Characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory. More recently, however, the total correlation or multipartite information measure has been helpful in accomplishing this goal, especially with the multipartite symmetric quantum (MSQ) discord (Piani et al. 2008 Phys. Rev. Lett. 100, 090502. ( doi:10.1103/PhysRevLett.100.090502 )) and the conditional entanglement of multipartite information (CEMI) (Yang et al. 2008 Phys. Rev. Lett. 101, 140501. ( doi:10.1103/PhysRevLett.101.140501 )). Here, we apply a recent and significant improvement of strong subadditivity of quantum entropy (Fawzi & Renner 2014 ( http://arxiv.org/abs/1410.0664 )) in order to develop these quantities further. In particular, we prove that the MSQ discord is nearly equal to zero if and only if the multipartite state for which it is evaluated is approximately locally recoverable after performing measurements on each of its systems. Furthermore, we prove that the CEMI is a faithful entanglement measure, i.e. it vanishes if and only if the multipartite state for which it is evaluated is a fully separable state. Along the way, we provide an operational interpretation of the MSQ discord in terms of the partial state distribution protocol, which in turn, as a special case, gives an interpretation for the original discord quantity. Finally, we prove an inequality that could potentially improve upon the Fawzi–Renner inequality in the multipartite context, but it remains an open question to determine whether this is so.
Characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory. More recently, however, the total correlation or multipartite information measure has been helpful in accomplishing this goal, especially with the multipartite symmetric quantum (MSQ) discord (Piani et al. 2008 Phys. Rev. Lett. 100, 090502. ( doi:10.1103/PhysRevLett.100.090502 )) and the conditional entanglement of multipartite information (CEMI) (Yang et al. 2008 Phys. Rev. Lett. 101, 140501. ( doi:10.1103/PhysRevLett.101.140501 )). Here, we apply a recent and significant improvement of strong subadditivity of quantum entropy (Fawzi & Renner 2014 ( http://arxiv.org/abs/1410.0664 )) in order to develop these quantities further. In particular, we prove that the MSQ discord is nearly equal to zero if and only if the multipartite state for which it is evaluated is approximately locally recoverable after performing measurements on each of its systems. Furthermore, we prove that the CEMI is a faithful entanglement measure, i.e. it vanishes if and only if the multipartite state for which it is evaluated is a fully separable state. Along the way, we provide an operational interpretation of the MSQ discord in terms of the partial state distribution protocol, which in turn, as a special case, gives an interpretation for the original discord quantity. Finally, we prove an inequality that could potentially improve upon the Fawzi–Renner inequality in the multipartite context, but it remains an open question to determine whether this is so.
The distribution of quantum correlations in multipartite systems plays a significant role in several aspects of quantum information theory. While it is well known that these quantum correlations cannot be freely distributed, the way that they are shared in a multipartite system is an open problem even for a small set of qubits. Based on monogamylike relations between entanglement and discord for n-partite systems, we show how these correlations are distributed in general, determining distinct equalities and inequalities to the quantum discord and the entanglement of formation for arbitrary multipartite pure states.
Quantum correlation is a key to our understanding of quantum physics. In particular, it is essential for the powerful applications to quantum information and quantum computation. There exist quantum correlations beyond entanglement, such as quantum discord (QD) and measurement-induced nonlocality (MiN) [Phys. Rev. Lett. \textbf{106}, 120401(2011)]. In [Phys. Rev. A \textbf{77}, 022113(2008)], a subclass of PPT states so-called strong positive partial transposition (SPPT) states was introduced and it was conjectured there that SPPT states are separable. However, it was illustrated with examples in [Phys. Rev. A \textbf{81}, 064101(2010)] that this conjecture is not true. Viewing the original SPPT as SPPT up to part B, in the present paper, we define SPPT state up to part A and B respectively and present a separable class of SPPT states, that is the super SPPT (SSPPT) states, in terms of local commutativity. In addition, classical-quantum (CQ) states and nullity of MiN are characterized via local commutativity. Based on CQ states, the geometric measure of quantum discord (GMQD) for infinite-dimensional case is proposed. Consequently, we highlight the relation among MiN, QD(GMQD), SSPPT and separability through a unified approach for both finite- and infinite-dimensional systems: zero MiN implies zero QD(GMQD), zero QD(GMQD) signals SSPPT and SSPPT guarantees separability, but the converses are not valid.
Within the unified framework of exploiting the relative entropy as a distance measure of quantum correlations, we make explicit the hierarchical structure of quantum coherence, quantum discord, and quantum entanglement in multipartite systems. On this basis, we define a basis-independent measure of quantum coherence and prove that it is exactly equivalent to quantum discord. Furthermore, since the original relative entropy of coherence is a basis-dependent quantity, we investigate the local and nonlocal unitary creation of quantum coherence, focusing on the two-qubit unitary gates. Intriguingly, our results demonstrate that nonlocal unitary gates do not necessarily outperform the local unitary gates. Finally, the additivity relationship of quantum coherence in tripartite systems is discussed in detail, where the strong subadditivity of von Neumann entropy plays an essential role.
Both coherence and entanglement stem from the superposition principle, capture quantumness of a physical system, and play a central role in quantum physics. In a multipartite quantum system, coherence and quantum correlations are closely connected. In particular, it has been established that quantum coherence of a bipartite state is an important resource for its conversion to entanglement [A. Streltsov et al., Phys. Rev. Lett. 115, 020403 (2015)] and to quantum discord [J. Ma et al., Phys. Rev. Lett. 116, 160407 (2016)]. We show here that there is a very close association between partial coherence introduced by Luo and Sun [S. Luo and Y. Sun, Phys. Rev. A 96, 022136 (2017)] and quantum correlations (quantified by quantum discord) in both directions. Furthermore, we propose families of coherence measures in terms of quantum correlations and quantum Fisher information.
This paper defines the fidelity of recovery of a tripartite quantum state on systems A,B, and C as a measure of how well one can recover the full state on all three systems if system A is lost and a recovery operation is performed on system C alone. The surprisal of the fidelity of recovery (its negative logarithm) is an information quantity which obeys nearly all of the properties of the conditional quantum mutual information I(A;B∣C), including non-negativity, monotonicity with respect to local operations, duality, invariance with respect to local isometries, a dimension bound, and continuity. We then define a (pseudo) entanglement measure based on this quantity, which we call the ``geometric squashed entanglement.'' We prove that the geometric squashed entanglement is a 1-LOCC monotone (i.e., monotone nonincreasing with respect to local operations and classical communication from Bob to Alice), that it vanishes if and only if the state on which it is evaluated is unentangled, and that it reduces to the geometric measure of entanglement if the state is pure. We also show that it is invariant with respect to local isometries, subadditive, continuous, and normalized on maximally entangled states. We next define the surprisal of measurement recoverability, which is an information quantity in the spirit of quantum discord, characterizing how well one can recover a share of a bipartite state if it is measured. We prove that this discordlike quantity satisfies several properties, including non-negativity, faithfulness on classical-quantum states, invariance with respect to local isometries, a dimension bound, and normalization on maximally entangled states. This quantity, combined with a recent breakthrough of Fawzi and Renner, makes it possible to characterize states with discord nearly equal to zero as being approximate fixed points of entanglement-breaking channels (equivalently, they are recoverable from the state of a measuring apparatus). Finally, we discuss a multipartite fidelity of recovery and several of its properties.
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.