Joseph M. Renes

Quantum information theory, particularly its entropic formulations, has made remarkable strides in characterizing quantum systems and tasks. However, a critical dimension remains underexplored: computational efficiency. While classical computational entropies integrate complexity and feasibility into information measures, analogous concepts have yet to be rigorously developed in the quantum setting. In this joint submission, we advance a quantum computational information theory through two complementary works. The first introduces the quantum computational unpredictability entropy, a natural generalization of the min entropy for classical-quantum states and of the classical unpredictability entropy that quantifies the guessing probability of classical randomness using quantum side information and bounded computational power. The second work extends this to the fully quantum setting by defining fully quantum computational min- and max-entropies. The computational min-entropy generalizes unpredictability entropy and retains essential properties, including data processing, a fully quantum leakage chain rule, and it satisfies a novel purification chain rule. The computational max-entropy is defined via a canonical duality relation and it captures a notion of efficient entanglement distillation under bounded quantum circuits. With the introduction of these computational entropies and their analysis, this work marks a critical step toward a quantum information theory that incorporates computational elements.

Abstract Noise in quantum metrology reduces the sensitivity to which one can determine an unknown parameter in the evolution of a quantum state, such as time. Here, we consider a probe system prepared in a pure state that evolves according to a given Hamiltonian. We study the resulting local sensitivity of the probe to time after the application of a given noise channel. We show that the decrease in sensitivity due to the noise is equal to the sensitivity that the environment gains with respect to the energy of the probe. We obtain necessary and sufficient conditions for when the probe does not suffer any sensitivity loss; these conditions are analogous to, but weaker than, the Knill-Laflamme quantum error correction conditions. New upper bounds on the sensitivity of the noisy probe are obtained via our uncertainty relation, by applying known sensitivity lower bounds on the environments system. Our time-energy uncertainty relation also generalizes to any two arbitrary parameters whose evolutions are generated by Hermitian operators. This uncertainty relation asserts a general trade-off between the sensitivities that two parties can achieve for any two respective parameters of a single quantum system, in terms of the commutator of the associated generators. We consider applications to strongly interacting many-body probes. We find probe states for general interaction graphs of Ising and Heisenberg interactions that are robust to any single located error. For a 1D spin chain with nearest-neighbor interactions subject to amplitude damping noise on each site, we verify numerically that our probe state does not lose any sensitivity to first order in the noise parameter.

Due to Csiszar and Koerner, the capacity of classical wiretap channels has a single-letter characterization in terms of the private information. For quantum wiretap channels, however, it is known that regularization of the private information is necessary to reach the capacity. Here, we study hybrid classical-quantum wiretap channels in order to resolve to what extent quantum effects are needed to witness non-additivity phenomena in quantum Shannon theory. For wiretap channels with quantum inputs but classical outputs, we prove that the characterization of the capacity in terms of the private information stays single-letter. Hence, entangled input states are of no asymptotic advantage in this setting. For wiretap channels with classical inputs, we show by means of explicit examples that the private information already becomes non-additive when either one of the two receivers becomes quantum (with the other receiver staying classical). This gives non-additivity examples that are not caused by entanglement and illustrates that quantum adversaries are strictly different from classical adversaries in the wiretap model.

Recently, Anshu et al. introduced “partially” smoothed information measures and used them to derive tighter bounds for several information-processing tasks, including quantum state merging and privacy amplification against quantum adversaries [arXiv:1807.05630 [quant-ph]]. Yet, a tight second-order asymptotic expansion of the partially smoothed conditional min-entropy in the i.i.d. setting remains an open question. Here we establish the second-order term in the expansion for pure states, and find that it differs from that of the original “globally” smoothed conditional min-entropy. Remarkably, this reveals that the second-order term is not uniform across states, since for other classes of states the second-order terms for partially and globally smoothed quantities coincide. In view of the tight bounds on the entanglement cost of state merging in terms of the partially-smoothed conditional min-entropy by Anshu et al., this indicates that the second-order asymptotic rate for that task will not separate so neatly into a term depending on the state (the variance) and a term depending on the error (the quantile). Finally, by relating the task of quantum compression to that of quantum state merging, our derived ex- pansion allows us to determine the second-order asymptotic expansion of the optimal rate of quantum data compression. This closes a gap in the bounds determined by Datta and Leditzky [IEEE Trans. Inf. Theory 61, 582 (2015)], and shows that the straightforward compression protocol of cutting off the eigenspace of least weight is indeed asymptotically optimal at second order.