We study the optimal rates of emulation (also called interconversion) between quantum channels. When the source and the target channels are idempotent, we give a single-letter expression for the zero-error emulation capacity in terms of structural properties of the range of the two channels. This expression shows that channel emulation is not reversible for general idempotent channels. Furthermore, we establish a strong converse rate that matches with the zero-error emulation capacity when the source or the target channel is either an identity or a completely dephasing channel.

We consider a quantum system described by a quantum channel $Φ$ that is applied at every time step and study the time evolution of its information capacities. When $Φ$ is a GNS-symmetric channel (this includes Pauli channels, for example), we give explicit exponential convergence bounds for the classical and quantum capacities. These bounds are in terms of entropic properties of $Φ$. We further illustrate how these results help quantify the performance of active versus passive error-correction setups.

Uhlmann’s theorem is a cornerstone of quantum information theory, stating that for any quantum state ρ<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AB</i> and any state σ<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i>, there exists an extension σ<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AB</i> of σ<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> such that the fidelity between ρ<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AB</i> and σ<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AB</i> equals the fidelity between their marginals ρ<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> and σ<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i>. This property underpins many results and applications in quantum information science. In this work, we generalize Uhlmann’s theorem to a broad class of measured <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i>-divergences, including the measured α-Rényi divergences for all α ≥ 0. The well-known Uhlmann’s theorem for the fidelity corresponds to the special case α = 1/2. Since most commonly used quantum Rényi divergences, including the Petz and sandwiched Rényi divergences, cannot satisfy this property (except for degenerate cases), this fundamentally distinguishes measured <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i>-divergences from other quantum divergences and highlights their unique mathematical structure.

International audience

International audience

Understanding the noise affecting a quantum device is of fundamental importance for scaling quantum technologies. A particularly important class of noise models is that of Pauli channels, as randomized compiling techniques can effectively bring any quantum channel to this form and are significantly more structured than general quantum channels. In this paper, we show fundamental lower bounds on the sample complexity for learning Pauli channels in diamond norm. We consider strategies that may not use auxiliary systems entangled with the input to the unknown channel and have to perform a measurement before reusing the channel. For non-adaptive algorithms, we show a lower bound of Ω(23nε−2) to learn an n-qubit Pauli channel. In particular, this shows that the recently introduced learning procedure by [1] is essentially optimal. In the adaptive setting, we show a lower bound of Ω(22.5nε−2) for ε = O(2−n), and a lower bound of Ω(22nε−2) for any ε > 0. This last lower bound holds even in a stronger model where in each step, before performing the measurement, the unknown channel may be used arbitrarily many times sequentially interspersed with unital operations.

We study the effect of shared non-signaling correlations for the problem of simulating a channel using noiseless communication in the one-shot setting. For classical channels, we show how to round any non-signaling-assisted simulation strategy - which exactly corresponds to the meta-converse for channel simulation - to a strategy that only uses shared randomness. For quantum channels, we round any non-signaling-assisted simulation strategy to a strategy that only uses shared entanglement. As our main result, we prove a guarantee on the ratio of success probabilities of at least$(1-\frac{-1}{\mathbf{e}})$, for both the classical and the quantum setting. We further - show this ratio to be optimal. It can be improved to$(1-\frac{1}{t})$using$o$(ln$(t)$) additional bits (qubits) of communication.

23 pages

We address the problem of coding for classical broadcast channels, which entails maximizing the success probability that can be achieved by sending a fixed number of messages over a broadcast channel. For point-to-point channels, Barman and Fawzi found a$(1-e^{-1})$-approximation algorithm running in polynomial time, and showed that it is NP-hard to achieve a strictly better approximation ratio. Furthermore, these algorithmic results were at the core of the limitations they established on the power of non-signaling assistance for point-to-point channels. It is natural to ask if similar results hold for broadcast channels, exploiting links between approximation algorithms of the channel coding problem and the non-signaling assisted capacity region. In this work, we make several contributions on algorithmic aspects and non-signaling assisted capacity regions of broadcast channels. For the class of deterministic broadcast channels, we describe a$(1-e^{-1})^{2}$-approximation algorithm running in polynomial time, and we show that the capacity region for that class is the same with or without non-signaling assistance. Finally, we show that in the value query model, we cannot achieve a better approximation ratio than$\Omega \left ({{\frac {1}{\sqrt {m}}}}\right)$in polynomial time for the general broadcast channel coding problem, with m the size of one of the outputs of the channel.

In the problem of quantum channel certification, we have black box access to a quantum process and would like to decide if this process matches some predefined specification or is $\eps$-far from this specification. The objective is to achieve this task while minimizing the number of times the black box is used. Note that the state certification problem is a special case where the black box has no input. Here, we focus on two relevant extreme cases. The first one is when the predefined specification is a unitary channel, e.g., a gate in a quantum circuit. In this case, we show that testing whether the black box is described by a fixed unitary or $\eps$-far from it in the trace norm requires $\Theta(d/\eps^2)$ uses of the black box. The second setting we consider is when the predefined specification is a completely depolarizing channels with input dimension $\din$ and output dimension $\dout$. In this case, we prove that, in the non-adaptive setting, $\Tilde{\Theta}(\din^2\dout^{1.5}/\eps^2)$ uses of the channel are necessary and sufficient to verify whether it is equal to the depolarizing channel or $\eps$-far from it in the diamond norm. Finally, we prove a lower bound of $\Omega(\din^2\dout/\eps^2)$ for this problem in the adaptive setting. Note that the special case $\din = 1$ corresponds to the well-studied quantum identity testing problem.

We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature $T = 0$) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature $T > 0$ in subpolynomial time, i.e., in time $O((\frac{1}{\varepsilon})^{c})$ for any constant $c > 0$ where $\varepsilon$ is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in $\frac{1}{\varepsilon}$. Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.

The min-entropy of a quantum system A conditioned on another quantum system E describes how much randomness can be extracted from A with respect to an adversary in possession of E. This quantity plays a crucial role in quantum cryptography: the security proofs of many quantum cryptographic protocols reduce to showing a lower bound on such a min-entropy. Here, we develop a new tool, called generalised entropy accumulation, for computing such bounds. Concretely, we consider a sequential process in which each step outputs a system Aiand updates a side information register E. We prove that if this process satisfies a natural “non-signalling” condition between past outputs and future side information, the min-entropy of the outputs $A_{1},\ldots,\ A_{n}$ conditioned on the side information E at the end of the process can be bounded from below by a sum of von Neumann entropies associated with the individual steps. This is a generalisation of the entropy accumulation theorem (EAT) [1], which deals with a more restrictive model of side information: there, past side information cannot be updated in subsequent rounds, and newly generated side information has to satisfy a Markov condition.Due to its more general model of side-information, our generalised EAT can be applied more easily and to a broader range of cryptographic protocols. In particular, it is the first general tool that is applicable to mistrustful device-independent cryptography. To demonstrate this, we give the first security proof for blind randomness expansion [2] against general adversaries. Furthermore, our generalised EAT can be used to give improved security proofs for quantum key distribution [3], and also has applications beyond quantum cryptography.

There is a large and important collection of Ramsey-type combinatorial problems, closely related to central problems in complexity theory, that can be formulated in terms of the asymptotic growth of the size of the maximum independent sets in powers of a fixed small hypergraph, also called the Shannon capacity. An important instance of this is the corner problem studied in the context of multiparty communication complexity in the Number On the Forehead (NOF) model. Versions of this problem and the NOF connection have seen much interest (and progress) in recent works of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC 2021). We introduce and study a general algebraic method for lower bounding the Shannon capacity of directed hypergraphs via combinatorial degenerations, a combinatorial kind of "approximation" of subgraphs that originates from the study of matrix multiplication in algebraic complexity theory (and which play an important role there) but which we use in a novel way. Using the combinatorial degeneration method, we make progress on the corner problem by explicitly constructing a corner-free subset in F₂ⁿ × F₂ⁿ of size Ω(3.39ⁿ/poly(n)), which improves the previous lower bound Ω(2.82ⁿ) of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us closer to the best upper bound 4^{n - o(n)}. Our new construction of corner-free sets implies an improved NOF protocol for the Eval problem. In the Eval problem over a group G, three players need to determine whether their inputs x₁, x₂, x₃ ∈ G sum to zero. We find that the NOF communication complexity of the Eval problem over F₂ⁿ is at most 0.24n + 𝒪(log n), which improves the previous upper bound 0.5n + 𝒪(log n).

The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation stating that arbitrarily long quantum computations can be performed with a polylogarithmic overhead provided the noise level is below a constant level. A recent work by Fawzi, Grospellier and Leverrier (FOCS 2018) building on a result by Gottesman (QIC 2013) has shown that the space overhead can be asymptotically reduced to a constant independent of the circuit provided we only consider circuits with a length bounded by a polynomial in the width. In this work, using a minimal model for quantum fault tolerance, we establish a general lower bound on the space overhead required to achieve fault tolerance. For any non-unitary qubit channel $\mathcal{N}$ and any quantum fault tolerance schemes against $\mathrm{i.i.d.}$ noise modeled by $\mathcal{N}$, we prove a lower bound of $\max\left\{\mathrm{Q}(\mathcal{N})^{-1}n,α_\mathcal{N} \log T\right\}$ on the number of physical qubits, for circuits of length $T$ and width $n$. Here, $\mathrm{Q}(\mathcal{N})$ denotes the quantum capacity of $\mathcal{N}$ and $α_\mathcal{N}>0$ is a constant only depending on the channel $\mathcal{N}$. In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and we allow classical computation to be free and perfect. This improves upon results that assumed classical computations to be also affected by noise, and that sometimes did not allow for fresh qubits to be added. Along the way, we prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude damping noise resolving a conjecture by Ben-Or, Gottesman, and Hassidim (2013).

We address the problem of coding for multiple-access channels (MACs) with the assistance of non-signaling correlations between parties. It is well-known that non-signaling assistance does not change the capacity of point-to-point channels. However, it was recently observed that one can construct MACs from two-player non-local games while relating the winning probability of the game to the capacity of the MAC. By considering games for which entanglement (a special kind of non-signaling correlation) increases the winning probability (e.g., the Magic Square game), this shows that for some specific kinds of channels, entanglement between the senders can increase the capacity.Here, we show that the increase in capacity from non-signaling assistance goes beyond such special channels and applies even to a simple deterministic MAC: the binary adder channel. In particular, we show that, with non-signaling assistance, a sum-rate of $\frac{{{{\log }_2}(72)}}{4} \simeq 1.5425$ can be reached with zero error, which beats the maximum classical sum-rate capacity of $\frac{3}{2}$.In order to achieve this, we show that efficient linear programs can be formulated to compute the success probability of the best non-signaling assisted code for a finite number of copies of a multiple-access channel. In particular, this can be used to give lower bounds on the zero-error non-signaling assisted capacity of multiple-access channels.

Optimal rates for achieving an information processing task are often characterized in terms of regularized information measures. In many cases of quantum tasks, we do not know how to compute such quantities. Here, we exploit the symmetries in the recently introduced$\mathrm {D}^{\#}$in order to obtain a hierarchy of semidefinite programming bounds on various regularized quantities. As applications, we give a general procedure to give efficient bounds on the regularized Umegaki channel divergence as well as the classical capacity and two-way assisted quantum capacity of quantum channels. In particular, we obtain slight improvements for the capacity of the amplitude damping channel. We also prove that for fixed input and output dimensions, the regularized sandwiched Rényi divergence between any two quantum channels can be approximated up to an$\epsilon $accuracy in time that is polynomial in$1/\epsilon $.

In a recent landmark result [Ji et al., arXiv:2001.04383 (2020)], it was shown that approximating the value of a two-player game is undecidable when the players are allowed to share quantum states of unbounded dimension. In this paper, we study the computational complexity of two-player games when the dimension of the quantum systems is bounded by T. More specifically, we give a semidefinite program of size exp(𝒪(T^{12}(log²(AT)+log(Q)log(AT))/ε²)) to compute additive ε-approximations on the value of two-player free games with T× T-dimensional quantum entanglement, where A and Q denote the number of answers and questions of the game, respectively. For fixed dimension T, this scales polynomially in Q and quasi-polynomially in A, thereby improving on previously known approximation algorithms for which worst-case run-time guarantees are at best exponential in Q and A. For the proof, we make a connection to the quantum separability problem and employ improved multipartite quantum de Finetti theorems with linear constraints that we derive via quantum entropy inequalities.

Optimal rates for achieving an information processing task are often characterized in terms of regularized information measures. In many cases of quantum tasks, we do not know how to compute such quantities. Here, we exploit the symmetries in the recently introduced divergence$\mathrm{D}^{\#}$in order to obtain a hierarchy of semidefinite programming bounds on various regularized quantities. As an application, we describe a general procedure to give efficient bounds on the regularized Umegaki channel divergence and on the classical capacity of quantum channels, and in particular we obtain slight improvements for the capacity of the amplitude damping channel.

What advantage do sequential procedures provide over batch algorithms for testing properties of unknown distributions? Focusing on the problem of testing whether two distributions $\mathcal{D}_1$ and $\mathcal{D}_2$ on $\{1,\dots, n\}$ are equal or $\epsilon$-far, we give several answers to this question. We show that for a small alphabet size $n$, there is a sequential algorithm that outperforms any batch algorithm by a factor of at least $4$ in terms sample complexity. For a general alphabet size $n$, we give a sequential algorithm that uses no more samples than its batch counterpart, and possibly fewer if the actual distance between $\mathcal{D}_1$ and $\mathcal{D}_2$ is larger than $\epsilon$. As a corollary, letting $\epsilon$ go to $0$, we obtain a sequential algorithm for testing closeness (with no a priori bound on the distance between $\mathcal{D}_1$ and $\mathcal{D}_2$) with a sample complexity $\tilde{\mathcal{O}}(\frac{n^{2/3}}{TV(\mathcal{D}_1, \mathcal{D}_2)^{4/3}})$: this improves over the $\tilde{\mathcal{O}}(\frac{n/\log n}{TV(\mathcal{D}_1, \mathcal{D}_2)^{2} })$ tester of [Daskalakis and Kawase 2017] and is optimal up to multiplicative constants. We also establish limitations of sequential algorithms for the problem of testing closeness: they can improve the worst case number of samples by at most a constant factor.

In the maximum coverage problem, we are given subsets $T_1, \ldots, T_m$ of a universe $[n]$ along with an integer $k$ and the objective is to find a subset $S \subseteq [m]$ of size $k$ that maximizes $C(S) := \Big|\bigcup_{i \in S} T_i\Big|$. It is a classic result that the greedy algorithm for this problem achieves an optimal approximation ratio of $1-e^{-1}$. In this work we consider a generalization of this problem wherein an element $a$ can contribute by an amount that depends on the number of times it is covered. Given a concave, nondecreasing function $φ$, we define $C^φ(S) := \sum_{a \in [n]}w_aφ(|S|_a)$, where $|S|_a = |\{i \in S : a \in T_i\}|$. The standard maximum coverage problem corresponds to taking $φ(j) = \min\{j,1\}$. For any such $φ$, we provide an efficient algorithm that achieves an approximation ratio equal to the Poisson concavity ratio of $φ$, defined by $α_φ := \min_{x \in \mathbb{N}^*} \frac{\mathbb{E}[φ(\text{Poi}(x))]}{φ(\mathbb{E}[\text{Poi}(x)])}$. Complementing this approximation guarantee, we establish a matching NP-hardness result when $φ$ grows in a sublinear way. As special cases, we improve the result of [Barman et al., IPCO, 2020] about maximum multi-coverage, that was based on the unique games conjecture, and we recover the result of [Dudycz et al., IJCAI, 2020] on multi-winner approval-based voting for geometrically dominant rules. Our result goes beyond these special cases and we illustrate it with applications to distributed resource allocation problems, welfare maximization problems and approval-based voting for general rules.

Lyapunov exponents describe the asymptotic behavior of the singular values of large products of random matrices. A direct computation of these exponents is however often infeasible. By establishing a link between Lyapunov exponents and an information theoretic tool called entropy accumulation theorem we derive an upper and a lower bound for the maximal and minimal Lyapunov exponent, respectively. The bounds assume independence of the random matrices, are analytical, and are tight in the commutative case as well as in other scenarios. They can be expressed in terms of an optimization problem that only involves single matrices rather than large products. The upper bound for the maximal Lyapunov exponent can be evaluated efficiently via the theory of convex optimization.

We introduce a decoder for quantum CSS codes that is based on linear programming. Our definition is a priori slightly different from the one proposed by Li and Vontobel as we have a syndrome oriented approach instead of an error oriented one, but we show that the success condition is equivalent. Although we prove that this decoder fails for quantum codes that do not have good soundness property (i.e., having large errors with syndrome of small weight) such as the toric code, we obtain good results from simulations. We run our decoder for hypergraph products of two random LDPC codes, showing that it performs better than belief propagation, even combined with the small-set-flip decoder that can provably correct a constant fraction of random errors.

We derive converging hierarchies of efficiently computable semidefinite programming outer bounds on the optimal fidelity for the transmission of quantum information over noisy quantum channels. Based on positive partial transpose conditions we give a sufficient criterion for the exact convergence at any given level of the hierarchies. The worst case convergence speed of our hierarchies is quantified via positive semidefinite representable outer approximations on the set of separable Choi states, which are based on novel finite de Finetti theorems for quantum channels.

We study the problem of finding the optimal code of size k for a given classical-quantum channel, with input space X and output space of dimension d, from an algorithmic point of view. We show that unlike the classical case, the relevant function is not submodular, and does not have the diminishing returns property. We also study a semidefinite programming relaxation, which corresponds to the setting where the sender and receiver have a non-signalling box, and show that it can be rounded in two different ways. If the SDP gives a value of p for the success probability, then a simple rounding strategy returns a code of size k with success probability at least p/O(log IXI log d). The second rounding method is p based on the pretty good measurement and returns a code with success probability close to p, but the code is smaller: it has size k · Ω(p2).

Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of semi-algebraic proof systems that manipulate polynomial inequalities via elementary inference rules that infer new inequalities from the premises. These proof systems are known to be very powerful, but searching for proofs remains a major difficulty. In this work, we introduce a machine learning based method to search for a dynamic proof within these proof systems. We propose a deep reinforcement learning framework that learns an embedding of the polynomials and guides the choice of inference rules, taking the inherent symmetries of the problem as an inductive bias. We compare our approach with powerful and widely-studied linear programming hierarchies based on static proof systems, and show that our method reduces the size of the linear program by several orders of magnitude while also improving performance. These results hence pave the way towards augmenting powerful and well-studied semi-algebraic proof systems with machine learning guiding strategies for enhancing the expressivity of such proof systems.

The entropy accumulation theorem states that the smooth min-entropy of an n-partite system A = (A1, ..., An) is lower-bounded by the sum of the von Neumann entropies of suitably chosen conditional states up to corrections that are sublinear in n. This theorem is particularly suited to proving the security of quantum cryptographic protocols, and in particular so-called device-independent protocols for randomness expansion and key distribution, where the devices can be built and preprogrammed by a malicious supplier. However, while the bounds provided by this theorem are optimal in the first order, the second-order term is bounded more crudely, in such a way that the bounds deteriorate significantly when the theorem is applied directly to protocols where parameter estimation is done by sampling a small fraction of the positions, as is done in most QKD protocols. The objective of this paper is to improve this second-order sublinear term and remedy this problem. On the way, we prove various bounds on the divergence variance, which might be of independent interest.

We study the robustness of classifiers to various kinds of random noise models. In particular, we consider noise drawn uniformly from the $\ell_p$ ball for $p ∈[1, ∞]$ and Gaussian noise with an arbitrary covariance matrix. We characterize this robustness to random noise in terms of the distance to the decision boundary of the classifier. This analysis applies to linear classifiers as well as classifiers with locally approximately flat decision boundaries, a condition which is satisfied by state-of-the-art deep neural networks. The predicted robustness is verified experimentally.

The threshold theorem is a seminal result in the field of quantum computing asserting that arbitrarily long quantum computations can be performed on a faulty quantum computer provided that the noise level is below some constant threshold. This remarkable result comes at the price of increasing the number of qubits (quantum bits) by a large factor that scales polylogarithmically with the size of the quantum computation we wish to realize. Minimizing the space overhead for fault-tolerant quantum computation is a pressing challenge that is crucial to benefit from the computational potential of quantum devices. In this paper, we study the asymptotic scaling of the space overhead needed for fault-tolerant quantum computation. We show that the polylogarithmic factor in the standard threshold theorem is in fact not needed and that there is a fault-tolerant construction that uses a number of qubits that is only a constant factor more than the number of qubits of the ideal computation. This result was conjectured by Gottesman who suggested to replace the concatenated codes from the standard threshold theorem by quantum error-correcting codes with a constant encoding rate. The main challenge was then to find an appropriate family of quantum codes together with an efficient classical decoding algorithm working even with a noisy syndrome. The efficiency constraint is crucial here: bear in mind that qubits are inherently noisy and that faults keep accumulating during the decoding process. The role of the decoder is therefore to keep the number of errors under control during the whole computation. On a technical level, our main contribution is the analysis of the SMALL-SET-FLIP decoding algorithm applied to the family of quantum expander codes . We show that it can be parallelized to run in constant time while correcting sufficiently many errors on both the qubits and the syndrome to keep the error under control. These tools can be seen as a quantum generalization of the BIT-FLIP algorithm applied to the (classical) expander codes of Sipser and Spielman.

Despite achieving impressive performance, state-of-the-art classifiers remain highly vulnerable to small, imperceptible, adversarial perturbations. This vulnerability has proven empirically to be very intricate to address. In this paper, we study the phenomenon of adversarial perturbations under the assumption that the data is generated with a smooth generative model. We derive fundamental upper bounds on the robustness to perturbations of any classification function, and prove the existence of adversarial perturbations that transfer well across different classifiers with small risk. Our analysis of the robustness also provides insights onto key properties of generative models, such as their smoothness and dimensionality of latent space. We conclude with numerical experimental results showing that our bounds provide informative baselines to the maximal achievable robustness on several datasets.

We show that quantum expander codes, a constant-rate family of quantum low-density parity check (LDPC) codes, with the quasi-linear time decoding algorithm of Leverrier, Tillich and Zémor can correct a constant fraction of random errors with very high probability. This is the first construction of a constant-rate quantum LDPC code with an efficient decoding algorithm that can correct a linear number of random errors with a negligible failure probability. Finding codes with these properties is also motivated by Gottesman’s construction of fault tolerant schemes with constant space overhead.

A central question in information theory is to determine the maximum success probability that can be achieved in sending a fixed number of messages over a noisy channel. This was first studied in the pioneering work of Shannon, who established a simple expression characterizing this quantity in the limit of multiple independent uses of the channel. Here, we consider the general setting with only one use of the channel. We observe that the maximum success probability can be expressed as the maximum value of a submodular function. Using this connection, we establish the following results: 1) There is a simple greedy polynomial-time algorithm that computes a code achieving a (1 - e-1)-approximation of the maximum success probability. The factor (1-e-1) can be improved arbitrarily close to 1 at the cost of slightly reducing the number of messages to be sent. Moreover, it is NP-hard to obtain an approximation ratio strictly better than (1 - e-1) for the problem of computing the maximum success probability. 2) Shared quantum entanglement between the sender and the receiver can increase the success probability by a factor of at most (1/(1 - e-1)). In addition, this factor is tight if one allows an arbitrary non-signaling box between the sender and the receiver. 3) We give tight bounds on the one-shot performance of the meta-converse of Polyanskiy-Poor-Verdú.

Given a state-of-the-art deep neural network classifier, we show the existence of a universal (image-agnostic) and very small perturbation vector that causes natural images to be misclassified with high probability. We propose a systematic algorithm for computing universal perturbations, and show that state-of-the-art deep neural networks are highly vulnerable to such perturbations, albeit being quasi-imperceptible to the human eye. We further empirically analyze these universal perturbations and show, in particular, that they generalize very well across neural networks. The surprising existence of universal perturbations reveals important geometric correlations among the high-dimensional decision boundary of classifiers. It further outlines potential security breaches with the existence of single directions in the input space that adversaries can possibly exploit to break a classifier on most natural images.

Quantum-proof randomness extractors are an important building block for classical and quantum cryptography as well as device independent randomness amplification and expansion. Furthermore, they are also a useful tool in quantum Shannon theory. It is known that some extractor constructions are quantum-proof whereas others are provably not [Gavinsky et al., STOC'07]. We argue that the theory of operator spaces offers a natural framework for studying to what extent extractors are secure against quantum adversaries: we first phrase the definition of extractors as a bounded norm condition between normed spaces, and then show that the presence of quantum adversaries corresponds to a completely bounded norm condition between operator spaces. From this, we show that very high min-entropy extractors as well as extractors with small output are always (approximately) quantum-proof. We also study a generalization of extractors called randomness condensers. We phrase the definition of condensers as a bounded norm condition and the definition of quantum-proof condensers as a completely bounded norm condition. Seeing condensers as bipartite graphs, we then find that the bounded norm condition corresponds to an instance of a well-studied combinatorial problem, called bipartite densest subgraph. Furthermore, using the characterization in terms of operator spaces, we can associate to any condenser a Bell inequality (two-player game), such that classical and quantum strategies are in one-to-one correspondence with classical and quantum attacks on the condenser. Hence, we get for every quantum-proof condenser (which includes in particular quantum-proof extractors) a Bell inequality that cannot be violated by quantum mechanics.

A central question in information theory is to determine the maximum success probability that can be achieved in sending a fixed number of messages over a noisy channel. This was first studied in the pioneering work of Shannon who established a simple expression characterizing this quantity in the limit of multiple independent uses of the channel. Here, we consider the general setting with only one use of the channel. We observe that the maximum success probability can be expressed as the maximum value of a submodular function. Using this connection, we establish the following results: 1) There is a simple greedy polynomial-time algorithm that computes a code achieving a (1 - e-1)-approximation of the maximum success probability. Moreover, for this problem it is NP-hard to obtain an approximation ratio strictly better than (1 - e-1). 2) Shared quantum entanglement between the sender and the receiver can increase the success probability by a factor of at most 1/1-e-1. In addition, this factor is tight if one allows an arbitrary non-signaling box between the sender and the receiver. 3) We give tight bounds on the one-shot performance of the meta-converse of Polyanskiy-Poor-Verdú.

The relative entropy is the basic concept underlying various information measures like entropy, conditional entropy and mutual information. Here, we discuss how to make use of variational formulas for measured relative entropy and quantum relative entropy for understanding the additivity properties of various entropic quantities that appear in quantum information theory. In particular, we show that certain lower bounds on quantum conditional mutual information are superadditive.

A natural measure for the amount of quantum information that a physical system E holds about another system A = A1, .. . , Anis given by the min-entropy Hmin(A|E). In particular, the min-entropy measures the amount of entanglement between E and A, and is the relevant measure when analyzing a wide variety of problems ranging from randomness extraction in quantum cryptography, decoupling used in channel coding, to physical processes such as thermalization or the thermodynamic work cost (or gain) of erasing a quantum system. As such, it is a central question to determine the behavior of the minentropy after some process M. is applied to the system A. Here, we introduce a new generic tool relating the resulting min-entropy to the original one, and apply it to several settings of interest. A simple example of such a process is the one of sampling, where a subset S of the systems A1, ... , An is selected at random. Our tool allows us to quantify the entanglement that E has with the selected systems AS, i.e., Hmin(AS|ES) as a function of the original Hmin(A|E). We give two applications of this result. First, it directly provides the first local quantum-to-classical randomness extractors for use in quantum cryptography, as well as decoupling operations acting on only a small fraction AS of the input A. Moreover, it gives lower bounds on the dimension of k-out-of-n fully quantum random access encodings. Another natural example of such a process is a measurement in, e.g., BB84 bases commonly used in quantum cryptography. We establish the first entropic uncertainty relations with quantum side information that are nontrivial whenever E is not maximally entangled with A. As a consequence, we are able to prove optimality of quantum cryptographic schemes in the noisy-storage model. This model allows for the secure implementation of two-party cryptographic primitives under the assumption that the adversary cannot store quantum information perfectly. A special case is the bounded-quantum-storage model (BQSM), which assumes that the adversary's quantum memory device is noise free but limited in size. Ever since the inception of the BQSM, it has been a vexing open question to determine whether the security is possible as long as the adversary can only.

Many constructions of randomness extractors are known to work in the presence of quantum side information, but there also exist extractors which do not [Gavinsky et al., STOC'07]. Here we find that spectral extractors with a bound on the second largest eigenvalue - considered as an operator on the Hilbert-Schmidt class - are quantum-proof. We then discuss fully quantum extractors and call constructions that also work in the presence of quantum correlations decoupling. As in the classical case we show that spectral extractors are decoupling. The drawback of classical and quantum spectral extractors is that they always have a long seed, whereas there exist classical extractors with exponentially smaller seed size. For the quantum case, we show that there exists an extractor with extremely short seed size d = O(log(1/ε)), where ε > 0 denotes the quality of the randomness. In contrast to the classical case this is independent of the input size and min-entropy and matches the simple lower bound d ≥ log(1/ε).

The goal of randomness extraction is to distill (almost) perfect randomness from a weak source of randomness. When the source yields a classical string X, many extractor constructions are known. Yet, when considering a physical randomness source, X is itself ultimately the result of a measurement on an underlying quantum system. When characterizing the power of a source to supply randomness, it is hence natural to ask how much classical randomness we can extract from a quantum system. To tackle this question, we here take on the study of quantum-to-classical randomness extractors (QC-extractors). We provide constructions of QC-extractors based on measurements in a full set of mutually unbiased bases (MUBs), and certain single qubit measurements. The latter are particularly appealing since they are not only easy to implement, but also appear throughout quantum cryptography. We proceed to prove an upper bound on the maximum amount of randomness that we could hope to extract from any quantum state. Some of our QC-extractors almost match this bound. We show two applications of our results. First, we show that any QC-extractor gives rise to entropic uncertainty relations with respect to quantum side information. Such relations were previously only known for two measurements. In particular, we obtain strong relations in terms of the von Neumann (Shannon) entropy as well as the min-entropy for measurements in (almost) unitary two-designs, a full set of MUBs, and single qubit measurements in three MUBs each. Second, we resolve the central open question in the noisy-storage model by linking security to the quantum capacity of the adversary's storage device. More precisely, we show that any two party cryptographic primitives can be implemented securely as long as the adversary's storage device has sufficiently low quantum capacity. Our protocol does not need any quantum storage to implement, and is technologically feasible using present-day technology.

A natural measure for the amount of quantum information that a physical system E holds about another system A = A 1 ,..., A n is given by the min-entropy H min ( A | E ). Specifically, the min-entropy measures the amount of entanglement between E and A , and is the relevant measure when analyzing a wide variety of problems ranging from randomness extraction in quantum cryptography, decoupling used in channel coding, to physical processes such as thermalization or the thermodynamic work cost (or gain) of erasing a quantum system. As such, it is a central question to determine the behaviour of the min-entropy after some process M is applied to the system A . Here we introduce a new generic tool relating the resulting min-entropy to the original one, and apply it to several settings of interest, including sampling of subsystems and measuring in a randomly chosen basis. The results on random measurements yield new high-order entropic uncertainty relations with which we prove the optimality of cryptographic schemes in the bounded quantum storage model. This is an abridged version of the paper; the full version containing all proofs and further applications can be found in [13]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the encoding complexity for quantum error correcting codes with large rate and distance. We prove that random Clifford circuits with O(nlog2n) gates can be used to encode k qubits in n qubits with a distance d provided k/n23-h(d/n). In addition, we prove that such circuits typically have a depth of O(log3n).

In the context of network information theory, one often needs a multiparty probability distribution to be typical in several ways simultaneously. When considering quantum states instead of classical ones, it is in general difficult to prove the existence of a state that is jointly typical. Such a difficulty was recently emphasized and conjectures on the existence of such states were formulated. In this paper, we consider a one-shot multiparty typicality conjecture. The question can then be stated easily: is it possible to smooth the largest eigenvalues of all the marginals of a multipartite state ρ simultaneously while staying close to ρ? We prove the answer is yes whenever the marginals of the state commute. In the general quantum case, we prove that simultaneous smoothing is possible if the number of parties is two or more generally if the marginals to optimize satisfy some non-overlap property.

The existence of quantum uncertainty relations is the essential reason that some classically unrealizable cryptographic primitives become realizable when quantum communication is allowed. One operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking [DiVincenzo et al. 2004]. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n -bit message is encoded in a quantum system using a classical key of size much smaller than n . Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message, in which case the message is said to be “locked”. Furthermore, knowing the key, it is possible to recover, that is “unlock”, the message. In this article, we make the following contributions by exploiting a connection between uncertainty relations and low-distortion embeddings of Euclidean spaces into slightly larger spaces endowed with the ℓ 1 norm. We introduce the notion of a metric uncertainty relation and connect it to low-distortion embeddings of ℓ 2 into ℓ 1 . A metric uncertainty relation also implies an entropic uncertainty relation. We prove that random bases satisfy uncertainty relations with a stronger definition and better parameters than previously known. Our proof is also considerably simpler than earlier proofs. We then apply this result to show the existence of locking schemes with key size independent of the message length. Moreover, we give efficient constructions of bases satisfying metric uncertainty relations. The bases defining these metric uncertainty relations are computable by quantum circuits of almost linear size. This leads to the first explicit construction of a strong information locking scheme. These constructions are obtained by adapting an explicit norm embedding due to Indyk [2007] and an extractor construction of Guruswami et al. [2009]. We apply our metric uncertainty relations to exhibit communication protocols that perform equality testing of n -qubit states. We prove that this task can be performed by a single message protocol using O (log 2 n ) qubits and n bits of communication, where the computation of the sender is efficient.

Calculating the capacity of interference channels is a notorious open problem in classical information theory. Such channels have two senders and two receivers, and each sender would like to communicate with a partner receiver. The capacity of such channels is known exactly in the settings of “very strong” and “strong” interference, while the Han-Kobayashi coding strategy gives the best known achievable rate region in the general case. Here, we introduce and study the quantum interference channel, a natural generalization of the interference channel to the setting of quantum information theory. We restrict ourselves for the most part to channels with two classical inputs and two quantum outputs in order to simplify the presentation of our results (though generalizations of our results to channels with quantum inputs are straightforward). We are able to determine the exact classical capacity of this channel in the settings of “very strong” and “strong” interference, by exploiting Winter's successive decoding strategy and a novel two-sender quantum simultaneous decoder, respectively. We provide a proof that a Han-Kobayashi strategy is achievable with Holevo information rates, up to a conjecture regarding the existence of a three-sender quantum simultaneous decoder. This conjecture holds for a special class of quantum multiple-access channels with average output states that commute, and we discuss some other variations of the conjecture that hold. Finally, we detail a connection between the quantum interference channel and prior work on the capacity of bipartite unitary gates.

Quantum uncertainty relations are at the heart of many quantum cryptographic protocols performing classically impossible tasks. One operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message (the message is "locked"). Furthermore, knowing the key, it is possible to recover (or "unlock") the message.