Entanglement in the energy-constrained prepare-and-measure scenario: applications to randomness certification and channel discrimination

TL;DR

This work investigates energy-constrained semi-device-independent prepare-and-measure scenarios, showing that sharing entanglement between preparation and measurement devices enlarges the set of achievable correlations beyond the separable case. It demonstrates that entanglement can dramatically reduce certifiable randomness for fixed inputs and, under energy constraints, can substantially boost the advantage in binary channel discrimination beyond known bounds. The authors establish both numerical and analytical evidence that nonunitary channels are required for entanglement advantage in this setting, and they develop SDP and Lasserre-hierarchy methods to quantify these effects. Overall, the results highlight the critical role of entanglement in SDI analyses under realistic energy constraints, with implications for randomness generation and process-discrimination tasks.

Abstract

Quantum information tasks are often analyzed under varying trust assumptions about the devices involved. The semi-device-independent (SDI) framework offers a balance between needed assumptions and experimental feasibility. In this work, we study the energy-constrained SDI scenario, where the only assumption in a prepare-and-measure setup is an upper bound on the energy of the prepared quantum states. In contrast to previous studies that restricted the preparation and measurement devices to be classically correlated, we show that allowing entanglement strictly enlarges the set of achievable correlations. We identify two operational consequences of this result. The first concerns randomness certification, where we show that allowing the adversary to employ entangled strategies may significantly reduce the amount of certifiable randomness. This includes situations where the amount of randomness drops to zero in the presence of entanglement, while it remains positive when entanglement is excluded. Second, for the task of distinguishing an arbitrary quantum channel from the identity, we show that the known dimension-independent bound on the advantage conferred by entanglement is violated under an energy constraint.

Figures (6)

• Figure 1: PM scenario. Given an input $x$, a quantum channel $\Lambda_x$ is applied to prepare a state $\rho_S^x$. Subsequently, upon receiving another input $y$, a measurement $\{M^{b|y}_{S}\}_{b}$ is performed on the state, yielding outcome $b$.
• Figure 2: Entanglement-assisted PM scenario. Starting from a shared state $\sigma_{PM}$, channel $\Lambda_x$ acts on $\Tr_M[\sigma_{PM}]$ upon input $x$ to produce $\rho_S^x$, which is then measured with $\{\Pi_{SM}^{b|y}\}_{b}$ upon input $y$, yielding outcome $b$.
• Figure 3: Seesaw violations of Eq. \ref{['maxIclasscorr']}. In blue, the classically correlated bound $I^{\omega,\mathrm{sep}}_{\mathrm{corr}}$ from Eq. \ref{['maxIclasscorr']} is shown. Curves in green and orange display violations of this bound when a qubit communication ($\dim \mathcal{H}_S=2$) is assisted by $2\times 2$ and $3\times 3$ shared entanglement between the PM devices, respectively.
• Figure 4: $H_\mathrm{min}$ lower bound given a violation $I_\mathrm{corr}^{\omega,\mathrm{sep}}$. In blue, the certifiable randomness $H^{*,\mathrm{sep}}_\mathrm{min}$ under a no-entanglement assumption coming from van2019correlations. In orange, a seesaw upper bound on the certifiable randomness $H^{*}_\mathrm{min}$ in the entanglement-assisted scenario.
• Figure 5: Entangled violations of Eq. \ref{['eq:deterministic-ineq-for-fixed-input-0']}. In blue, the lower bound $I^{\omega,\mathrm{sep}}_\mathrm{det}$ in Eq. \ref{['eq:deterministic-ineq-for-fixed-input-0']}. In orange, seesaw upper bounds to Eq. \ref{['eq:minimization-of-E1']}. We observe that entangled PM devices can generate behaviors deterministic for one of the inputs and, yet, violate the inequality in Eq. \ref{['eq:deterministic-ineq-for-fixed-input-0']}.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Lemma 4
• proof