Complementarity Reveals Entanglement Sharing in Sequential Quantum Measurements

TL;DR

The paper addresses how entanglement can be shared across multiple observers performing sequential measurements on a two-qubit system, using three complementarity-based classical correlation metrics: $\mathcal{I}$ (mutual information), $\mathcal{S}$ (sum of conditional probabilities), and $\mathcal{C}$ (Pearson correlation). It develops a theoretical framework with a Bell-like state $|\psi\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$, mutually unbiased bases for measurements, and two measurement strategies—weak measurements and probabilistic projective measurements (PPM)—to explore unilateral and bilateral sharing. The key finding is that the Pearson correlation $\mathcal{C}$ is the most robust and informative criterion across all scenarios, while weak measurements more readily enable entanglement sharing; mutual information can fail to detect sharing in bilateral setups, although $\mathcal{S}$ and $\mathcal{C}$ often succeed, with nuanced behavior under asymmetries. These results elucidate a fundamental trade-off between measurement disturbance and complementary-correlation recovery and have implications for designing quantum-resource reuse protocols in sequential measurement contexts.

Abstract

We investigate entanglement sharing in a two-qubit sequential measurement scenario using three complementary classical correlation metrics: mutual information (I), sum of conditional probabilities (S), and the Pearson correlation coefficient (C). By investigating both weak measurement and probabilistic projective measurement (PPM) strategies in unilateral and bilateral scenarios, the phenomenon of entanglement sharing is conclusively certified when multiple pairs of classical correlation metrics simultaneously exceed their thresholds. Our investigation reveals that weak measurement strategies are more favorable than PPM for exhibiting entanglement sharing, regardless of the scenario. Furthermore, the mutual information criterion fails to characterize entanglement sharing in the bilateral scenario. While, the Pearson correlation criterion (C) is proven to be the most robust across all strategies and scenarios. These findings unveil a critical trade-off between measurement disturbance and complementary correlation recovery, which is essential for quantum resource reuse problems.

Figures (5)

• Figure 1: The general bilateral sequential scenario: Schematic of a quantum scenario where a pair of qubits emit particles to sequential observers on both sides. Each observer performs measurements with two complementary choices: for each $k\in(1,2,...,n)$, Alice$_k$ measures with observables $\hat{A}_{k,1}$ or $\hat{A}_{k,2}$, and Bob$_k$ with $\hat{B}_{k,1}$ or $\hat{B}_{k,2}$. Here, $A_{k,1}$ and $B_{k,1}$ are defined in the computational basis, while $A_{k,2}$ and $B_{k,2}$ are defined in the Fourier basis. Arrows indicate the transfer of qubits. The black dice symbolize that each observer's measurement choice is made probabilistically.
• Figure 2: Unilateral entanglement sharing via weak measurement strategy: (a) Case 1: the initial state is the maximal entangled state. (b) Case 2: the initial state is the partial entangled state with $G_1 = 1$. (c) Case 3: the initial state is the partial entangled state with $G_2 = 1$. (d) Case 4: the initial state is the partial entangled state with symmetric sharpness parameters $G_i=G$, $i\in\{1,2\}$.
• Figure 3: Unilateral entanglement sharing via PPM strategy: (a) Case 1: the initial state is the maximal entangled state. (b) Case 2: the initial state is the partial entangled state with $G_1 = 1$. (c) Case 3: the initial state is the partial entangled state with $G_2 = 1$. (d) Case 4: the initial state is the partial entangled state with symmetric sharpness parameters $G_i=G$, $i\in\{1,2\}$.
• Figure 4: Bilateral entanglement sharing via weak measurement strategy: Subfigures (A), (B), and (C) correspond to the criteria $\{\mathcal{I}_{k} , \mathcal{S}_{k}, \mathcal{C}_{k}\}$, respectively. Top panel: The case where $G_1=1$, $G_3=1$. Bottom panel: All parameters equal.
• Figure 5: Bilateral entanglement sharing via PPM strategy: Subfigures (A), (B), and (C) correspond to the criteria $\{\mathcal{I}_{k} , \mathcal{S}_{k}, \mathcal{C}_{k}\}$, respectively. Top panel: The case where $G_1=1$, $G_3=1$. Bottom panel: All parameters equal.