In this paper, we explore the value of measures of mixedness in witnessing entanglement. While all measures of mixedness may be used to witness entanglement, we show that all such entangled states must have a negative partial transpose (NPT). Where the experimental resources needed to determine this negativity scale poorly at high dimension, we compare different measures of mixedness over both Haar-uniform and uniform-purity ensembles of joint quantum states at varying dimension to gauge their relative success at witnessing entanglement. In doing so, we find that comparing joint and marginal purities is overwhelmingly (albeit not exclusively) more successful at identifying entanglement than comparing joint and marginal von Neumann entropies, in spite of requiring fewer resources. We conclude by showing how our results impact the fundamental relationship between correlation and entanglement and related witnesses.
In this paper, we explore the value of measures of mixedness in witnessing entanglement. While all measures of mixedness may be used to witness entanglement, we show that all such entangled states must have a negative partial transpose (NPT). Where the experimental resources needed to determine this negativity scale poorly at high dimension, we compare different measures of mixedness over both Haar-uniform and uniform-purity ensembles of joint quantum states at varying dimension to gauge their relative success at witnessing entanglement. In doing so, we find that comparing joint and marginal purities is overwhelmingly (albeit not exclusively) more successful at identifying entanglement than comparing joint and marginal von Neumann entropies, in spite of requiring fewer resources. We conclude by showing how our results impact the fundamental relationship between correlation and entanglement and related witnesses.
In this work, we prove that while all measures of mixedness can be used to witness entanglement, no measure of mixedness is more sensitive than the negativity of the partial transpose. However, computing either the negativity or differences between von Neumann entropies to witness entanglement requires complete knowledge of the joint density matrix (and is therefore not practical at high dimension). In light of this, we examine joint vs marginal purities as a witness of entanglement, (which can be obtained directly through interference measurements) and find that comparing purities is actually more sensitive at witnessing entanglement than using von Neumann entropies while also providing tight upper and lower bounds to it in the high-entanglement limit.
In the general framework of d 1 × d 2 mixed states, we derive an explicit bound for bipartite negative partial transpose (NPT) entanglement based on the mixedness characterization of the physical system. The derived result is very general, being based only on the assumption of finite dimensionality. In addition, it turns out to be of experimental interest since some purity-measuring protocols are known. Exploiting the bound in the particular case of thermal entanglement, a way to connect thermodynamic features to the monogamy of quantum correlations is suggested, and some recent results on the subject are given a physically clear explanation.
Abstract. Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or in general by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two–mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p−entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixedstate entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions. 1.
Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.
Maximally entangled mixed states are those states that, for a given mixedness, achieve the greatest possible entanglement. For two-qubit systems and for various combinations of entanglement and mixedness measures, the form of the corresponding maximally entangled mixed states is determined primarily analytically. As measures of entanglement, we consider entanglement of formation, relative entropy of entanglement, and negativity; as measures of mixedness, we consider linear and von Neumann entropies. We show that the forms of the maximally entangled mixed states can vary with the combination of (entanglement and mixedness) measures chosen. Moreover, for certain combinations, the forms of the maximally entangled mixed states can change discontinuously at a specific value of the entropy. Along the way, we determine the states that, for a given value of entropy, achieve maximal violation of Bell's inequality.
One aspect of complexity for quantum many-body systems resides in their ability to exhibit entanglement, uniquely quantum correlations central to quantum computation. To quantitatively explore the diminishing effect that quantum noise has on entanglement resources, we focus on the negativity witness, an entanglement measure based on local time reversal transformations that provides a perfect witness in simple physical contexts. Beyond calculation of the negativity, we analyze the associated eigenvalue spectra of partially transposed density matrices for qudit-qudit bipartitions, finding that higher negativity rank provides greater(reduced) robustness to targeted(depolarizing) noise. In particular, we extend the maximum number of separable states that can be arbitrarily mixed into an entangled pure ensemble while retaining negativity to be exactly equal to the original negativity rank. Furthermore, we report how understanding the role of the negativity rank with respect to the completely depolarizing channel allows direct calculation of (1) certain noise boundaries for the Absolutely Positive Partial Transpose (APPT) subspace beyond which no unitary operator is capable of generating negativity and (2) the maximum qutrit-qutrit APPT purity. Beyond proving tight bounds and contributing to the ongoing search for APPT entanglement, these results expand our understanding of the nuanced relationship between computation with entanglement and practical environments for operating quantum devices.