Excluding false negative error in certification of quantum channels

Certification of quantum channels is based on quantum hypothesis testing and involves also preparation of an input state and choosing the final measurement. This work primarily focuses on the scenario when the false negative error cannot occur, even if it leads to the growth of the probability of false positive error. We establish a condition when it is possible to exclude false negative error after a finite number of queries to the quantum channel in parallel, and we provide an upper bound on the number of queries. On top of that, we found a class of channels which allow for excluding false negative error after a finite number of queries in parallel, but cannot be distinguished unambiguously. Moreover, it will be proved that parallel certification scheme is always sufficient, however the number of steps may be decreased by the use of adaptive scheme. Finally, we consider examples of certification of various classes of quantum channels and measurements.

Quantum hypothesis testing in many-body systems

One of the key tasks in physics is to perform measurements in order to determine the state of a system. Often, measurements are aimed at determining the values of physical parameters, but one can also ask simpler questions, such as ``is the system in state A or state B?''. In quantum mechanics, the latter type of measurements can be studied and optimized using the framework of quantum hypothesis testing. In many cases one can explicitly find the optimal measurement in the limit where one has simultaneous access to a large number n of identical copies of the system, and estimate the expected error as n becomes large. Interestingly, error estimates turn out to involve various quantum information theoretic quantities such as relative entropy, thereby giving these quantities operational meaning. In this paper we consider the application of quantum hypothesis testing to quantum many-body systems and quantum field theory. We review some of the necessary background material, and study in some detail the situation where the two states one wants to distinguish are parametrically close. The relevant error estimates involve quantities such as the variance of relative entropy, for which we prove a new inequality. We explore the optimal measurement strategy for spin chains and two-dimensional conformal field theory, focusing on the task of distinguishing reduced density matrices of subsystems. The optimal strategy turns out to be somewhat cumbersome to implement in practice, and we discuss a possible alternative strategy and the corresponding errors.

On Composite Quantum Hypothesis Testing

We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states $$\rho ^{\otimes n}$$ ρ ⊗ n against convex combinations of quantum states $$\sigma ^{\otimes n}$$ σ ⊗ n can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.

Optimal provable robustness of quantum classification via quantum hypothesis testing

Quantum machine learning models have the potential to offer speedups and better predictive accuracy compared to their classical counterparts. However, these quantum algorithms, like their classical counterparts, have been shown to also be vulnerable to input perturbations, in particular for classification problems. These can arise either from noisy implementations or, as a worst-case type of noise, adversarial attacks. In order to develop defense mechanisms and to better understand the reliability of these algorithms, it is crucial to understand their robustness properties in the presence of natural noise sources or adversarial manipulation. From the observation that measurements involved in quantum classification algorithms are naturally probabilistic, we uncover and formalize a fundamental link between binary quantum hypothesis testing and provably robust quantum classification. This link leads to a tight robustness condition that puts constraints on the amount of noise a classifier can tolerate, independent of whether the noise source is natural or adversarial. Based on this result, we develop practical protocols to optimally certify robustness. Finally, since this is a robustness condition against worst-case types of noise, our result naturally extends to scenarios where the noise source is known. Thus, we also provide a framework to study the reliability of quantum classification protocols beyond the adversarial, worst-case noise scenarios.

Experimental quantum reading with photon counting

The final goal of quantum hypothesis testing is to achieve quantum advantage over all possible classical strategies. In the protocol of quantum reading, this is achieved for information retrieval from an optical memory, whose generic cell stores a bit of information in two possible lossy channels. We show, theoretically and experimentally, that quantum advantage is obtained by practical photon-counting measurements combined with a simple maximum-likelihood decision. In particular, we show that this receiver combined with an entangled two-mode squeezed vacuum source is able to outperform any strategy based on statistical mixtures of coherent states for the same mean number of input photons. Our experimental findings demonstrate that quantum entanglement and simple optics are able to enhance the readout of digital data, paving the way to real applications of quantum reading and with potential applications for any other model that is based on the binary discrimination of bosonic loss.