Entangled Measurement Outcomes for Optimal Postselection in Quantum Batteries

Abstract

Postselection can enhance the performance of a selected quantum battery branch, raising the question of whether the optimum can always be achieved by product measurements on the auxiliary subsystems. For battery auxiliary states obtained by mixing a pure state with global white noise of strength p, the conditional ergotropy reduces to a generalized Rayleigh quotient. For 0<p<1, product postselection attains the joint optimum if and only if the eigenspace associated with the largest generalized eigenvalue contains a product vector. For two auxiliary qubits, a strict product-joint gap therefore occurs precisely when this eigenspace is one dimensional and spanned by an entangled vector. Noisy W and GHZ states display qualitatively different behavior: the three qubit W state exhibits a strict gap throughout 0<p<1, whereas the three qubit GHZ state admits a product optimum. For interacting two qubit batteries, the optimal postselection can switch from a Bell outcome to a product outcome as the coupling strength is varied. By contrast, the outcome averaged capacity and ergotropy are independent of the measurement basis, showing that the joint measurement advantage is genuinely postselective.

The paper asks when optimizing a selected quantum-battery branch requires an entangled measurement outcome on two auxiliaries, rather than product postselection. It considers supplied resource states $\rho_{TM}=(1-p)|\Psi\rangle\langle\Psi|+p\mathbb{1}/(d_Td_M)$ and compares arbitrary joint rank-one projective outcomes with all product vectors. Each conditional battery state is an isotropic background plus a rank-one term. For $0<p<1$, conditional ergotropy becomes a generalized Rayleigh quotient, giving an exact joint optimum. A product outcome attains it precisely when the largest generalized-eigenvalue eigenspace contains a product vector. For two auxiliary qubits, a strict gap therefore occurs exactly when this eigenspace is one-dimensional and entangled. The criterion follows from the Rayleigh principle, compactness of the product-vector set, and the existence of a product vector in every two-dimensional two-qubit subspace. Analytic three- and four-qubit examples distinguish noisy W and GHZ resources. W$_3$ requires a Bell outcome throughout $0<p<1$, whereas GHZ$_3$ admits a product optimum. For an interacting two-qubit battery, GHZ$_4$ requires an entangled optimum whenever $J>0$; W$_4$ switches from Bell to product postselection at $J=\epsilon_A+\epsilon_B$. Exact product optimizations also distinguish conditional ergotropy from its probability-weighted contribution. Completeness gives measurement-basis-independent outcome averages: capacity $(1-p)(E_{\max}-E_{\min})$ and ergotropy $(1-p)[\operatorname{Tr}(\rho_T^\Psi H_T)-E_{\min}]$. Thus joint measurements redistribute these quantities among branches rather than increasing their averages. The results rely on global white noise and rank-one projective measurements. Preparation, measurement, and reset costs are excluded; more general noise, measurements, and cost-inclusive protocols remain open.