Entanglement detection and classification of multi-qubit GHZ state, <inline-formula><tex-math id="Z-20251002193700">\begin{document}${\mathrm{W}}\overline{{\mathrm{W}}} $\end{document}</tex-math></inline-formula> state, and SGT state under one-axis twisting model

Entanglement detection and classification of multipartite systems remain the key topics in the field of quantum information and science. In this work, we take advantage of the nature that quantum Fisher information (QFI) can witness multipartite entanglement to comprehensively investigate the entanglement detection and classification of multi-qubit <inline-formula><tex-math id="Z-20250918160713">\begin{document}$W{\overline{W}} $\end{document}</tex-math></inline-formula> states immersed in a white noise environment. In the situation of local operation, by combining the information of the known quantum state, we have presented a criterion with visibility for witnessing the genuine multipartite entanglement and another for identifying the presence of quantum entanglement. Specifically, with respect to the 5-qubit <inline-formula><tex-math id="Z-20250918160719">\begin{document}$W{\overline{W}} $\end{document}</tex-math></inline-formula> state and 6-qubit <inline-formula><tex-math id="Z-20250918160726">\begin{document}$W{\overline{W}} $\end{document}</tex-math></inline-formula> state, due to the fact that the maximum QFI of their splitting-structure states exceeds that of the original states, it is infeasible to strictly establish a criterion for detecting the genuine multipartite entanglement. However, we delineate the scope for inferring the possible entanglement structures. Furthermore, it is found that as the number of qubits increases, the conditions for witnessing the genuine multipartite entanglement become increasingly strict, while those for detecting the existence of entanglement grow relatively more relaxed. Taking into account the likelihood of the crosstalk between neighboring qubits during the local operations on the multipartite systems in experiments, we employ the Lipkin-Meshkov-Glick (LMG) model to explore the entanglement classification of diverse multi-qubit multipartite states. It is found that with the increasing interaction strength, even for the strong white noise, the <inline-formula><tex-math id="Z-20250918160731">\begin{document}$W{\overline{W}} $\end{document}</tex-math></inline-formula> states can still be distinguished, thereby resolving the challenge of managing the entanglement classification under local operation. Besides, as the interaction strength continues to increase, the task of entanglement classification becomes more straightforward. This fully shows the superiority of nonlocal operations over local operations in the aspect of entanglement classification.

Quantum metrology is a subject of studying quantum measurement and quantum statistical deduction, and the precision of parameter estimation can be enhanced by quantum properties. In general, the process of parameter estimation includes four steps:preparation of probe state, parameterization process, measurement, and data processing. Of these four steps, the preparation of probe state is the most crucial. However, in practical applications, in the process of preparing quantum probe state, the probe system will couple to its environment, which will inevitably cause the quantum properties of the probe system to deteriorate, and thus reducing the precision of quantum parameter estimation. The dynamics of quantum Fisher information (QFI) for W state and Greenberger-Horne-Zeilinger (GHZ) state have been studied in decoherence channels. Because W state and GHZ state have different entanglement properties, the studies of the dynamics of QFI for the superposition of W state and GHZ state are of practical significance in quantum metrology field. In this paper, the dynamics of QFIs for the superposition of W state and GHZ state in three typical decoherence channels (depolarization channel, amplitude damping channel and phase damping channel) are studied. In the four steps of quantum parameter estimation, our major attention is paid to the first step (i.e., the preparation of probe state). For comparison, the QFIs of different probe states are studied, with the other three steps fixed, i.e., all the probe states will undergo the same parameterization, measurement and estimation process. The parameterization process involved here is a quantum spin operation (specified by the spin rotation direction), which is chosen to maximize the QFI of the probe state. The initial probe states under consideration are the superpositions of W state and GHZ state of three-particle and five-particle systems, and the QFI dynamics of those probe states are studied in the three different typical decoherence channels. By using the operator-sum (Kraus) representation of those three typical decoherence channels, the QFI dynamics of the probe state can be analytically derived in three different decoherence channels. The results show that in the depolarization channel, the maximum QFI of the probe state decreases with the decoherence evolving to zero in the end; in the amplitude damping channel, the QFI of the probe state decreases to the minimum with the decoherence evolution and then increases to the shot noise limit; in the phase damping channel, the QFI of the probe state decreases with the evolution of decoherence, but the final stable value is not zero. Further analyses show that W state component of the superposition plays a role in resisting phase damping and the GHZ state component plays a role in resisting amplitude damping. These results can help us to choose the optimal probe state for maximizing the estimation precision in practice.

In recent years, quantifying non-Markovian effect in open quantum system has become an important subject in the quantum decoherence control field. In this paper, a non-Markovian measure independent of the initial state of open system is proposed, thereby extending non-Markovian measure based on quantum Fisher information from the case where the initial state of the system is a pure state to the case where the initial state of the system is an arbitrary mixed state. As its application, the non-Markovian process is quantified by quantum Fisher information about a two-level system undergoing the three well-known dissipative channels, i.e. amplitude dissipative channel, phase damping channel, and random unitary channel. The results show that the conditions of non-Markovian processes in the three dissipative channels are independent of the selection of the initial state of the system by means of the quantum Fisher information of a phase parameter. Further, for amplitude dissipation channel and phase damping channel, the conditions for the non-Markovian processes to occur are equivalent to those given by trace distance, divisibility, quantum mutual information, quantum Fisher-information matrix, et al. As expected, for the case of amplitude dissipation channel, the corresponding results can reduce to the one in other paper (Lu X M, Wang X G, Sun C P <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1103/PhysRevA.82.042103">2010 <i>Phys. Rev. A</i><italic/> <b>82</b> 042103</ext-link>) by selecting the initial state of the system as an optimal pure state. However, for random unitary channel, the conditions of non-Markovian process are not equivalent to those for other measures. In addition, we also obtain an interesting relationship between quantum Fisher information and quantum coherence of the open system in the three dissipative channels, namely the square of quantum <inline-formula><tex-math id="M2">\begin{document}$l_1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20221053_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20221053_M2.png"/></alternatives></inline-formula> coherence for the evolved state of system is exactly equal to the quantum Fisher information of the phase parameter. In a word, the obtained results not only improve the application scope of using the quantum Fisher information to detect non-Markovian effects in open systems, but also further highlight its important role in quantum information processing.

Quantum Fisher information of an [formula omitted]-qubit maximal sliced state in decoherence channels and Ising-type interacting model

Quantum Fisher information (QFI) is a central concept in quantum sciences used to quantify the ultimate precision limit of parameter estimation, detect quantum phase transitions, witness genuine multipartite entanglement, or probe nonlocality. Despite this widespread range of applications, computing the QFI value of quantum many-body systems is, in general, a very demanding task. Here we combine ideas from functional theories and quantum information to develop a functional framework for the QFI of fermionic and bosonic ground states. By relying upon the constrained-search approach, we demonstrate that the QFI matrix terms can universally be determined by the one-body reduced density matrix (1-RDM), thus avoiding the use of exponentially large wave functions. Furthermore, we show that QFI functionals can be determined from the universal 1-RDM functional by calculating its derivatives with respect to the coupling strengths, thus becoming the generating functional of the QFI. We showcase our approach with the Bose-Hubbard model and present exact analytical and numerical QFI functionals. Our results provide the first connection between the one-body reduced density matrix functional theory and the quantum Fisher information. Published by the American Physical Society 2024

Phase damping destroys quantum Fisher information of W states

In this work, there are two parties, Alice on Earth and Bob on the satellite, which initially share an entangled state, and some open problems, which emerge during quantum steering that Alice remotely steers Bob, are investigated. Our analytical results indicate that all entangled pure states and maximally entangled evolution states (EESs) are steerable, and not every entangled evolution state is steerable and some steerable states are only locally correlated. Besides, quantum steering from Alice to Bob experiences a “sudden death” with increasing decoherence strength. However, shortly after that, quantum steering experiences a recovery with the increase of decoherence strength in bit flip (BF) and phase flip (PF) channels. Interestingly, while they initially share an entangled pure state, all EESs are steerable and obey Bell nonlocality in PF and phase damping channels. In BF channels, all steerable states can violate Bell-CHSH inequality, but some EESs are unable to be employed to realize steering. However, when they initially share an entangled mixed state, the outcome is different from that of the pure state. Furthermore, the steerability of entangled mixed states is weaker than that of entangled pure states. Thereby, decoherence can induce the degradation of quantum steering, and the steerability of state is associated with the interaction between quantum systems and reservoirs.

Quantum Fisher information of an open and noisy system in the steady state

The theory of multipartite entanglement provides a robust framework for understanding complex phenomena in quantum many-body systems, while multipartite entanglement also serves as a key resource in numerous quantum information processing tasks. Using density matrix renormalization group calculations, we study the multipartite entanglement in one-dimensional quantum long-range ferromagnetic and antiferromagnetic Ising chains with algebraically decaying interactions $1/r^\alpha$, respectively. To quantitatively probe the system’s multipartite entanglement structure, we employ the quantum Fisher information (QFI). Our results reveal two key behaviors: as the system size increases, the QFI rises, while the multipartite entanglement density decreases and converges toward a constant value. The latter trend points to a fundamental constraint on entanglement distribution in the thermodynamic limit. We further discuss the effects of the magnetic field and the exponent $\alpha$ on the QFI. We find that the peak values of the QFI increase with increasing $\alpha$. Additionally, the discontinuity in the QFI can be regarded as a powerful tool for characterizing quantum phase transitions in the systems.

Effect of quantum decoherence in a three-player quantum Kolkata restaurant problem is investigated using tripartite entangled qutrit states. Amplitude damping, depolarizing, phase damping, trit-phase flip and phase flip channels are considered to analyze the behaviour of players payoffs. It is seen that Alice's payoff is heavily influenced by the amplitude damping channel as compared to the depolarizing and flipping channels. However, for higher level of decoherence, Alice's payoff is strongly affected by depolarizing noise. Whereas the behaviour of phase damping channel is symmetrical around 50 % decoherence. It is also seen that for maximum decoherence (p=1), the influence of amplitude damping channel dominates over depolarizing and flipping channels. Whereas, phase damping channel has no effect on the Alice's payoff. Therefore, the problem becomes noiseless one at maximum decoherence in case of phase damping channel. Furthermore, the Nash equilibrium of the problem does not change under decoherence.

All open quantum systems are affected by environmental noises due to their interactions with the external environment and inevitably suffer from decoherence. Hence, it is fundamentally important and necessary to investigate decoherence suppression for open quantum systems via proper control strategies. Inspired by feed-forward control in the classical control theory, this paper proposes a novel decoherence suppression scheme via weak measurement and environment-assisted measurement. We first take the single-qubit system as an example to illustrate steps of the proposed scheme. To be specific, the single-qubit system is transferred to a state that is more robust to environmental noises via pre-weak measurement operators and feed-forward control operators before the decoherence channel, a measurement is performed on the environment coupled to the protected qubit during the decoherence channel, and the initial state is recovered via reversed feed-forward control operators and post-weak measurement operators after the decoherence channel. The optimum post-weak measurement strength is derived by setting the normalized final state equal to the initial state. By considering the optimum post-weak measurement strength, analytical formulas of the total success probability and the total fidelity are deduced. The proposed scheme is applicable for protecting quantum states from arbitrary decoherence channels with at least one invertible Kraus operator although only the amplitude damping channel and the phase damping channel are taken into account. Provided that the decay rate of the amplitude or phase damping channel is completely known, one can always achieve unit fidelity even for heavy damping cases, which is the biggest advantage of the proposed scheme. Influences of several parameters including strengths of weak measurements, the initial state and the decay rate of the decoherence channel on the performance of decoherence suppression are analyzed, and detailed procedures of a single-qubit pure and mixed state protection are presented on the Bloch sphere, respectively. Subsequently, the Kronecker product is employed to construct operators of dimension <inline-formula><tex-math id="M1">\begin{document}$ 2^N \times 2^N$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20220760_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20220760_M1.png"/></alternatives></inline-formula>, the proposed scheme is extended to the general <i>N</i>-qubit case, and unified analytical formulas of the total success probability and the total fidelity are deduced. By applying the proposed scheme to the protection of two-qubit entangled states, it is demonstrated that post-weak measurement operators are not necessary sometimes because of the particular structure of two-qubit entangled states. Furthermore, two numerical simulations are designed to enhance the concurrence of two-qubit entangled states and improve the average fidelity of the standard quantum teleportation in a noisy environment. Analytical formulas of the improvement of concurrence and the average teleportation fidelity are deduced, and the superiority of the proposed scheme is highlighted in comparison with unprotected scenarios.

In this paper we propose a scheme by using weak-measurement-based pre- and post-flips (WMPPF) to protect the average quantum Fisher information (QFI) in the independent amplitude-damping channel (ADC) for N-qubit GHZ state and generalized N-qubit GHZ states. We also discuss the weak measurement and quantum measurement reversal (WMQMR) with the same ADC. Based on the analytical and numerical results we obtain the main result: the WMPPF can reduce the effect of dissipation on the average QFI of the phase or the frequency for GHZ state and some generalized GHZ states, and the WMQMR can reduce the effect of dissipation on the average fidelity for GHZ state and generalized GHZ states in ADC. Comparing QFI with fidelity for WMPPF or for WMQMR, a scheme protecting the average fidelity does not necessarily protect the average QFI, even with the same parameters, and vice versa. We also focus on the average QFI versus N in the phase estimation and the frequency estimation of WMPPF, both of which show the advantages over the do-nothing (DN) case. From the investigation of the QFI of weight factor, we find that increasing qubit number can protect it both for WMPPF and for DN.

Enhancing quantum Fisher information by utilizing uncollapsing measurements

The unavoidable interaction of a real quantum system with the environment leads to the degradation of quantum coherence. Consequently, this paper investigates how such interaction can influence the quantum Fisher information (QFI) with the Hawking effect in the Schwarzschild black hole. We have considered two models of quantum decoherence: amplitude damping and phase damping channels; both can be experimentally induced. We found that Quantum information is more robust to phase flip channel than to the amplitude damping channel. It is observed that under the influence of phase damping channel, Quantum information does not depend on Hawking temperature making the parameter estimation immune to the Hawking effect but susceptible to decoherence. We found that QFI dwindles with increasing Hawking temperature and becomes more attenuated with decoherence which implies that the parameter estimation can be achieved with higher precision when the black hole is not evaporating.

Considering a pair of Unruh–DeWitt detectors, when one of them is kept inertial and the other one is accelerated and coupled to a scalar field, we address the teleportation of a two-qubit entangled state $$ \left( |\psi _\mathrm{in}\rangle =\text {cos}~\theta /2 |10\rangle +e^{i\varphi }~\text {sin}~\theta /2 |01\rangle \right) $$ through the quantum channel created by the above system and investigate how thermal noise induced by Unruh effect affects the quantum resources and quantum Fisher information (QFI) teleportation. Our results showed while the teleported quantum resources and QFI with respect to phase parameter $$ \varphi $$ $$\left( F_{\text {out}}\left( \varphi \right) \right) $$ reduce with increasing acceleration and effective coupling, QFI with respect to weight parameter $$ \theta $$ $$\left( F_{\text {out}}\left( \theta \right) \right) $$ interestingly increases after a specified value of acceleration and effective coupling. We also find that the teleported quantum resources and the precision of estimating phase parameter $$ \varphi $$ can be improved by a more entangled input state and more entangled channel. Moreover, the precision of estimating weight parameter $$ \theta $$ increases for a maximally entangled input state only in large acceleration regime, while it does not change considerably for both maximally and partially entangled states of the channel. The influence of Unruh effect on fidelity of teleportation is also investigated. We showed that for small effective coupling the average fidelity is always larger than $$ \frac{2}{3} $$ .