Schedule for: 24w5274 - Towards Infinite Dimension and Beyond in Quantum Information

Noncommutative functional inequalities have recently found many applications in quantum information. In this talk, we will introduce the noncommutative analog of Sobolev inequality for quantum Markov semigroups and discuss its relation to other functional inequalities such as log-Sobolev inequality and Poincaré inequality. In particular, we will talk about the equivalence between Sobolev inequality, Nash inequality, and ultra-contractivity in the noncommutative setting, which answers a question of Kastoryano and Temme. If time permits, we will also discuss the completely bounded case and tensorization property. This is based on a joint work with Marius Junge and Bang Xu.

Quantum state tomography, the task of reconstructing a quantum state from measurement data, stands as the gold standard for verifying quantum devices. While extensively studied for finite-dimensional systems, rigorous performance guarantees remain almost unexplored for continuous variable systems within the trace distance metric, which is considered the most meaningful metric for distinguishing quantum states. In this talk, we will address this gap. First, we discover that learning energy-constrained n-mode states without any additional prior assumption is extremely inefficient: The number of copies needed for achieving a trace distance e-approximation must necessarily scale with e as e^(-2n) - in sharp contrast with the n-qudit case, where the e-scaling is e^(-2). Second, we establish that Gaussian states can be efficiently learned within the trace distance metric. To do that, we prove an upper bound on the trace distance between two Gaussian states, determined by the norm difference of their covariance matrices and first moments. Third, we show how to efficiently learn t-doped bosonic Gaussian states, i.e., states prepared by Gaussian unitaries reflecting non-interacting Hamiltonians and at most t local non-Gaussian evolutions, unveiling more of the structure of these slightly non-Gaussian systems. The end of the talk will be a coda on other recent group work on continuous variable systems. Joint work with Francesco A. Mele, Salvatore F. E. Oliviero, Lennart Bittel, Vittorio Giovannetti, Ludovico Lami, Lorenzo Leone, and Antonio A. Mele

A wide variety of bosonic quantum information processing tasks requires non-Gaussian quantum states, from entanglement distillation to universal quantum computing and quantum error-correction. Characterizing and understanding the properties of these states is therefore of major importance. In this talk, I will present the stellar representation, which is a phase-space formalism based on analytic (Segal-Bargmann) functions allowing to characterise non-Gaussian quantum states in bosonic quantum information processing. The main quantity of interest in this context is the so-called stellar rank, which provides a measure of the non-Gaussian character of quantum states. I will explain its operational properties, how it can be used to assess state preparation and state conversion protocols, how it can be measured and how it quantifies the computational usefulness of non-Gaussian quantum states.

We address the following question: Given a quantum channel and a quantum state which is almost a fixed point of the channel, can we find a new channel and a new state, which are respectively close to the original ones, such that they satisfy an exact fixed point equation? This question can be asked under many interesting constraints in which the original channel and state are assumed to have certain structures which the new channel and state are supposed to satisfy as well. We answer this question in the affirmative under fairly general assumptions on aforementioned structures through a compactness argument. We then concentrate on specific structures of states and chan- nels and establish explicit bounds on the approximation errors between the original- and new states and channels respectively. We find a particularly desirable form of these approximation errors for a variety of interesting examples, including the structure of general quantum states and general quantum channels, unitary channels and unital channels. On the other hand, for the setup of bipartite quantum systems for which the considered channels are demanded to act locally, we are able to lower bound the pos- sible approximation errors. Here, we show that these approximation errors necessarily scale in terms of the dimension of the quantum system in an undesirable manner. We apply our results to the robustness question of quantum Markov chains (QMC) and establish the following: For a tripartite quantum state we show the existence of a dimension- dependent upper bound on the distance to the set of QMCs, which decays as the conditional mutual information of the state vanishes.

We propose new entanglement measures as the detection performance based on the hypothesis testing setting. We clarify how our measures work for detecting an entangled state by extending the quantum Sanov theorem. Our analysis covers the finite-length setting. Exploiting this entanglement measure, we present how to derive an entanglement witness to detect the given entangled state by using the geometrical structure of this measure. We derive their calculation formulas for maximally correlated states, and propose their algorithms that work for general entangled states. In addition, we investigate how our algorithm works for solving the membership problem for separability.

The Asymptotic Equipartition Property (AEP) establishes the Shannon entropy as the relevant quantity for various problems involving inde- pendent and identically distributed (i.i.d.) random variables. In the quantum setting, the von Neumann entropy plays a similar role for i.i.d quantum states. In this talk, we will discuss how this fundamental property holds in quantum systems modelled by von Neumann algebras, thereby establishing this property in systems beyond finite dimensions. We also discuss a generalization of this property, which is known as the Entropy Accumulation Theorem in this general framework.

I will survey recent developments in the analysis of quantum correlations and their deep interconnections with the multi-faceted Connes-Kirchberg Problem, leading also to infinite dimensional phenomena in QIT.

For rank one games the operator space version of Grothendieck’s inequality implies that the non-signaling and quantum value of a bipartite game only differ by a universal multiplicative constant. We show that arbitrary games can very well distinguish between quantum and non-signaling strategies in the sense that no such universal constant can exists. In the language of operator algebras this means that counterexamples for separating non-sginaling and quantum values disprove a version of a tripartite Grothendieck inequality formulated by Pisier. We also show that if the value of the game tested against sub-tracial non-signaling strategies is bound by a certain value if and only if the game can be dominated by the sum of two games Ga and Gb resilient again quantum communication from Alice to Bob, respectively from Bob to Alice. This indicates that at least the sub-nonsignaling strategies have a quantum mechanical interpretation.