Negativity vs. purity and entropy in witnessing entanglement

Abstract

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.

Figure 1

(Top) Scatterplots of $S_2(A|B)$ vs $S_1(A|B)$ and respective purity histograms for ${10}^6$ 2-quDit systems for $D=(2,3,5,8,10)$ with each plot labeled $2\otimes 2$, $3\otimes 3$, $5\otimes 5$, $8\otimes 8$,and $10\otimes 10$, respectively. The light orange scatterplots are from the fully uniform ensemble (abbreviated UE) while the blue scatterplots are from the ensemble uniform with respect to purity (abbreviated UP). The inset histograms are of the joint purity of the fully uniform ensemble). The red dotted line in each plot is where $S_1(A|B)=S_2(A|B)$. The set of all $D\otimes D$ pure states is within the green serrated blade region in the lower left quadrant (or is only a single curve for $2\otimes 2$), while the set of all $D\otimes D$ states with maximally mixed marginals corresponds to the large magenta serrated blade spanning three quadrants of the plot. The regions enclosed between the two blades also correspond to valid density matrices. (Bottom) This table gives the percentages of the total number of generated states whose entanglement was witnessed with the function in the first column.

Figure 2

Plots showing different conditional entropy functions (and the negativity) for uniform-purity ensembles as a function of joint purity. (Left) Case of $3\otimes 3$ systems. (Right) Case of $10\otimes 10$ systems. Narrow curves of the same color plot the corresponding function of the Werner state whose purity is varied by changing the mixing parameter p. Note that the Werner state curves for negative the log negativity $-E_{\mathcal{N}}(\widehat{\rho })$ coincide with the conditional min entropy $S_{\infty }(A|B)$ where entanglement is witnessed.

Figure 3

Plot showing percentages of 60-dimensional joint systems sampled according to the uniform-purity ensemble, whose entanglement was witnessed by comparing different forms of joint and marginal entropies, and by the negativity of the partial transpose. These systems were bipartitioned into subsystems of dimensions $D_A$ and $D_B$ respectively, such that $D_A{D}_B=60$. Starting with ${10}^6$ random diagonal density matrices, different random unitaries were performed for each possible bipartition to generate the full ensemble of density matrices analyzed here. The entanglement-success percentages are plotted as a function of $D_B$. The vertical axis denotes the different entanglement witnesses used, and in particular, that we are conditioning on subsystem B.