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We show that the communication cost of quantum broadcast channel simulation under free entanglement assistance between the sender and the receivers is asymptotically characterized by an efficiently computable single-letter formula in terms of the channel's multipartite mutual information. Our core contribution is a new one-shot achievability result for multipartite quantum state splitting via multipartite convex splitting. As part of this, we face a general instance of the quantum joint typicality problem with arbitrarily overlapping marginals. The crucial technical ingredient to sidestep this difficulty is a conceptually novel multipartite mean-zero decomposition lemma, together with employing recently introduced complex interpolation techniques for sandwiched R\'enyi divergences. Moreover, we establish an exponential convergence of the simulation error when the communication costs are within the interior of the capacity region. As the costs approach the boundary of the capacity region moderately quickly, we show that the error still vanishes asymptotically., Comment: The idea of the mean-zero decomposition lemma is independently and concurrently discovered for multipartite decoupling by Pau Colomer Saus and Andreas Winter (arXiv:2304.12114). v2: References updated

We present an end-to-end and practical randomness amplification and privatisation protocol based on Bell tests. This allows the building of device-independent random number generators which output (near-)perfectly unbiased and private numbers, even if using an uncharacterised quantum device potentially built by an adversary. Our generation rates are linear in the repetition rate of the quantum device and the classical randomness post-processing has quasi-linear complexity - making it efficient on a standard personal laptop. The statistical analysis is also tailored for real-world quantum devices. Our protocol is then showcased on several different quantum computers. Although not purposely built for the task, we show that quantum computers can run faithful Bell tests by adding minimal assumptions. In this semi-device-independent manner, our protocol generates (near-)perfectly unbiased and private random numbers on today's quantum computers., Comment: As published in the journal Quantum, 33+23 pages (15 figures and 2 tables)

We study quantum interior point methods (QIPMs) for second-order cone programming (SOCP), guided by the example use case of portfolio optimization (PO). We provide a complete quantum circuit-level description of the algorithm from problem input to problem output, making several improvements to the implementation of the QIPM. We report the number of logical qubits and the quantity/depth of non-Clifford T-gates needed to run the algorithm, including constant factors. The resource counts we find depend on instance-specific parameters, such as the condition number of certain linear systems within the problem. To determine the size of these parameters, we perform numerical simulations of small PO instances, which lead to concrete resource estimates for the PO use case. Our numerical results do not probe large enough instance sizes to make conclusive statements about the asymptotic scaling of the algorithm. However, already at small instance sizes, our analysis suggests that, due primarily to large constant pre-factors, poorly conditioned linear systems, and a fundamental reliance on costly quantum state tomography, fundamental improvements to the QIPM are required for it to lead to practical quantum advantage., Comment: 38 pages, 15 figures

We study channel simulation under common randomness-assistance in the finite-blocklength regime and identify the smooth channel max-information as a linear program one-shot converse on the minimal simulation cost for fixed error tolerance. We show that this one-shot converse can be achieved exactly using no-signaling assisted codes, and approximately achieved using common randomness-assisted codes. Our one-shot converse thus takes on an analogous role to the celebrated meta-converse in the complementary problem of channel coding, and find tight relations between these two bounds. We asymptotically expand our bounds on the simulation cost for discrete memoryless channels, leading to the second-order as well as the moderate deviation rate expansion, which can be expressed in terms of the channel capacity and channel dispersion known from noisy channel coding. Our techniques extend to discrete memoryless broadcast channels. In stark contrast to the elusive broadcast channel capacity problem, we show that the reverse problem of broadcast channel simulation under common randomness-assistance allows for an efficiently computable single-letter characterization of the asymptotic rate region in terms of the broadcast channel's multi-partite mutual information. Finally, we present a Blahut-Arimoto type algorithm to compute the rate region efficiently., Comment: 32 pages, 10 figures

Topological invariants of a dataset, such as the number of holes that survive from one length scale to another (persistent Betti numbers) can be used to analyse and classify data in machine learning applications. We present an improved quantum algorithm for computing persistent Betti numbers, and provide an end-to-end complexity analysis. Our approach provides large polynomial time improvements, and an exponential space saving, over existing quantum algorithms. Subject to gap dependencies, our algorithm obtains an almost quintic speedup in the number of datapoints over rigorous state-of-the-art classical algorithms for calculating the persistent Betti numbers to constant additive error - the salient task for applications. However, this may be reduced to closer to quadratic when compared against heuristic classical methods and observed scalings. We discuss whether quantum algorithms can achieve an exponential speedup for tasks of practical interest, as claimed previously. We conclude that there is currently no evidence that this is the case., Comment: 10 + 28 pages

Brascamp-Lieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theory -- including entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of `geometric' Brascamp-Lieb inequalities., Comment: v3: 24 pages, minor changes, to appear in Commun. Math. Phys

We show that Araki and Masuda's weighted non-commutative vector valued $L_p$-spaces [Araki \& Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)] correspond to an algebraic generalization of the sandwiched R��nyi divergences with parameter $��= \frac{p}{2}$. Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data-processing inequality and monotonicity in $��$. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases $��\to \{\frac{1}{2},1,\infty\}$ leading to minus the logarithm of Uhlmann's fidelity, Umegaki's relative entropy, and the max-relative entropy, respectively. As a contribution that might be of independent interest, we derive a Riesz-Thorin theorem for Araki-Masuda $L_p$-spaces and an Araki-Lieb-Thirring inequality for states on von Neumann algebras., v2: 20 pages, published version

Quantum information measures such as the entropy and the mutual information find applications in physics, e.g., as correlation measures. Generalizing such measures based on the R��nyi entropies is expected to enhance their scope in applications. We prescribe R��nyi generalizations for any quantum information measure which consists of a linear combination of von Neumann entropies with coefficients chosen from the set {-1,0,1}. As examples, we describe R��nyi generalizations of the conditional quantum mutual information, some quantum multipartite information measures, and the topological entanglement entropy. Among these, we discuss the various properties of the R��nyi conditional quantum mutual information and sketch some potential applications. We conjecture that the proposed R��nyi conditional quantum mutual informations are monotone increasing in the R��nyi parameter, and we have proofs of this conjecture for some special cases., 9 pages, related to and extends the results from arXiv:1403.6102

In [Berta et al., J. Math. Phys. 56, 022205 (2015)], we recently proposed Renyi generalizations of the conditional quantum mutual information of a tripartite state on $ABC$ (with $C$ being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system $B$ (with it being left open to show that the Renyi quantity is monotone with respect to local operations on system $A$). Here we define a Renyi squashed entanglement and a Renyi quantum discord based on a Renyi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Renyi conditional quantum mutual information is monotone with respect to local operations on both systems $A$ and $B$, we prove that the Renyi squashed entanglement and the Renyi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work [Berta et al., Phys. Rev. A 91, 022333 (2015)], we also detailed a procedure to obtain Renyi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set $\{-1,0,1\}$. Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Renyi generalizations in the Renyi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound., v3: 41 pages, 2 tables, final version

We consider an unknown quantum state shared between two parties, Alice and Bob, and ask how much quantum communication is needed to transfer the full state to Bob. This problem is known as state merging and was introduced in [Horodecki et al., Nature, 436, 673 (2005)]. It has been shown that for free classical communication the minimal number of quantum bits that need to be sent from Alice to Bob is given by the conditional von Neumann entropy. However this result only holds asymptotically (in the sense that Alice and Bob share initially many identical copies of the state) and it was unclear how much quantum communication is necessary to merge a single copy. We show that the minimal amount of quantum communication needed to achieve this single-shot state merging is given by minus the smooth conditional min-entropy of Alice conditioned on the environment. This gives an operational meaning to the smooth conditional min-entropy., Comment: 33 pages, Diploma thesis, ETH Zurich, February 2008

Determining which quantum tasks we can perform with currently available tools and devices is one of the most important goals of quantum information science today. To achieve this requires careful investigation of the capability of current quantum tools as well as development of classical protocols which can assist quantum tasks and amplify their abilities. In this thesis, we approach this problem through two different topics in quantum information theory: Gaussian resource theories and semidefinite programming hierarchies. In the first part of this thesis, we examine the possibility of implementing quantum information processing tasks in the Gaussian platform through the eyes of quantum resource theories. Gaussian states and operations are primary tools for the study of continuous-variable quantum information processing due to their easy accessibility and concise mathematical descriptions, although it has been discovered that they are subject to a number of limitations for advanced quantum information processing tasks. We explore the capability of the Gaussian platform further in the first part of this thesis. Firstly, we investigate whether introducing convex structure to the Gaussian framework can circumvent the known no-go theorem of Gaussian resource distillation. Surprisingly, we find that resource distillation becomes possible — albeit in a limited fashion — when convexity is introduced. Then, we consider the quantum resource theory of Gaussian thermal operations when catalysts are allowed, and examine the abilities of catalytic Gaussian thermal operations by characterising all possible state transformations under them. In the second part of this thesis, we address the problem of characterising quantum cor- relations via semidefinite programming hierarchies. In particular, we focus on characterising quantum correlations of fixed dimension, which is practically relevant to the field of semi- device-independent quantum information processing. Semidefinite programming is a special type of mathematical optimisation, and it is known that some important but difficult problems in quantum information theory admit semidefinite programming relaxations; these include the characterisation of general quantum correlations in the context of non-locality and the distinction of quantum separable states from entangled states. In this second part, we show how to construct a hierarchy of semidefinite programming relaxations for quantum correlations of fixed dimension and derive analytical bounds on the convergence speed of the hierarchy. For the proof, we make a connection to a variant of quantum separability problem and employ multipartite quantum de Finetti theorems with linear constraints. Open Access

The main topic of this thesis is the study of de Finetti methods and their applications in quantum information theory. Those methods include de Finetti representation theorems and De Finetti reductions. The primary motivation of a de Finetti representation theorem is to represent, or approximate, a mathematical object symmetric under permutation of its components, into a probabilistic ensemble of elementary independent and identically distributed (i.i.d.) constituents. De Finetti reductions are another class of techniques that are used to take advantage of permutation symmetries. For example, a quantum de Finetti reduction provides an upper bound to a symmetric quantum state in the form of an integral superposition of product states, weighted by a factor that is polynomial in terms of the number of copies and exponential in terms of the local dimensionality. Our findings include: 1- The development of general mathematical techniques that can be used to obtain concrete constrained de Finetti representation theorems for the desired application. 2- The application of those methods to the problem of approximate quantum error correction. In particular, we use our framework to develop asymptotically converging SDP hierarchies that can be used to study the average and worst error cases, as given by the quantum channel fidelity and a channel distance based on the diamond norm, respectively. 3- A new de Finetti reduction in the presence of an additional system carrying side information, that can handle various types of linear constraints. 4- The development of entropic techniques that can be used to generate de Finetti representation theorems from a starting de Finetti reduction. In particular, we use those methods to obtain a new proof for finite quantum de Finetti theorems. Open Access