Abstract:
This thesis summarises and, in some instances, extends results arising from six projects situated at the intersection of quantum information theory and many-body systems. The first two projects concern the derivation of continuity bounds for entropy functionals.
More specifically, the first project establishes a tight upper bound for the difference in von Neumann entropy between two quantum states ρ and ρ′. This bound is expressed in terms of the entropy difference of the Jordan-Hahn decomposition of (ρ − ρ′), corrected by a binary entropy term involving their trace distance. The resulting inequality not only recovers but strengthens the well-known Audenaert-Fannes inequality. As further consequences, the project yields continuity bounds for the conditional entropy when the marginal on the conditioning system agrees, as well as for the Umegaki relative entropy.
The second project concerns continuity bounds for functionals of the form
ρ ↦ inf{D̃(ρ ‖ σ) : σ ∈ C},
where D̃ denotes the sandwiched Rényi divergence and C a convex, compact set of quantum states containing at least one positive-definite element. Such quantities naturally arise in quantum resource theories as resource monotones. Building upon earlier work by Marwah and Dupuis and, independently, Beigi and Goodarzi on the sandwiched Rényi conditional entropy, we improve and generalise their strategies to this broader context. One approach uses the super-/subadditivity and convexity/concavity of the exponential of the divergence; the other rewrites the divergence as a norm in an interpolation space. However, the existing scope of interpolation theory is insufficient for establishing the necessary norm properties. Thus, instead of relying on abstract theory, we base our analysis on Lieb’s concavity theorem and Sion’s minimax principle. As a consequence of these continuity bounds, we also obtain a general continuity estimate for the sandwiched Rényi divergence itself.
The third project establishes a super-exponential decay of the Belavkin-Staszewski conditional mutual information for Gibbs states of one-dimensional, local, translation-invariant interactions at arbitrary positive temperature. Using that this quantity forms an upper bound to the recovery error of a reconstruction map, we construct a matrix-product-operator (MPO) approximation of marginals of such Gibbs states. This approximation achieves a bond dimension sub-polynomial in N/ε, where N is the size of the marginal and ε the trace-distance reconstruction error. By combining local tomography with this MPO reconstruction, the project culminates in an efficient algorithm for recovering such marginals from measurement data, with runtime and sample complexity polynomial in N/ε.
The fourth project investigates the mixing time of the Davies semigroup for local, commuting interactions at positive temperature. We show that combining a uniform lower bound on local gaps with the decay of what we term the marginal conditional mutual information in the corresponding Gibbs state yields an exponentially improved mixing-time estimate compared to relying on the uniform bound on local gaps alone. Furthermore, this improved mixing time leads to an exponentially enhanced global spectral gap and establishes a large-time entropy-contraction coefficient for the semigroup.
The fifth project develops the framework of quantum Sobolev spaces and Sobolev-preserving semigroups on a single-mode bosonic system. Specifically, we show that if an unbounded operator (i) takes GKLS form on the finite-rank operators in the Fock basis, with constituents polynomial in creation and annihilation operators, and (ii) increases the moments of the number operator in a controlled manner on this domain, then it is the core of a generator of a quantum Markov semigroup that additionally preserves quantum Sobolev spaces. Depending on the strength of condition (ii), we further establish time-independent Sobolev-norm bounds and even regularisation properties, whereby initially irregular states acquire finite Sobolev norms after any positive evolution time. Applying these results to concrete examples --- including single-mode Gaussian semigroups and generators modelling cat-code gates --- we demonstrate the utility of the framework in generation and perturbation theory.
The sixth and final project proves a generalised quantum Stein’s lemma for subalgebra resources. This means the alternative hypothesis in the nth test is given by the fixed points of an n-fold tensor product of the Hilbert-Schmidt adjoint of a conditional expectation onto a von Neumann subalgebra. Building on the work of Gao and Rahaman, who proved the lemma for Hilbert-Schmidt-symmetric conditional expectations, we generalise their approach and prove the result for arbitrary conditional expectations whose Hilbert-Schmidt dual has a positive-definite fixed point.
