We simulate quantum spin systems and measure entanglement using circuits tailored for near-term quantum computers. Traditional tools like entanglement entropy are limited to pure states and require full state tomography, making them impractical on current hardware. Instead, we employ the novel approach, Positive Partial Transpose (PPT) criterion to efficiently detect pairwise entanglement from two-spin reduced density matrices, applicable to both pure and mixed states. This method enables scalable entanglement detection, providing a practical route to study quantum correlations, phase transitions, and benchmark quantum devices.
We simulate quantum spin systems and measure entanglement using circuits tailored for near-term quantum computers. Traditional tools like entanglement entropy are limited to pure states and require full state tomography, making them impractical on current hardware. Instead, we employ the novel approach, Positive Partial Transpose (PPT) criterion to efficiently detect pairwise entanglement from two-spin reduced density matrices, applicable to both pure and mixed states. This method enables scalable entanglement detection, providing a practical route to study quantum correlations, phase transitions, and benchmark quantum devices.
Quantum phase transitions in many-body systems give rise to highly entangled states, and understanding their quantum correlations is crucial for characterizing quantum materials. However, traditional entanglement measures such as entanglement entropy are difficult to interpret for noisy or mixed states and require complex circuits to evaluate. Therefore, we explore the Positive Partial Transpose (PPT) criterion, coupled with overlapping state tomography, as an efficient and scalable spin-spin entanglement witness. It detects pairwise entanglement from reduced density matrices, distinguishes quantum from classical correlations, and applies to both pure and mixed states. It is ideal for studying condensed matter systems prepared on noisy quantum devices as well as future extensions to finite temperatures. We demonstrate the approach on quantum hardware, using variational circuits to prepare quantum critical states with up to 20 qubits and completely map their two-spin entanglement across various quantum phase transitions.
Chapter 13 delves into the complex terrain of mixed-state entanglement, extending the discourse from pure-state entanglement to encompass the broader and more practical scenarios encountered in quantum systems. The chapter systematically explores the detection of entanglement in mixed states, introducing criteria and methods such as the Positive Partial Transpose (PPT) criterion and entanglement witnesses, which serve as diagnostic tools for identifying entanglement in a mixed quantum state. Furthermore, it addresses the quantification of entanglement in mixed states, discussing various measures like entanglement cost and distillable entanglement. These concepts highlight the operational aspects of entanglement, including its creation and extraction, within mixed-state frameworks. The chapter also introduces the notion of entanglement conversion distances, providing a quantitative approach to understanding the transformations between different entangled states.
For the certification and benchmarking of medium-size quantum devices, efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partially transposed density matrix. The first method is based on the Hankel matrices and provides a family of entanglement criteria, of which the lowest order reduces to the known p_{3}-positive-partial-transpose criterion proposed in A. Elben et al. [Phys. Rev. Lett. 125, 200501 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.200501]. The second method is optimal and gives necessary and sufficient conditions for entanglement based on some moments of the partially transposed density matrix.
In the realm of quantum information theory, the detection and quantification of quantum entanglement stand as paramount tasks. The relative entropy of entanglement (REE) serves as a prominent measure of entanglement, with extensive applications spanning numerous related fields. The positive partial transpose (PPT) criterion, while providing an efficient method for the computation of REE, unfortunately, falls short when dealing with bound entanglement. In this study, we propose a method termed “pure bosonic extension” to enhance the practicability of k -bosonic extensions, which approximates the set of separable states from the “outside”, through a hierarchical structure. It enables efficient characterization of the set of k -bosonic extendible states, facilitating the derivation of accurate lower bounds for REE. Compared to the semi-definite programming (SDP) approach, such as the symmetric/bosonic extension function in QETLAB, our algorithm supports much larger dimensions and higher values of extension k . Published by the American Physical Society 2024
Entanglement plays a crucial role in quantum information science and many-body physics, yet quantifying it in mixed quantum many-body systems has remained a notoriously difficult problem. Here, we introduce families of quantitative entanglement witnesses, constructed from partial transpose and realignment moments, which provide rigorous bounds on entanglement monotones as well as entanglement dimensionality. Our witnesses can be efficiently measured using SWAP tests or variants of Bell measurements, thus making them directly implementable on current hardware. Leveraging our witnesses, we present several novel results on entanglement properties of mixed states, both in quantum information and many-body physics. We develop efficient algorithms to test whether mixed states with bounded entropy have low or high entanglement, which previously was only possible for pure states. We also provide an efficient algorithm to test the Schmidt rank using only two-copy measurements, and the operator Schmidt rank using four-copy measurements. Further, our witnesses robustly certify the quantum circuit depth in the presence of noise, as well as the Schmidt rank of mixed states. Finally, we show that the entanglement phase diagram of Haar random states, quantified by the partial transpose negativity, can be fully established solely by computing our witness, a result that also applies to any state 4 -design. Our witnesses can also be efficiently computed for matrix product states, thus enabling the characterization of entanglement in extensive many-body systems. Finally, we make progress on the entanglement required for quantum cryptography, establishing rigorous limits on pseudoentanglement and pseudorandom density matrices with bounded entropy. Our work opens new avenues for quantifying entanglement in large and noisy quantum systems.
The efficient detection of quantum entanglement is a central problem in quantum information processing. This paper systematically proposes a quantum circuit implementation scheme based on the Positive Partial Transpose (PPT) and the Computable Cross-Norm Realignment (CCNR) criteria, providing a complete quantum algorithmic pathway for efficient and computable entanglement detection. By encoding quantum states into specific forms and utilizing SWAP operations, complex matrix operations such as partial transpose and realignment are transformed into executable quantum circuits. Furthermore, by integrating an improved variational quantum singular value decomposition subroutine, the scheme enables the efficient estimation of the trace norm, thereby determining the existence of entanglement. Designed to operate within a hybrid quantum-classical framework, this scheme exhibits excellent scalability and practicality, offering a theoretical tool and methodological support for analyzing the entanglement structure of complex quantum systems on future intermediate-scale quantum devices.