Abstract Quantum entanglement and quantum entropy are crucial concepts in the study of multipartite quantum systems. In this work, we show how the notion of concurrence vector, re-expressed in a particularly useful form, provides new insights and computational tools for the analysis of both. In particular, using this approach for a general multipartite pure state, one can easily prove known relations in an easy way and to build up new relations between the concurrences associated with the different bipartitions. The approach is also useful to derive sufficient conditions for genuine entanglement in generic multipartite systems that are computable in polynomial time. From an entropy-of-entanglement perspective, the approach is powerful to prove properties of the Tsallis-2 entropy, such as the subadditivity, and to derive new ones, e.g., a modified version of the strong subadditivity which is always fulfilled; thanks to the purification theorem these results hold for any multipartite state, whether pure or mixed.
Abstract Quantum entanglement and quantum entropy are crucial concepts in the study of multipartite quantum systems. In this work, we show how the notion of concurrence vector, re-expressed in a particularly useful form, provides new insights and computational tools for the analysis of both. In particular, using this approach for a general multipartite pure state, one can easily prove known relations in an easy way and to build up new relations between the concurrences associated with the different bipartitions. The approach is also useful to derive sufficient conditions for genuine entanglement in generic multipartite systems that are computable in polynomial time. From an entropy-of-entanglement perspective, the approach is powerful to prove properties of the Tsallis-2 entropy, such as the subadditivity, and to derive new ones, e.g., a modified version of the strong subadditivity which is always fulfilled; thanks to the purification theorem these results hold for any multipartite state, whether pure or mixed.
Abstract In a recent work by us Ge et al [Phys. Rev. A 110, L010402 (2024)], we have derived a series of polygon relations of bipartite entanglement measures that is useful to reveal entanglement properties of discrete, continuous, and even hybrid multipartite quantum systems. In this work, with the information-theoretical measures of Rényi and Tsallis entropies, we study the relationship between the polygon relation and the subadditivity of entropy. In particular, the entropy-polygon relations are derived for pure multi-qubit states and then generalized to multi-mode Gaussian states, by utilizing the known results from the quantum marginal problem. Then the equivalence between the polygon relation and subadditivity is established, in the sense that for all discrete or continuous multipartite states, the polygon relation holds if and only if the underlying entropy is subadditive. As a byproduct, the subadditivity of Rényi and Tsallis entropies is proven for all bipartite Gaussian states. Finally, the difference between polygon relations and monogamy relations is clarified, and generalizations of our results are discussed. Our work provides a better understanding of the rich structure of multipartite states, and hence is expected to be helpful for the study of multipartite entanglement.
The paper is an elaborate comparative assessment of the most relevant entropy-based quantities to be applied in quantifying quantum entanglement in pure, mixed, bipartite, and multipartite quantum systems. The study fills the existing gap in the literature in which most of the previous literature examines individual measures, but a lot of knowledge is lacking on what these measures are regarding their strengths, weaknesses, and the operational usefulness against each other. The study can prove the behaviour of each measure under noise, decoherence, dimensional scaling, and variation of state-type by analysing von Neumann entropy, the Tsallis entropy, entropy of entanglement of formation, concurrence and logarithmic negativity. Numerical studies with pure two-qubit states, Werner states, GHZ states, W states and mixed random states demonstrate specific sensitivity properties and computational limitations which ensure that no single measure can be a universal quantifier. The paper goes on to say that entropy-only measures do not work in mixed regimes because of a dominance in classical correlations. The results have several important implications to quantum communication, quantum simulation, and quantum computing in the NISQ-era, where the choice of measurement has a direct impact on benchmarking, error-correction scheme, and protocol performance based on entanglement. The paper ends with a recommendation of contextual, purpose-based scheme of selecting entanglement measures and suggests a further study on scalable benchmarking, hybrid quantifiers, and operational performance-based entanglement measurements. These observations reinforce the theoretical premise of entanglement studies and enable the advancement of more effective quantum technologies in the field of practice.
Multipartite entanglement is the premier resource for quantum technologies. Yet, its exact quantification in the laboratory is notoriously challenging, typically requiring the full knowledge of high dimensional quantum states. Here, we construct observable bounds to multipartite entanglement for systems of arbitrary size, which are functions of the local and global state purities, and correlation functions. First, we derive experimentally accessible upper and lower limits to both the bipartite entanglement of formation and the squashed entanglement of bipartite systems, by leveraging cornerstone results of quantum information theory: the entropy strong subadditivity inequality and the Koashi-Winter monogamy relation. Then, we convert them into bounds to the entanglement up to degree k for arbitrary states, and to the genuine k-partite entanglement, by employing a recently proposed method. Finally, we analytically and numerically test these results, by bounding the multipartite entanglement of several relevant states and mixtures, including the important classes of GHZ, Dicke, W states, and random pure states.
The quantum entanglement as one of very important resources has been widely used in quantum information processing. In this work, we present a new kind of genuine multipartite entanglement. It is derived from special geometric feature of entangled systems compared with quantum multisource networks. We prove that any symmetric entangled pure state shows stronger nonlocality than the genuinely multipartite nonlocality in the biseparable model. Similar results hold for other entangled pure states with local dimensions no larger than 3. We further provide computational conditions for witnessing the new nonlocality of noisy states. These results suggest that the present model is useful characterizing a new kind of generic quantum entanglement.
Abstract Multipartite entanglement has a much more complex structure than bipartite entanglement. A state that lacks generic multipartite entanglement is 2-producible, i.e. it can be written as a tensor product of at most 2-partite entangled states. Recently, it has been proved that a tripartite pure state is 2-producible if and only if the gap between the entanglement of purification and its lower bound vanishes. Here, we show that the entanglement of purification gap is insufficient to detect more than tripartite entanglement in 4-partite stabilizer states. We then generalize entanglement of purification to the multipartite cases, and demonstrate that a multipartite pure state is 2-producible if and only if all the generalized entanglement of purification gaps vanish. The generalized entanglement of purification gap quantifies the quantum communication cost for redistributing one part of the system to the others, and also relates to the local recoverability of a multipartite state and the relative entropy between that state and 2-producible states. Moreover, we calculate the generalized entanglement of purification for states satisfying the general Schmidt decomposition, which implies that 4-partite stabilizer states do not necessarily have a general Schmidt decomposition. Our results provide a quantitative characterization of multipartite entanglement in multipartite system, which will promote further investigations and understanding of multipartite entanglement.
What would be a reasonable definition of the conditional entropy of bipartite quantum processes, and what novel insight would it provide? We develop this notion using four information-theoretic axioms and define the corresponding quantitative formulas. Our definitions of the conditional entropies of channels are based on the generalized state and channel divergences, for instance, quantum relative entropy. We find that the conditional entropy of quantum channels has potential to reveal insights for quantum processes that aren't already captured by the existing entropic functions, entropy or conditional entropy, of the states and channels. The von Neumann conditional entropy S[A∣B]N of the channel NA′B′→AB is based on the quantum relative entropy, with system pairs A′,A and B′,B being nonconditioning and conditioning systems, respectively. We identify a connection between the underlying causal structure of a bipartite channel and its conditional entropy. In particular, we provide a necessary and sufficient condition for a bipartite quantum channel NA′B′→AB in terms of its von Neumann conditional entropy S[A∣B]N, to have no causal influence from A′ to B. As a consequence, if S[A∣B]N<−log∣A∣ then the channel necessarily has causal influence (signaling) from A′ to B. Our definition of the conditional entropy establishes the strong subadditivity of the entropy for quantum channels. We also study the total amount of correlations possible due to quantum processes by defining the multipartite mutual information of quantum channels.