Beyond Hoeffding and Chernoff: Trading conclusiveness for advantages in quantum hypothesis testing

TL;DR

This work fundamentally broadens quantum hypothesis testing by allowing abstentions (inconclusive outcomes) and mapping how varying degrees of inconclusiveness reshape optimal error-exponent trade-offs. It delivers a comprehensive asymptotic landscape, showing that arbitrarily small or exponentially vanishing inconclusiveness can surpass the quantum Hoeffding and Chernoff bounds in both asymmetric and symmetric settings, with precise characterizations via sandwiched and anti-divergences. A key advance is the unified treatment that connects sequential hypothesis testing, pinching techniques, and postselected frameworks to quantify achievable exponents and their limits, including strong-converse-type results. The findings have practical implications for designing discrimination protocols with minimal overhead while achieving unprecedented error decay rates, and open avenues for extensions to channels and composite hypotheses.

Abstract

The ultimate limits of quantum state discrimination are often thought to be captured by asymptotic bounds that restrict the achievable error probabilities, notably the quantum Chernoff and Hoeffding bounds. Here we study hypothesis testing protocols that are permitted a probability of producing an inconclusive discrimination outcome, and investigate their performance when this probability is suitably constrained. We show that even by allowing an arbitrarily small probability of inconclusiveness, the limits imposed by the quantum Hoeffding and Chernoff bounds can be significantly exceeded. This completely circumvents the conventional trade-offs between error exponents in hypothesis testing while incurring only a vanishingly small overhead over conventional approaches. Such improvements over standard state discrimination are robust and can be obtained even when an exponentially vanishing probability of inconclusive outcomes is demanded. Relaxing the constraints on the inconclusive probability can enable even larger advantages, but this comes at a price. We show a 'strong converse' property of this setting: targeting error exponents beyond those achievable with vanishing inconclusiveness necessarily forces the probability of inconclusive outcomes to converge to one. By exactly quantifying the rate of this convergence, we give a complete characterisation of the trade-offs between error exponents and rates of conclusive outcome probabilities. Overall, our results provide a comprehensive asymptotic picture of how the allowance for inconclusive measurement outcomes reshapes optimal quantum hypothesis testing.

Figures (4)

• Figure 1: Beyond Hoeffding with low inconclusiveness. The schematic plot shows the achievable ranges of exponents of the conditional errors of quantum hypothesis testing, $\overline{\alpha}_n \sim \exp(-nA)$ and $\overline{\beta}_n \sim \exp(-nB)$, in the different regimes studied in this work. The blue region demarcated by the curve $B = H_A(\sigma\|\rho)$, where $H_A$ is the Hoeffding divergence, shows all of the exponents that can be achieved in conventional hypothesis testing using deterministic protocols. All error exponents in the green region, delineated by $B = D(\rho\|\sigma)$ and $A = D(\sigma\|\rho)$, can be achieved with an arbitrarily small --- or asymptotically vanishing --- probability of inconclusive outcomes (Proposition \ref{['prop:highprob_exponents']}). The orange region below the curve defined by $B = D(\rho\|\sigma) - H^*_A(\sigma\|\rho)$, with $H^*_A$ denoting the Han--Kobayashi anti-divergence, can be achieved by tests that are highly conclusive only for the null hypothesis $\rho$ (Proposition \ref{['prop:onesided']}).
• Figure 2: Achievable region of exponents with high inconclusiveness. The plot shows the relations and trade-offs between exponents of different types characterized by our results in the 'strong converse' setting of high inconclusive probability. Here, $A$ and $B$ are the exponents of the type I and type II conditional errors, $\overline{\alpha}_n\sim\exp(-nA)$ and $\overline{\beta}_n\sim\exp(-nB)$, and $K$ and $L$ the exponents of the probabilities of conclusive outcomes under the two hypotheses, $\pi_n(\rho)\sim\exp(-nK)$ and $\pi_n(\sigma)\sim\exp(-nL)$. The central plot shows the three-dimensional region in which the exponents $A$, $B$, and $L$ are simultaneously achievable for some fixed $K = K_0$. The red rectangle within the achievable region is its intersection with the plane $L=L_0$, representing the achievable error exponents for fixed exponents of conclusiveness, as characterized in Proposition \ref{['prop:conclusive']}. The three plots surrounding the central one depict projections of the region, from which one can deduce different trade-offs: between the conditional type II error exponent $B$ and its corresponding conclusive probability $L$ (leftmost plot); between the two conditional errors themselves when $L$ is unconstrained (top right), with $K_{A,B}^*(\rho\|\sigma)$ defined in Eq. \ref{['eq:kstar']}; and between the conditional type I error exponent $A$ and the conclusiveness exponent $L$ (bottom).
• Figure 3: Trade-offs in hypothesis testing with exponentially low inconclusiveness. The shaded region represents the range of achievable exponents of the conditional errors $\overline{\alpha}_n \sim \exp(-nA)$ and $\overline{\beta}_n \sim \exp(-nB)$ of hypothesis testing of the classical probability distributions $P$ and $Q$ with probability of inconclusive discrimination converging to $0$ exponentially fast --- with exponent $K$ for the distribution $P$, and exponent $L$ for $Q$ (Lemma \ref{['lem:exponentially_classical']}).
• Figure S4: The regimes of asymmetric hypothesis testing. Here we show the ranges of exponents of $\overline{\alpha}_n \sim \exp(-nA)$ and $\overline{\beta}_n \sim \exp(-nB)$ achievable in the different settings studied in this work. Region I is the conventional Hoeffding regime, achievable in hypothesis testing with no inconclusive outcomes audenaert_2008. Region II shows the regime of exponents achievable in inconclusive hypothesis testing with low inconclusiveness, where the asymptotic probability of inconclusive outcomes is an arbitrary constant or converges to 0 (Sections \ref{['app:typicality']} and \ref{['app:sequential']}). Regions III are the parameter ranges achievable when the inconclusiveness of one of the states is fixed as an arbitrary constant, while the inconclusiveness of the other is unconstrained --- region III${}_\rho$ fixes low inconclusiveness on $\rho$, while region III${}_\sigma$ on $\sigma$ (Section \ref{['app:special']}). Finally, region IV represents hypothesis testing with high inconclusiveness (Section \ref{['app:conclusive_bigsection']}), where the inconclusive probability converges to 1 and we study how imposing constraints on the speed of this affects the achievable exponent trade-offs. (The states $\rho$, $\sigma$ considered in the plot are classical Bernoulli distributions with parameters $0.9$ and $0.2$, respectively, and the logarithm is to base two.)

Theorems & Definitions (32)

• Proposition 1
• Corollary 2
• Proposition 3
• Proposition 4
• Lemma 5
• Proposition 6
• Corollary 7
• Proposition 8
• Proposition 9
• Proposition S10: Formal statement of Proposition \ref{['prop:conclusive']}