Separation between entanglement criteria and entanglement detection protocols

Entanglement detection is essential in quantum information science and quantum many-body physics. It has been proved that entanglement exists almost surely for a random quantum state, while the realizations of effective entanglement criteria usually consume exponentially many resources with regard to system size or qubit number, and efficient criteria often perform poorly without prior knowledge. This fact implies a fundamental limitation might exist in the detectability of entanglement. In this work, we formalize this limitation as a fundamental trade-off between the efficiency and effectiveness of entanglement criteria via a systematic method to evaluate the detection capability of entanglement criteria theoretically. For a system coupled to an environment, we prove that any entanglement criterion needs exponentially many observables to detect the entanglement effectively when restricted to single-copy operations. Otherwise, the detection capability of the criterion will decay double exponentially. Furthermore, if multicopy joint measurements are allowed, the effectiveness of entanglement detection can be exponentially improved, which implies a quantum advantage in entanglement detection problems. Our results may shed light on why quantum phenomena are difficult to observe in large noisy systems.

Multipartite entanglement plays an essential role in both quantum information science and many-body physics. Because of the exponentially large dimension and complex geometric structure of the state space, the detection of entanglement in many-body systems is extremely challenging in reality. Conventional means, like entanglement witness and entropy criterion, either highly depend on the prior knowledge of the studied systems or the detection capability is relatively weak. In this Letter, we propose a framework for designing multipartite entanglement criteria based on permutation moments, which have an effective implementation with either the generalized control-swap quantum circuits or the random unitary techniques. As an example, in the bipartite scenario, we develop an entanglement criterion that can detect bound entanglement and show strong detection capability in the multiqubit Ising model with a long-range XY Hamiltonian. In the multipartite case, the permutation-moment-based criteria can detect entangled states that are not detectable by any criteria extended from the bipartite case. Our framework also shows potential in entanglement quantification and entanglement structure detection.

Improved bounds on some entanglement criteria in bipartite quantum systems

A universal programmable discriminator can perform the discrimination between\ntwo unknown states, and the optimal solution can be approached via the\ndiscrimination between the two averages over the uniformly distributed unknown\ninput pure states, which has been widely discussed in previous works. In this\npaper, we consider the success probabilities of the optimal universal\nprogrammable unambiguous discriminators when applied to the pure input states.\nMore precisely, the analytic results of the success probabilities are derived\nwith the expressions of the optimal measurement operators for the universal\ndiscriminators and we find that the success probabilities have nothing to do\nwith the dimension d while the amounts of the copies in the two program\nregisters are equal. The success probability of programmable unambiguous\ndiscriminator can asymptoticly approach to that of usual unambiguous\ndiscrimination (state comparison) as the number of copies in program registers\n(data register) goes to infinity.\n

Experimental procedures are presented for the rapid detection of entanglement of unknown arbitrary quantum states. The methods are based on the entanglement criterion using accessible correlations and the principle of correlation complementarity. Our first scheme essentially establishes the Schmidt decomposition for pure states, with few measurements only and without the need for shared reference frames. The second scheme employs a decision tree to speed up entanglement detection. We analyze the performance of the methods using numerical simulations and verify them experimentally for various states of two, three, and four qubits.

Detection of polarization-spatial classical optical entanglement through implementation of partial transpose on measured intensities is explored. A sufficient criterion for polarization-spatial entanglement in partially coherent light fields based on intensities measured at various orientations of the polarizer, as implied through partial transpose, is outlined. Detection of polarization-spatial entanglement using the outlined method is demonstrated experimentally through a Mach-Zehnder interferometer setup.

Information arises out of a communication game played between a sender and receiver of signals. For physical systems, the players may be designated as Nature and the scientist. The average information obtained from a quantum system is given by the von Neumann measurement scheme, which represents a generalization of thermodynamic entropy that is in perfect accord with common sense when considering a mixed state. While testing multiple copies of such a state can reveal information about the choice made by the sender, entropy for an unknown pure state is zero. A related puzzle is how the total information in the universe appears to be increasing over time.1 Zero entropy for an unknown pure state is reasonable inasmuch as once that state has been identified, no further information may be gained from examining its copies. But such an idea will not be reasonable if the game between sender and receiver consists of the former choosing one of a certain number of polarization states (say, for a photon) and supplying several copies to the receiver. In this case, measurements made on the copies reveal information regarding the sender’s choices. If the set of choices is infinite, the ‘information’ generated by the source is unbounded. From the point of view of the preparer of the states, information in the pure state is limited by the ‘relationship’ between source and receiver, and by the precision of the receiver’s measurement apparatus. If the sender chose a polarization state with which the apparatus was synchronized, the receiver could readily recognize the state. In recent papers,2, 3 I have been investigating information obtainable from an unknown pure state within the framework of communication between source and receiver. I propose a measure of entropy that covers both pure and mixed states. In general, entropy has two components. One is informational, related to the pure components of the quantum state, which can vary by receiver. The other component is thermodynamic and independent of the receiver. The increase of information over time is a consequence of the interplay between unitary and non-unitary evolution, which makes it possible to transform one type of information into the other. Such complementarity indicates that a fundamental duality is essential for information. For a two-component elementary mixed state, the most information to be obtained from each measurement is one bit. Each further measurement of identically prepared states will also yield one bit. For an unknown pure state, information represents the choice made by the source from an infinity of choices related to the values of probability amplitudes with respect to the basis components of the receiver’s measurement apparatus. For a two-component pure state, each measurement also provides a maximum of one bit of information. If the source has made available an unlimited number of identically prepared states, the receiver can obtain additional information from each measurement until the probability amplitudes have been correctly estimated. Once that has occurred, unlike the case with a mixed state, no further information will be obtained from testing additional copies of this pure state. These ideas have potential application in the field of quantum computing. Estimates by the receiver can be made by adjusting the basis vectors to move closer to the unknown pure state. In the most general case, the information obtainable from such a state in repeated experiments is potentially infinite. But if the observer learns the value of the pure state, the information associated with the states vanishes. This suggests a fundamental divide between objective and subjective information. This approach is consistent with the positivist view that one cannot speak of information associated with a system except in relation to an experimental arrangement together with the protocol for measurement. The experimental arrangement is thus integral to the amount of information that can be obtained. The informational measure I have proposed resolves the puzzle of entropy increase. We can suppose that, in the begin-

Quantum entanglement plays a key role in quantum computation and quantum information processing. It is of great significance to find efficient and experimentally friend separability criteria to detect entanglement. In this paper, we firstly propose two easily used entanglement criteria based on matrix moments. The first entanglement criterion only uses the first two realignment moments of a density matrix. The second entanglement criterion is based on the moments related to the partially transposed matrix. By detailed examples we illustrate the effectiveness of these criteria in detecting entanglement. Moreover, we provide an experimentally measurable lower bound of concurrence based on these moments. Finally, we present both bipartite and genuine tripartite entanglement measures based on the moments of the reduced states. By detailed examples, we show that our entanglement measures characterize the quantum entanglement in a more fine ways than the existing measures.

Quantum entanglement detection is one of the fundamental tasks in quantum information science. Conventional methods for quantum state tomography exhibit limitations in scalability as the number of qubits increases, leading to exponential growth in the number of unknown parameters and required measurements. Consequently, the accuracy enhancement achieved by these methods is constrained. In response to this challenge, we developed a tailored convolutional neural network (CNN) model capable of effectively detecting entanglement in two-qubit quantum states, achieving an accuracy exceeding 97.5%. Notably, even in the presence of noise, this model retains its robust performance, displaying resilience against a tolerable level of noise contamination. Furthermore, the inherent generalization power of CNNs allows our model, which was initially trained on a specific spectrum of quantum states, to extend its applicability to wider states, positioning it as an outstanding tool for the further application of machine learning in the field of quantum computing, opening up new pathways for solving entanglement detection problems in quantum information.

The minimum requirements for entanglement detection are discussed for a spin chain in which the spins cannot be individually accessed. The methods presented detect entangled states close to a cluster state and a many-body singlet state, and seem to be viable for experimental realization in optical lattices of two-state bosonic atoms. The entanglement criteria are based on entanglement witnesses and on the uncertainty of collective observables.

Entanglement detection and quantification usually require the knowledge of correlations and thus is a global issue. For both theoretical and practical reasons, it is desirable to characterize entanglement via local quantities. Of course, this can only be achieved in special setups with some underlying structure encoding the correlations. For the often encountered bilinear interactions such as beamsplitter experiments, we propose an operational and friendly entanglement criterion for the two-mode Gaussian output states. Specifically, we show that in order to detect and evaluate the entanglement of the output states of a bilinear interaction involving uncorrelated Gaussian input states, we do not need to know the joint covariance matrix of the whole two-mode output states. Instead, we only need the respective covariances of the reduced states of the two input modes and that of the two output modes, which are all local quantities pertaining to a single mode and can be easily measured locally without explicitly considering the correlations between the modes. This is achieved by exploiting the input-output characteristics of bilinear interactions. The result recasts the celebrated entanglement criteria of Simon (Phys. Rev. Lett., 84 (2000) 2726) and Duan et al. (Phys. Rev. Lett., 84 (2000) 2722) into a more convenient and simple form.

Quantum entanglement is a critical physical process in quantum mechanics and quantum information theory. It is a required process in quantum computing, quantum teleportation, and quantum cryptography. Entanglement detection affects the performance of quantum information processing tasks. Entanglement detection has grown in popularity over the years, and various entanglement detection methods are available, though some have application and system scale limitations. This scoping review sought to identify various measurement methods for entanglement detection in both bipartite and multipartite entanglement systems. Secondary resource indexed literatures were selected based on specific keywords from literatures published between 2017 and 2021. The goal of this study is to present a proposed conceptual framework of entanglement detection based on previous work as a guidance and reference founded on one’s specific requirements.

Multipartite entanglement detection via generalized Wigner–Yanase skew information

Entanglement detection and classification of different kinds of entangled states in quantum many-body systems have always been a key topic in quantum information and quantum computation. In this work, we investigate the entanglement detection and classification of three special entangled states: 4-qubit GHZ state, 4-qubit <inline-formula><tex-math id="M6">\begin{document}$ {\mathrm{W}}\overline{{\mathrm{W}}} $\end{document}</tex-math></inline-formula> state, and 4-qubit SGT state, which cannot be distinguished by the general quantum Fisher information (QFI) under the usual local operations. By utilizing the experimentally mature and controllable one-axis twisting model, along with optimized rotations and adjustable interaction strength, we successfully classify the three states by QFI. Additionally, we also study the effects of four types of environmental noise on entanglement detection, namely, bit-flip channel, amplitude-damping channel, phase-damping channel, and depolarizing channel. The results show that under local operations, the changes of the QFI from the 4-qubit GHZ state with decoherence parameter <i>p</i> in four noise channels are significantly different from those of the <inline-formula><tex-math id="M7">\begin{document}$ {\mathrm{W}}\overline{{\mathrm{W}}} $\end{document}</tex-math></inline-formula> state and SGT state, and thus making them distinguished. However, the QFI about the <inline-formula><tex-math id="M8">\begin{document}$ {\mathrm{W}}\overline{{\mathrm{W}}} $\end{document}</tex-math></inline-formula> state and the QFI about the SGT state exhibit the same variations and cannot be classified. In the one-axis twisting model, the variation curves of the QFI of the three states under the four noise channels are different from each other, which can be clearly observed. It should be noted that in the bit-flip channel, the QFI curves of the <inline-formula><tex-math id="M9">\begin{document}$ {\mathrm{W}}\overline{{\mathrm{W}}} $\end{document}</tex-math></inline-formula> state and the SGT state overlaps in the middle region (<inline-formula><tex-math id="M10">\begin{document}$ p\approx0.5 $\end{document}</tex-math></inline-formula>), which prevents their classification. Our work provides a new method for entanglement detection and classification in quantum many-body systems, which will contribute to future research in quantum science and technology.

We show that the decoherence, which in the long run destroys quantum features\nof a system, can be used to reveal the entanglement in a two-qubit system. To\nthis end, we consider a criterion that formally resembles the\nClauser-Horne-Shimony-Holt (CHSH) inequality. In our case the local observables\nare set by the coupling of each qubit to the environmental noise, controlled\nwith the dynamical decoupling method. We demonstrate that the constructed\ninequality is an entanglement criterion---it can only be violated by\nnon-separable initial two-qubit states, provided that the local noises are\ncorrelated. We also show that for a given initial state, this entanglement\ncriterion can be repurposed as a method of discriminating between Gaussian and\nnon-Gaussian noise generated by the environment of the qubits. The latter\napplication is important for ongoing research on using qubits to characterize\nthe dynamics of environment that perturbs them and causes their decoherence\n