum algorithm, called a variational quantum steering algorithm (VQSA), that tests for and quantifies the separability of a general bipartite quantum state by using the quantum steering effect. The VQSA approximates an interactive quantum protocol that is directly related to the fidelity of separability, a bona fide entanglement measure.

Abstract

This paper presents a quantum algorithm for testing and quantifying the separability of a general bipartite quantum state. The key insights are:

The authors develop a quantum interactive proof system that tests the separability of a mixed state by exploiting the quantum steering effect. They show that the acceptance probability of this protocol is directly related to the fidelity of separability, an entanglement measure.

To make this protocol practical, the authors modify it to a variational quantum steering algorithm (VQSA) that replaces the computationally unbounded prover with parameterized unitary circuits and classical optimization techniques. They prove that the maximum acceptance probability of the VQSA also equals the fidelity of separability.

The authors generalize their separability test and VQSA to the multipartite setting. They also analyze the computational complexity of estimating the fidelity of separability, showing it is contained in a new complexity class called QIPEB(2).

The authors simulate their VQSA on noisy quantum simulators and find favorable convergence properties. They also develop semidefinite programs to benchmark the VQSA results, as near-term quantum computers have limited scale and error tolerance.

The authors discuss potential applications of their VQSA beyond entanglement quantification, such as in distributed quantum algorithms over a quantum network. They also suggest the paradigm of parameterized mid-circuit measurements in VQAs may be helpful for a wide variety of computational problems in quantum information science.

Stats

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Quotes

"Entanglement is a unique feature of quantum mechanics, initially brought to light by Einstein, Podolsky, and Rosen [1]."
"Determining whether a general state ρAB is separable or entangled, known as the separability problem, is a fundamental problem of interest relevant to various fields of physics, including condensed matter [8, 9, 10], quantum gravity [11, 12, 13, 14, 15], quantum optics [16], and quantum key distribution [17, 18]."
"Our approach is distinct from recent work on quantum algorithms for estimating entanglement. For example, VQAs have been used to address this problem by estimating the Hilbert–Schmidt distance [45], by creating a zero-sum game using parameterized unitary circuits [46], by employing symmetric extendibility tests [31], by estimating logarithmic negativity [47], and using the positive map criterion [47]."

Key Insights Distilled From

Schrödinger as a Quantum Programmer

by Aby Philip,S... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2303.07911.pdf

Deeper Inquiries

How can the insights from the VQSA be extended to address other computational problems in quantum information science beyond entanglement quantification?

The insights gained from the Variational Quantum Steering Algorithm (VQSA) can be extended to tackle various other computational problems in quantum information science. One key application is in optimizing the decomposition of quantum states for different quantum information processing tasks. By leveraging the parameterized mid-circuit measurements and distributed variational quantum algorithms, similar techniques can be applied to optimize quantum states for tasks such as quantum error correction, quantum state preparation, and quantum algorithm design. Additionally, the concept of quantum steering can be utilized in developing new quantum protocols for secure communication, quantum key distribution, and quantum cryptography. The flexibility and adaptability of VQSA make it a valuable tool for addressing a wide range of computational challenges in quantum information science.

What are the limitations of the VQSA approach, and how can it be further improved to handle larger quantum systems with higher accuracy?

While VQSA offers a promising approach for estimating the fidelity of separability and addressing entanglement quantification, it does have certain limitations. One limitation is the susceptibility to barren plateaus, where the gradients become vanishingly small, especially as the number of qubits increases. To overcome this limitation, techniques such as using a local reward function instead of a global one can help mitigate barren plateaus and improve the convergence of the algorithm. Additionally, incorporating error mitigation strategies and optimizing the quantum circuit design can enhance the accuracy of VQSA for larger quantum systems. Furthermore, exploring hybrid classical-quantum optimization methods and leveraging advancements in quantum hardware can also contribute to improving the performance and scalability of VQSA for handling larger quantum systems with higher accuracy.

Can the complexity-theoretic placement of the fidelity of separability problem within QIPEB(2) lead to new insights about the hardness of other quantum information tasks?

The complexity-theoretic placement of the fidelity of separability problem within the class QIPEB(2) can indeed provide valuable insights into the computational complexity of other quantum information tasks. By establishing the separability problem as complete for QIPEB(2), it allows for a deeper understanding of the computational resources required to solve entanglement-related problems. This placement can lead to new insights into the hardness of tasks related to quantum interactive proofs, entanglement verification, and quantum communication protocols. Furthermore, the connections between QIPEB(2) and other complexity classes like QSZK and QAM can offer a framework for analyzing the computational complexity of a wide range of quantum information tasks, shedding light on the boundaries of efficient quantum computation and the challenges in solving complex quantum problems.