Bipartite quantum states admitting a causal explanation

TL;DR

The paper investigates when a bipartite quantum state can be explained by a direct temporal causal influence between two observers, under tomographically complete measurements. It proves that all separable states are temporally compatible and provides an explicit temporal decomposition $ ext{E}= ext{G}\circ ext{D}$ where $ ext{D}$ is a generalized dephasing channel and $ ext{G}$ a pretty good measure-and-prepare channel, with the two evolutions acting as Bayesian inverses rather than Petz recoveries. It further introduces a temporal PPT-like criterion that is both necessary and sufficient for temporal compatibility in finite dimensions and links temporal compatibility to state discrimination, thereby giving an operational interpretation of spatiotemporal correlations. The results illuminate how temporal and spatial quantum correlations interconvert via dephasing and discrimination steps and open avenues for extending to broader measurement families, multipartite scenarios, and infinite-dimensional settings.

Abstract

The statistics of local measurements of joint quantum systems can sometimes be used to distinguish the spatiotemporal structure in which they were measured. We first prove that every bipartite separable density matrix is temporally compatible with direct causal influence for arbitrary finite-dimensional quantum systems and measurements of a tomographically-complete class of observables, which includes all Pauli observables in the case of multi-qubit systems. Equivalently, if a bipartite density matrix is not temporally compatible with direct causal influence, then it must be entangled. We also provide an operational meaning for the two temporal evolutions consistent with such correlations in terms of generalized dephasing channels and pretty good measurements. The two temporal evolutions are Bayesian inverses of each other, which is different from them being Petz recovery maps of each other. Finally, we prove necessary and sufficient conditions for an arbitrary bipartite quantum state to be temporally compatible, thereby providing a temporal analogue of the positive partial transpose criterion valid for quantum systems of any dimension.

Figures (2)

• Figure 1: A Venn diagram depicting the sets of bipartite states, separable states, PPT states, and states over time, the last of which contains temporally compatible pseudo-density matrices (although the sets of bipartite states, separable states, and PPT states are convex, the set of states over time is not convex SNREG23). The containment of separable and PPT states inside states over time are proved in Theorem \ref{['thm:tempcompSB']} and Corollary \ref{['cor:PPTimpliestempcompat']}, respectively.
• Figure 2: A visualization of the decomposition $\mathcal{E}=\mathcal{G}\circ\mathcal{D}$ of the temporal channel in terms of a dephasing channel $\mathcal{D}$ followed by a measure-and-prepare channel $\mathcal{G}$ from Theorem \ref{['thm:splittemporalmap']}. To produce these exact images, set $\tau=\sum_{\theta}t_{\theta}\rho_{\mathpzc{A};\theta}\otimes\rho_{\mathpzc{B};\theta}$, with $\theta\in\{1,\dots,6\}$, $t_{1}=\frac{5}{8}$, $t_{2}=\cdots=t_{6}=\frac{3}{40}$, $\rho_{\mathpzc{A};1}=\rho_{\mathpzc{B};1}=|0\rangle\langle0|$, $\rho_{\mathpzc{A};2}=\rho_{\mathpzc{B};2}=|1\rangle\langle1|$, $\rho_{\mathpzc{A};3}=\rho_{\mathpzc{B};3}=|+\rangle\langle+|$, $\rho_{\mathpzc{A};4}=\rho_{\mathpzc{B};4}=|-\rangle\langle-|$, $\rho_{\mathpzc{A};5}=\rho_{\mathpzc{B};5}=|\!+i\rangle\langle+i|$, and $\rho_{\mathpzc{A};6}=\rho_{\mathpzc{B};6}=|\!-i\rangle\langle-i|$. Our convention here is that $|0\rangle=(1,0)$, $|1\rangle=(0,1)$, $|\pm\rangle=\frac{1}{\sqrt{2}}(|0\rangle\pm|1\rangle)$, and $|\pm i\rangle=\frac{1}{\sqrt{2}}(|0\rangle\pm i|1\rangle)$. The figure on the left is that of the Bloch ball $\mathpzc{S}$. The figure on top shows the image, $\mathcal{D}(\mathpzc{S})$, of $\mathpzc{S}$ under the map $\mathcal{D}$ as well as $\mathpzc{S}$ for comparison. The figure on the right shows the image of $\mathpzc{S}$ under $\mathcal{E}$ as well as $\mathpzc{S}$ and $\mathcal{D}(\mathpzc{S})$ for comparison (the figure on the right does not show $\mathcal{G}(\mathpzc{S})$ to avoid cluttering). The measure-and-prepare channel illustrates greater distinguishability in the vertical direction due to the higher weight given by $t_{1}+t_{2}$ as compared with $t_{3}+t_{4}=t_{5}+t_{6}$.

Theorems & Definitions (34)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Proposition 2.4
• Remark 2.5
• Definition 2.6
• Theorem 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['lem:invsumprob']}
• Lemma 3.3