Enhanced quantum state discrimination under general measurements with entanglement and nonorthogonality restrictions

TL;DR

This work addresses whether binary quantum-state discrimination can beat the Helstrom bound by employing non-positive operator-valued measurements (NPOVMs) on an extended system. By formulating a resource-constrained optimization over auxiliary systems and joint measurements, the authors show sub-Helstrom error probabilities can be achieved even when the extended state is not entangled, and that advantages persist across cases with varying coherence and entanglement. Through four detailed scenarios (pure and mixed qubit states, with and without entanglement in the extensions), they derive explicit analytic expressions in some cases and provide numerical evidence in others, demonstrating robust NPOVM advantages under bounds on local distinguishability $d$ and entanglement $E$. The results reveal that non-POVM strategies can enhance discrimination performance in practical settings where measurements are constrained, offering fundamental insight and potential applications in quantum sensing and communication where Helstrom-optimal POVMs are not accessible or desirable.

Abstract

The minimum error probability for distinguishing between two quantum states is bounded by the Helstrom limit, derived under the assumption that measurement strategies are restricted to positive operator-valued measurements. We explore scenarios in which the error probability for discriminating two quantum states can be reduced below the Helstrom bound under some constrained access of resources, indicating the use of measurement operations that go beyond the standard positive operator-valued measurements framework. We refer to such measurements as non-positive operator-valued measurements. While existing literature often associates these measurements with initial entanglement between the system and an auxiliary, followed by joint projective measurement and discarding the auxiliary, we demonstrate that initial entanglement between system and auxiliary is not necessary for the emergence of such measurements in the context of state discrimination. Interestingly, even initial product states can give rise to effective non-positive measurements on the subsystem, and achieve sub-Helstrom discrimination error when discriminating quantum states of the subsystem.

Figures (6)

• Figure 1: The minimum error probability, $P_{\text{error}}^{\text{NPOVM}}$, obtained if Bob pertains to generalized measurements on the received state is plotted along the vertical axes, versus the upper bound, $d$, on the distinguishability between the two states, $\rho_A$ and $\sigma_A$, which is plotted along the horizontal axes. The states which we aim to discriminate are $\rho_B=\ket{0}\bra{0}$ and $\sigma_B=\ket{+}\bra{+}$. The minimum error decreases monotonically with an increase in the value of $d$. The quantities plotted along both the axes are dimensionless.
• Figure 2: The difference in error probabilities between the NPOVM and standard POVM strategies, defined as $\Delta P_{\text{er}} = P_{\text{error}}^{\text{NPOVM}} - P_{\text{error}}^{\text{POVM}}$, under the constraint that the distinguishability between the local states $\rho_A$ and $\sigma_A$ does not exceed a fixed bound $d$. The scenario considered corresponds to Case I. The value of $d$ chosen is $0.4$. The overall error probability is minimized with respect to the parameters of subsystem $A$. Notably, regions where $\Delta P_{\text{er}} < 0$ indicate an operational advantage of NPOVMs, yielding a strictly lower error probability compared to what is achievable with standard POVMs. The quantities plotted along all the axes are dimensionless.
• Figure 3: The figure of merit, $\Delta P_{er}=P_{\text{error}}^{\text{NPOVM}} - P_{\text{error}}^{\text{POVM}}$, plotted as a function of the parameters of the state, $n$, $m$ and $p$, illustrating the advantage of NPOVMs under local indistinguishability, under the condition that that the local distinguishablity of the subsystem, $A$ does not exceed a bound, $d$. Here, $d$ is chosen to be $0.6$. The analysis is conducted within the framework of Case II. The quantities plotted along all the axes are dimensionless.
• Figure 4: The figure of merit, $\Delta P_{er}=P_{\text{error}}^{\text{NPOVM}} - P_{\text{error}}^{\text{POVM}}$, is plotted as a function of the state parameters, parameters $\lambda$ and $\mu$, illustrating the advantage of NPOVMs under restricted local indistinguishability ($d=0.3$) and global entanglement ($E=0.1$). The scenario considered corresponds to Case III. The quantities plotted along each of the axes are dimensionless.
• Figure 5: The figure of merit, defined as $\Delta P_{\text{er}} = P_{\text{error}}^{\text{NPOVM}} - P_{\text{error}}^{\text{POVM}}$, is plotted as a function of the state parameters $x$ and $y$, to illustrate the operational advantage of NPOVMs over POVMs. The analysis is conducted corresponds to Case IV where the extended joint states are entangled. The plot corresponds to fixed values $\lambda = 0.25$ and $\mu = 0.33$. All quantities displayed along the axes are dimensionless.

Theorems & Definitions (5)

• Example 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4