Generalized conditional entropy optimization for qudit-qubit states

We derive a general approximate solution to the problem of minimizing the conditional entropy of a qudit-qubit system resulting from a local measurement on the qubit, which is valid for general entropic forms and becomes exact in the limit of weak correlations. This entropy measures the average conditional mixedness of the postmeasurement state of the qudit, and its minimum among all local measurements represents a generalized entanglement of formation. In the case of the von Neumann entropy, it is directly related to the quantum discord. It is shown that at the lowest nontrivial order, the problem reduces to the minimization of a quadratic form determined by the correlation tensor of the system, the Bloch vector of the qubit and the local concavity of the entropy, requiring just the diagonalization of a $3\times 3$ matrix. A simple geometrical picture in terms of an associated correlation ellipsoid is also derived, which illustrates the link between entropy optimization and correlation access and which is exact for a quadratic entropy. The approach enables a simple estimation of the quantum discord. Illustrative results for two-qubit states are discussed.

Figure 1

Schematic representation of the correlation ellipsoid (20) depicting the possible Bloch vectors ${\mathit{r}}_{A/\mathit{k}}$ of the postmeasurement state of $A$. It is centered at ${\mathit{r}}_A-C{\tilde{\mathit{r}}}_B$, with ${\tilde{\mathit{r}}}_B={\mathit{r}}_B/(1-r_B^2)$. For a given direction $\mathit{k}$ on the unit sphere of $B$, the vectors ${\mathit{r}}_{A/\pm \mathit{k}}$ in $A$ are the end points of a chord running through ${\mathit{r}}_A$. If ${\mathit{r}}_B=\mathbf{0}$, the ellipsoid becomes centered at ${\mathit{r}}_A$.

Figure 2

Bloch vectors of the postmeasurement states of $\mathit{A}$ that minimize the quadratic conditional entropy (24). The left panel depicts the general case, whereas the right panel depicts the case ${\mathit{r}}_B=\mathbf{0}$. The increase $\delta {\mathit{r}}_A={\mathit{r}}_{A/\mathit{k}}-{\mathit{r}}_A$ is parallel to the largest semiaxis of the correlation ellipsoid and coincides with it when ${\mathit{r}}_B=\mathbf{0}$.

Figure 3

Plot of the factor $\eta \left(r_A\right)=\frac{r_A{h}^{\prime \prime }\left(r_A\right)}{h^{\prime }\left(r_A\right)}$ in Eq. (46) for the von Neumann entropy. Since it is an increasing function, differences with the quadratic entropy results [for which $\eta \left(r_A\right)=1$ $\forall$ $r_A]$ will increase as $r_A=\left|\langle {\mathit{\sigma }}_A\rangle \right|$ increases. Quantities plotted are dimensionless.

Figure 4

A measurement is performed in the direction of vector $\mathit{k}$ on qubit $B$ and the entropy of the postmeasurement state of $A$ as measured by $S_f$ is $h_f\left(\right|{\mathit{r}}_{A/\pm \mathit{k}}\left|\right)$. The measurement equivalent ${\Delta }_f$ is defined as the increase in the norm of vector ${\mathit{r}}_A$ that satisfies $h_f({\mathit{r}}_A+{\Delta }_f)=p_{\mathit{k}}{h}_f\left(\right|{\mathit{r}}_{A/\mathit{k}}\left|\right)+p_{-\mathit{k}}{h}_f\left(\right|{\mathit{r}}_{A/-\mathit{k}}\left|\right)$ $(h$ and $r$ dimensionless).

Figure 5

Comparison between the projective minimizing measurements for the von Neumann and quadratic conditional entropies, for $X$ states [Eq. (67)] with $r_B=0.25$ and $J_z=0.3$ (top left), 0.15 (top right), $-0.25$ (bottom left), $-0.5$ (bottom right). Yellow (sector A) and green (sector B) disks show the set of states where the minimizing measurement is the same for both entropies (along $\mathit{x}$ in A and along $\mathit{z}$ in B), while blue disks (C) show those states where the minimizing measurements differ $(J_x$ and $r_A$ dimensionless).

Figure 6

The entropy decrease $\Delta S_f=S\left(A\right)-S\left(A\right|B_{\mathit{k}})$ after a local measurement on $B$ along direction $\mathit{k}=(k_x,0,k_z)$ for the quadratic $\left(\Delta S_2\right)$ and von Neumann $\left(\Delta S\right)$ entropies, together with the quadratic approximation (61, 62, 63, 64, 65) for the latter (dashed lines). We have considered an $X$ state with $r_A=r_B=0.25,J_z=-0.25$, and $J_x=0.1,0.325,0.5$, corresponding to states in sectors B, C, and A, respectively, of the bottom left panel of Fig. 5. The bottom right panel depicts the quantum discord (54) and its quadratic approximation (65) and (66) (dashed line) for $J_x=0.1$ (quantities plotted dimensionless).