Characterising Two-Sided Quantum Correlations Beyond Entanglement via Metric-Adjusted f–Correlations

Abstract

We introduce an infinite family of quantifiers of quantum correlations beyond entanglement which vanish on both classical-quantum and quantum-classical states and are in one-to-one correspondence with the metric-adjusted skew informations. The ‘quantum f–correlations’ are defined as the maximum metric-adjusted f–correlations between pairs of local observables with the same fixed equispaced spectrum. We show that these quantifiers are entanglement monotones when restricted to pure states of qubit-qudit systems. We also evaluate the quantum f–correlations in closed form for two-qubit systems and discuss their behaviour under local commutativity preserving channels. We finally provide a physical interpretation for the quantifier corresponding to the average of the Wigner–Yanase–Dyson skew informations.

A   Monotonicity of Eq. (35) Under Local Unital Channels

We will here prove that, if $\varLambda _A$ is a unital channel on qubit A, then the following inequality holds:

$$\begin{aligned} \widetilde{\mathrm{Q}}^{f}(\varLambda _A(\rho ^{AB}))\le \widetilde{\mathrm{Q}}^{f}(\rho ^{AB}). \end{aligned}$$

(36)

In order to prevent the notation from becoming too cumbersome, in this Appendix we shall leave identity operators implicit wherever convenient: for example in the equation above we defined $\varLambda _A(\rho ^{AB})\equiv \varLambda _A\otimes \mathbb {I}_B(\rho ^{AB})$.

To begin our proof, let us assume that $\rho ^{ABC}$ is the optimal dilation of $\rho ^{AB}$ for the sake of Eq. (35), that is, $\widetilde{\mathrm{Q}}^{f}(\rho ^{AB})={\mathrm{Q}}^{f}_{AB}(\rho ^{ABC})$, where the subscript AB indicates what subsystems are involved in the calculation of the relevant quantum f–correlations. Consider now any dilation $\tau ^{ABCD}$ of $\varLambda _{A}(\rho ^{ABC})$ into a larger space, including a further ancillary system D. We note that $\tau ^{ABCD}$ is automatically also a dilation of $\varLambda _{A}(\rho ^{AB})$. Hence, the following inequality holds by definition:

$$\begin{aligned} \widetilde{\mathrm{Q}}^{f}(\varLambda _A(\rho ^{AB}))\le {\mathrm{Q}}^{f}_{AB}(\tau ^{ABCD}), \end{aligned}$$

(37)

Eq. (36) can then be proven by showing that ${\mathrm{Q}}^{f}_{AB}(\tau ^{ABCD})\le {\mathrm{Q}}^{f}_{AB}(\rho ^{ABC})$ for a particular choice of $\tau ^{ABCD}$.

To proceed, we use the fact that any unital qubit operation can be equivalently written as a random unitary channel [88], i.e.

$$\begin{aligned} \varLambda _{A}(\bullet ) = \sum _{k} q_{k}\, U_{A}^{(k)} \;\bullet \;( U_{A}^{(k)}) ^{\dagger }, \end{aligned}$$

(38)

for an appropriate collection of unitaries $\{U_{A}^{(k)}\}$ (acting on subsystem A) and probabilities $\{q_{k}\}$. A suitable dilation of $\varLambda _{A}(\rho ^{ABC})$ may then be chosen as

(39)

(40)

where is an orthonormal basis on system D. We shall now make use of Eqs. (7) and (28) to calculate the matrix ${M^f_\tau }$ corresponding to $\tau ^{ABCD}$, relating it to the matrix $M^f_\rho \equiv M^f$ of $\rho ^{ABC}$. We will then show that the maximum singular value of $M^f_\tau$ is smaller than that of $M^f$.

To do so we infer from Eq. (39) that the nonzero eigenvalues of $\tau ^{ABCD}$ are the same as those of $\rho ^{ABC}$, say $\{p_i\}$, while the associated eigenvectors are

(41)

being the eigenvectors of $\rho ^{ABC}$. Using the shorthand $\varvec{\sigma }=\{\sigma _1,\sigma _2,\sigma _3\}$ as in the main text, we can then write

(42)

where we have used the fact that . From the well known correspondence between the special unitary group $\mathsf{SU}(2)$ and special orthogonal group $\mathsf{SO}(3)$, it follows that for each k there exists an orthogonal matrix $R_{k}$ such that $(U_{A}^{(k)})^{\dagger }\varvec{\sigma }_{A} U_{A}^{(k)} = R_{k} \varvec{\sigma }_{A}$. Applying this idea to the last line in Eq. (42) we thus obtain

(43)

where $S=\sum _{k} q_{k} R_{k}$ is a real matrix such that $SS^T\le \mathbb I$, since it is a convex combination of orthogonal matrices. Since $M^f$ and $M^f_\tau$ are real matrices, their singular values are found as the square roots of the eigenvalues of $Q=M^f(M^f)^T$ and $Q_\tau =M^f_\tau (M^f_\tau )^T=SQS^T,$ respectively. Let $\varvec{v}$ be the normalised eigenvector of $Q_\tau$ corresponding to its largest eigenvalue. Then

$$\begin{aligned} \lambda _{\max }(Q_\tau )=\varvec{v}^T Q_\tau \varvec{v}=\varvec{v}^T SQS^T\varvec{v}\le \lambda _{\max }(Q)\underbrace{\Vert S^T\varvec{v}\Vert ^2}_{\le 1}\le \lambda _{\max }(Q), \end{aligned}$$

(44)

where we have used that $\varvec{v}$ is normalised and $SS^T\le \mathbb I$. This in turn implies that $s_{\max }(M_\tau ^f)\le s_{\max }(M^f)$, concluding our proof.

The proof can be repeated to show monotonicity under unital channels on qubit B as well. This proves that the quantity $\widetilde{\mathrm{Q}}^{f}(\rho ^{AB})$ defined in Eq. (35) is a bona fide quantifier of two-sided quantum correlations which obeys requirement (Q3) for any state $\rho ^{AB}$ of a two-qubit system.