Discriminating quantum states: the multiple Chernoff distance (1508.06624v2)

Abstract: We consider the problem of testing multiple quantum hypotheses ${\rho_1^{\otimes n},\ldots,\rho_r^{\otimes n}}$, where an arbitrary prior distribution is given and each of the $r$ hypotheses is $n$ copies of a quantum state. It is known that the average error probability $P_e$ decays exponentially to zero, that is, $P_e=\exp{-\xi n+o(n)}$. However, this error exponent $\xi$ is generally unknown, except for the case that $r=2$. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szko\l a's conjecture that $\xi=\min_{i\neq j}C(\rho_i,\rho_j)$. The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C(\rho_i,\rho_j):=\max_{0\leq s\leq 1}{-\log\operatorname{Tr}\rho_i^s\rho_j^{1-s}}$ has been previously identified as the optimal error exponent for testing two hypotheses, $\rho_i^{\otimes n}$ versus $\rho_j^{\otimes n}$. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szko\l a's lower bound. Specialized to the case $r=2$, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.

• The paper establishes that the multiple Chernoff distance is the optimal asymptotic error exponent for discriminating among more than two quantum states.
• It introduces a novel upper bound on the average error probability that generalizes the lower bound by Nussbaum and Szkoła with a state-dependent factor.
• The work also provides an alternative proof for the binary quantum Chernoff distance, reinforcing results based on the Holevo-Helstrom test.

Discriminating Quantum States: The Multiple Chernoff Distance

The paper "Discriminating Quantum States: The Multiple Chernoff Distance" by Ke Li addresses the complex problem of hypothesis testing within the domain of quantum mechanics, particularly extending previous results from binary to multiple hypothesis testing scenarios. The challenge at hand lies in determining the asymptotic decay rate of the minimal average error probability $P_e$ when testing among multiple quantum states. While for two hypotheses, the error exponent is given by the quantum Chernoff distance $C(P_1, P_2)$, the scenario becomes intricate when extended to $r > 2$ quantum states.

Key Contributions

1. Resolution of a Long-standing Conjecture: The paper successfully proves the conjecture posed by Nussbaum and Szkoła, establishing that the multiple quantum Chernoff distance $C(P_1, \ldots, P_r) = \min_{i,j} C(P_i, P_j)$ serves as the optimal asymptotic error exponent for discriminating among multiple quantum states. This result generalizes previously known outcomes for binary state discrimination to more complex, multi-hypothesis scenarios.
2. New Upper Bound on Error Probability: A significant achievement in this work is the derivation of a new upper bound for the average error probability in discriminating a set of finite-dimensional quantum states, which coincides with a multiple-state generalization of the lower bound by Nussbaum and Szkoła, up to a state-dependent factor. This leads to the establishment of the achievability of the multiple Chernoff distance as the asymptotic error exponent.
3. Alternative Proof for Binary Case: The paper also offers an alternative proof for the achievability of the binary quantum Chernoff distance, complementing the results derived by Audenaert et al., which were reliant on the Holevo-Helstrom test.

Implications and Future Work

Theoretical Implications: This advancement confirms the consistency of the generalization from classical statistical hypothesis testing to quantum settings. It supports the notion that quantum statistical methods, despite their complexities arising from non-commutative properties of states, can yield generalized bounds that mirror classical theory under quantum conditions.

Practical Implications: The explicit determination of error exponents has crucial implications in quantum computing, specifically in areas such as quantum communication and cryptography where multi-state discrimination is often encountered. The findings can lead to more efficient algorithms for quantum state classification, potentially improving error rates in quantum information processing.

Speculations on Future Research: The successful resolution of the conjecture opens avenues for exploring correlated states and composite hypothesis testing in quantum mechanics, which are often more intricate. Future research may focus on extending these results to systems exhibiting correlations, which are not simply products of independent states, by evaluating the asymptotic behavior of error probabilities for correlated quantum hypotheses.

Additionally, there remains an open question concerning the states-dependent factor $f(r, T)$ that affects the bounds. Whether this factor can be further optimized, or even replaced by a universal constant, presents an interesting challenge for researchers looking to refine the asymptotic analysis presented by the author.

In conclusion, the findings presented in this paper not only provide resolution to a significant theoretical problem but also set the stage for further expansion and application of quantum hypothesis testing in advanced quantum mechanics and information theory domains.