Observational entropy of quantum correlations and entanglement

TL;DR

This work advances the theory of observational entropy by analyzing entropy gaps under locality-restricted measurements (LO, LO*, LOCC, SEP) and relating them to quantum correlations. It establishes that for bipartite pure states all local gaps reproduce the entanglement entropy, while SEP gaps bound entanglement via the relative entropy of entanglement and LO* gaps equal the relative entropy of quantumness. The paper also shows a strict hierarchy among gaps across measurement classes, demonstrates separations with explicit states (e.g., CQ and Werner states), and extends the framework to multipartite systems to probe genuineness and robustness of correlations. Overall, observational-entropy gaps provide a operationally meaningful, scale-invariant set of quantities that capture both entanglement and discord-type resources, with potential for a joint resource-theoretic interpretation.

Abstract

The use of coarse graining to connect physical and information theoretic entropies has recently been given a precise formulation in terms of ``observational entropy'', describing entropy for observers with respect to a measurement. Here we consider observers with various locality restrictions, including local measurements (LO), measurements based on local operations with classical communication (LOCC), and separable measurements (SEP), with the idea that the ``entropy gap'' between the minimum locally measured observational entropy and the von Neumann entropy quantifies quantum correlations in a given state. After introducing entropy gaps for general classes of measurements and deriving their general properties, we specialize to LO, LOCC, SEP and other measurement classes related to the locality of subsystems. For those, we show that the entropy gap can be related to well-known measures of entanglement or non-classicality of the state (even though we point out that they are not entanglement monotones themselves). In particular, for bipartite pure states, all of the ``local'' entropy gaps reproduce the entanglement entropy, and for general multipartite states they are lower-bounded by the relative entropy of entanglement. The entropy gaps of the different measurement classes are ordered, and we show that in general (mixed and multipartite states) they are all different.

Figures (3)

• Figure 1: An example for which $E_{\textsc{lo}}(\rho) \neq E_{\textsc{lo}^*}(\rho)$ is given by classical-quantum "trine" state $\rho = \frac{1}{3} \sum_{k=0}^{2} {|{k}\rangle \! \langle{k}|} \otimes {|{\psi_k}\rangle \! \langle{\psi_k}|}$. Optimal LO/LO* measurements take the form $\Pi_{\textsc{cb}} \otimes N$ where $\Pi_{\textsc{cb}}$ measures system $A$ in the $\{|{0}\rangle,|{1}\rangle,|{2}\rangle\}$ basis. System $B$, a qubit, is depicted on the $xy$ plane in the Bloch sphere. For LO the optimal $N$ is the anti-trine POVM (red), while for LO* measuring on an aligned basis (blue dashed) is optimal.
• Figure 2: For Werner states $E_{\textsc{ppt}}(\rho_\lambda)=E_{\textsc{lo}^*}(\rho_\lambda)$, so all gaps in the local classes are equal. Here we plot $E_\chi(\rho_\lambda)$ for any $\text{PPT} \supset \chi \supset \text{LO*}$. Panel (a) shows the gap $E_\chi(\rho_\lambda)$ as a function of $\lambda$ for various Hilbert space dimensions. Panel (b) shows how $E_\chi(\rho_\lambda)$ arises from a combination of $S_M(\rho_\lambda)$ and $S(\rho_\lambda)$ for the case $d=3$. For $0 \leq \lambda \leq 0.5$ the Werner states are separable, otherwise they are entangled. However, the only Werner state with $E_\chi(\rho_\lambda) = 0$ is the maximally mixed state. All other Werner states, even the separable ones, exhibit a nonzero gap.
• Figure 3: Genuineness and robustness of quantum correlations in 4-partite GHZ, 2-Bell, and $W$ states. Plots show $E_{\textsc{lo}^*}$ in different partitions and when different subsystems are lost. Partitions are denoted by the type of splitting; e.g. "2+2" partitions include $AB|CD$, $AC|BD$, and so on, while "1+1+1+1" means $A|B|C|D$. Subsystem loss is denoted by the reduced density matrix for which $E_{\textsc{lo}^*}$ is calculated; the remaining subsystems are fully partitioned as in the $1+ \ldots +1$ case. Markers show values for each particular choice of partition/reduction, while dashed lines show the average over all partitions/reductions of a given type. (a) Correlations in the $W$ state are more genuinely multipartite, as it takes fully partitioning to reveal the full amount of correlations. (b) Correlations in the $W$ state are most robust, decreasing least on average when subsystems are lost.

Theorems & Definitions (51)

• Definition 1: Observational entropy (OE)
• Definition 2: Entropy gaps
• Proposition 3: General bounds
• proof
• Lemma 4: Optimal measurement
• proof
• Lemma 5: Convexity
• proof
• Definition 6: CPP
• Definition 7: QPP