Einstein-Podolsky-Rosen steering and quantum steering ellipsoids: Optimal two-qubit states and projective measurements

We identify the families of states that maximize some recently proposed quantifiers of Einstein-Podolsky-Rosen (EPR) steering and the volume of the quantum steering ellipsoid (QSE). The optimal measurements which maximize genuine EPR steering measures are discussed and we develop a way to find them using the QSE. We thus explore the links between genuine EPR steering and the QSE and introduce states that can be the most useful for one-sided device-independent quantum cryptography for a given amount of noise.

Figure 1

Distribution of the normalized volume of the QSEs (i.e., $V_A/(4\pi /3)$) versus linear entropy for ${10}^8$ random two-qubit states (green dots). The area of the graph is populated by random states (green), with X states shown in lighter shades. The three families of states which maximize the volume are shown. The Werner states (${\rho }^{\left(R1\right)}$, black dashed line) maximize the volume at high linear entropy. The maximally discordant states (${\rho }^{\left(R2\right)}$, blue solid line) maximize the volume up to a value of $2/3$ for the linear entropy. The third family ${\rho }^{\left(R3\right)}$, which results from the linear mixture of the first two, is shown in yellow. Inset: We show the functional relation between $V_A$ and $m_2$ for family ${\rho }^{\left(R_2\right)}$ and $m_2\in [1/2,1]$ (which corresponds to the region $S_l\in [0,2/3]$ where ${\rho }^{\left(R_2\right)}$ is maximal). Gray dots are the results of our numerical simulation, while the red solid curve is the analytic function linking the volume of Alice's QSE to $m_2$.

Figure 2

A graph of the SW versus the linear entropy for random states. The two families of states which maximize the SW are shown. The Werner states (black dashed line) are shown for the sake of comparison. The maximally discordant states (${\rho }^{\left(R2\right)}$, blue solid line) maximize the SW up to a value of $S_L=2/3$ for the linear entropy. The third family (${\rho }^{\left(R3\right)}$, yellow points), which results from a linear mixture of the first two, is shown as a distribution.

Figure 3

Graphs showing (a) the quantum discord and (b) the negativity for the two families which maximize the SW at a given linear entropy. The colors correspond to the same families as in the previous figures. The correlations are lower at low linear entropy counterintuitive to what one would expect.

Figure 4

A graph of the RS versus the linear entropy for random states. The three families of states which maximize the RS are shown. The Werner states (black dashed line) are shown for the sake of comparison. The doubly asymmetric Bell states (${\rho }^{\left(R4\right)}$, blue solid line) and the family that maximizes the concurrence (${\rho }^{\left(\mathrm{conc}\right)}$, purple dot-dashed line) are shown. The third family (${\rho }^{\left(R5\right)}$, yellow points), resulting from a linear mixture of the first two, is also shown.

Figure 5

Diagram of a QSE showing how measurement in the Pauli basis is suboptimal in the case of this two-qubit state. Using only three types of measurements the best we can do in terms of maximizing our usage of the QSE volume is to use the semiaxes of the QSE as the basis for our measurements.

Figure 6

Diagram of a QSE showing how measurement in the Pauli basis can be suboptimal. Using only three types of measurements the best we can do in terms of maximizing our usage of the QSE volume is to use the semiaxes of the QSE as the basis for our measurements.