We study the sample complexity of conditional independence testing. In this problem, given i.i.d. samples from a discrete distribution $P_{ABC}$, the goal is to distinguish whether $A$ and $C$ are conditionally independent with respect to $B$, i.e., $P_{ABC}=P_{A|B}P_BP_{C|B}$, or whether $A$ and $C$ are conditionally dependent, $\Delta(P_{ABC},P_{A|B}P_BP_{C|B})\geq \varepsilon$ for some fixed threshold $\varepsilon$ and distance measure $\Delta$. We are interested in the cases where $\Delta$ is either the $\ell_1$ distance or the KL-divergence. The study for the case of $\ell_1$ distance was initiated by (Canonne et al., STOC 2018), and the KL-divergence was recently studied by (Seyfried et al., COLT 2025). Both works design algorithms whose sample complexities scale sublinearly in the dimensions of the subsystems, and showed tight lower bounds in some parameter regimes. While Canonne et al. derived partial lower bounds for the remaining regimes as well, the problem of fully resolving the sample complexity in all parameters remained open. In this work, we settle these open questions and prove optimal sample complexity lower bounds for both of these problems, thereby completely settling the sample complexities up to polylogarithmic factors.

We investigate the sample complexity of mutual information and conditional mutual information testing. For conditional mutual information testing, given access to independent samples of a triple of random variables $(A, B, C)$ with unknown distribution, we want to distinguish between two cases: (i) $A$ and $C$ are conditionally independent, i.e., $I(A:C|B) = 0$, and (ii) $A$ and $C$ are conditionally dependent, i.e., $I(A:C|B) \geq \varepsilon$ for some threshold $\varepsilon$. We establish an upper bound on the number of samples required to distinguish between the two cases with high confidence, as a function of $\varepsilon$ and the three alphabet sizes. We conjecture that our bound is tight and show that this is indeed the case in several parameter regimes. For the special case of mutual information testing (when $B$ is trivial), we establish the necessary and sufficient number of samples required up to polylogarithmic terms. Our technical contributions include a novel method to efficiently simulate weakly correlated samples from the conditionally independent distribution $P_{A|B} P_{C|B} P_B$ given access to samples from an unknown distribution $P_{ABC}$, and a new estimator for equivalence testing that can handle such correlated samples, which might be of independent interest.

We show that Frenkel’s integral representation of the quantum relative entropy provides a natural framework to derive continuity bounds for quantum information measures. Our main general result is a dimension-independent semi-continuity relation for the quantum relative entropy with respect to the first argument. Using it, we obtain a number of results: (1) a tight continuity relation for the conditional entropy in the case where the two states have equal marginals on the conditioning system, resolving a conjecture by Wilde in this special case; (2) a stronger version of the Fannes–Audenaert inequality on quantum entropy; (3) better estimates on the quantum capacity of approximately degradable channels; (4) an improved continuity relation for the entanglement cost; (5) general upper bounds on asymptotic transformation rates in infinite-dimensional entanglement theory; and (6) a proof of a conjecture due to Christandl, Ferrara, and Lancien on the continuity of ’filtered’ relative entropy distances.

We say that a matrixPwith non-negative entries majorizes another such matrixQif there is a stochastic matrixTsuch thatQ=TP. We study matrix majorization in large samples and in the catalytic regime in the case where the columns of the matrices need not have equal support, as has been assumed in earlier works. We focus on two cases: either there are no support restrictions (except for requiring a non-empty intersection for the supports) or the final column dominates the others. Using real-algebraic methods, we identify sufficient and almost necessary conditions for majorization in large samples or when using catalytic states under these support conditions. These conditions are given in terms of multivariate divergences that generalize the Rényi divergences. We notice that varying support conditions dramatically affect the relevant set of divergences. Our results find an application in the theory of catalytic state transformation in quantum thermodynamics.

We study a noise model for linear stochastic bandits for which the subgaussian noise parameter vanishes linearly as we select actions on the unit sphere closer and closer to the unknown vector. We introduce an algorithm for this problem that exhibits a minimax regret scaling as $\log^3(T)$ in the time horizon $T$, in stark contrast the square root scaling of this regret for typical bandit algorithms. Our strategy, based on weighted least-squares estimation, achieves the eigenvalue relation $\lambda_{\min} ( V_t ) = \Omega (\sqrt{\lambda_{\max}(V_t ) })$ for the design matrix $V_t$ at each time step $t$ through geometrical arguments that are independent of the noise model and might be of independent interest. This allows us to tightly control the expected regret in each time step to be of the order $O(\frac1{t})$, leading to the logarithmic scaling of the cumulative regret.

Characterizing the minimal communication needed for quantum channel simulation is a fundamental task in the quantum information theory. In this paper, we show that, in fidelity, the quantum channel simulation can be directly achieved via quantum state splitting without using a technique known as the de Finetti reduction, and thus provide a pair of tighter one-shot bounds. This opens up new potentials for higher-order analysis. Using the bounds, we also recover the quantum reverse Shannon theorem in a much simpler way.

We introduce a new quantum decoder based on a variant of the pretty good measurement, but defined via an alternative matrix quotient. We then use this novel decoder to derive new lower bounds on the error exponent both in the one-shot and asymptotic regimes for the classical-quantum and the entanglement-assisted channel coding problems. Our bounds are expressed in terms of measured (for the one-shot bounds) and sandwiched (for the asymptotic bounds) channel Rényi mutual information of order between 1/2 and 1. The bounds are not comparable with some previously established bounds for general channels, yet they are tight (for rates close to capacity) when the channel is classical. Finally, we also use our new decoder to rederive Cheng’s recent tight bound on the decoding error probability, which implies that most existing asymptotic results also hold for the new decoder.

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 we 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 bounds imply the well-known fact that the optimal asymptotic rate of one channel to simulate another under common randomness assistance is given by the ratio of their respective capacities. Additionally, our higher-order asymptotic expansion shows that this reversibility falls apart in the second order. 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 multipartite mutual information.

One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix majorization. Solving an open problem raised by Muet al, we show that if certain monotones—namely multivariate extensions of Rényi divergences—are strictly ordered between the two tuples, then for sufficiently largen, there exists a stochastic matrix taking then-fold Kronecker power of each input distribution to then-fold Kronecker power of the corresponding output distribution. The same conditions, with non-strict ordering for the monotones, are also necessary for such matrix majorization in large samples. Our result also gives conditions for the existence of a sequence of statistical maps that asymptotically (with vanishing error) convert a single copy of each input distribution to the corresponding output distribution with the help of a catalyst that is returned unchanged. Allowing for transformation with arbitrarily small error, we find conditions that are both necessary and sufficient for such catalytic matrix majorization. We derive our results by building on a general algebraic theory of preordered semirings recently developed by one of the authors. This also allows us to recover various existing results on majorization in large samples and in the catalytic regime as well as relative majorization in a unified manner.

We study the problem of random-assisted simulation of discrete broadcast channel in one-shot and i.i.d. setups. We derive one-shot inner and outer bounds of the set of attainable message-size pairs for simulating WYZ|Xwithin some total variation distance (TVD) tolerance of ϵ. The inner bounds are based on the bipartite convex split lemma. Whereas the outer bounds are based on the properties of the multi-partite max information. Using these bounds, we establish a single-letter expression of the simulation region of a broadcast channel.

Bounds on information combining are a fundamental tool in coding theory, in particular when analyzing polar codes and belief propagation. They usually bound the evolution of random variables with respect to their Shannon entropy. In recent work this approach was generalized to Rényi α-entropies. However, due to the lack of a traditional chain rule for Rényi entropies the picture remained incomplete. In this work we establish the missing link by providing Rényi chain rules connecting different definitions of Rényi entropies by Hayashi and Arimoto. This allows us to provide new information combining bounds for the Arimoto Rényi entropy. In the second part, we generalize the chain rule to the quantum setting and show how they allow us to generalize results and conjectures previously only given for the von Neumann entropy. In the special case of α = 2 we give the first optimal information combining bounds with quantum side information.

The quadratic decaying property of the information rate function states that, given a fixed conditional distribution$p_{ \mathsf {Y}| \mathsf {X}}$, the mutual information between the (finite) discrete random variables$\mathsf {X}$and$\mathsf {Y}$decreases at least quadratically in the Euclidean distance as$p_{\mathsf {X}}$moves away from the capacity-achieving input distributions. It is a property of the information rate function that is particularly useful in the study of higher order asymptotics and finite blocklength information theory, where it was already implicitly used by Strassen (1962) and later, more explicitly, by Polyanskiy–Poor–Verdú (2010). However, the proofs outlined in both works contain gaps that are nontrivial to close. This comment provides an alternative, complete proof of this property.

The moderate deviation regime is concerned with the finite block length trade-off between communication cost and error for information processing tasks in the asymptotic regime, where the communication cost approaches a capacity-like quantity and the error vanishes at the same time. We find exact characterisations of these trade-offs for a variety of fully quantum communication tasks, including quantum source coding, quantum state splitting, entanglement-assisted quantum channel coding, and entanglement-assisted quantum channel simulation. The main technical tool we derive is a tight relation between the partially smoothed max-information and the hypothesis testing relative entropy. This allows us to obtain the expansion of the partially smoothed max-information for i.i.d. states in the moderate deviation regime.

Divergence chain rules for channels relate the divergence of a pair of channel inputs to the divergence of the corresponding channel outputs. An important special case of such a rule is the data-processing inequality, which tells us that if the same channel is applied to both inputs then the divergence cannot increase. Based on direct matrix analysis methods, we derive several Rényi divergence chain rules for channels in the quantum setting. Our results simplify and in some cases generalise previous derivations in the literature.

We study the problem of one-shot channel simulation of DMCs with unlimited shared randomness. For any fixed tolerance measured in total variational distance, we propose an achievability bound and a converse bound on the size of the code to simulate the channel. The achievability bound utilizes the convex split lemma, whereas the converse bound is the result of the relationships between smoothed max-divergences and the max-mutual information. The achievability proof does not rely on a "universal state" (compared with some previous related works), and provides a tighter bound. Using the two bounds, we also provide an alternative proof to the reverse Shannon theorem.

We consider the sequential quantum channel discrimination problem using adaptive and non-adaptive strategies. In this setting the number of uses of the underlying quantum channel is not fixed but a random variable that is either bounded in expectation or with high probability. We show that, by using adaptive strategies for the discrimination problem, both types of error probabilities decrease to zero exponentially fast and the rates are characterized by the measured relative entropy between two quantum channels. Allowing for quantum memory, we see that the optimal rates are given by the regularized channel relative entropy. We also characterize the error exponents in the discrimination problem if non-adaptive strategies are used.

We consider the problem of learning N identical copies of an unknown n-qubit quantum graph state with product measurements. These graph states have corresponding graphs where every vertex has exactly d neighboring vertices. Here, we detail an explicit algorithm that uses product measurements on multiple identical copies of such graph states to learn them. When n ≫ d and N = O(d log(1/ϵ) + d2log n), this algorithm correctly learns the graph state with probability at least 1 – ϵ. From channel coding theory, we find that for arbitrary joint measurements on graph states, any learning algorithm achieving this accuracy requires at least Ω(log(1/ϵ) + d log n) copies when $d = o\left( {\sqrt n } \right)$. We also supply bounds on N when every graph state encounters identical and independent depolarizing errors on each qubit.

We introduce and analyse the problem of encoding classical information into different resources of a quantum state. More precisely, we consider a general class of communication scenarios characterised by encoding operations that commute with a unique resource destroying map and leave free states invariant. Our motivating example is given by encoding information into coherences of a quantum system with respect to a fixed basis (with unitaries diagonal in that basis as encodings and the decoherence channel as a resource destroying map), but the generality of the framework allows us to explore applications ranging from super-dense coding to thermodynamics. For any state, we find that the number of messages that can be encoded into it using such operations in a one-shot scenario is upper bounded in terms of the information spectrum relative entropy between the given state and its version with erased resources. Furthermore, if the resource destroying map is the twirling channel over some unitary group, we find matching one-shot lower bounds as well. In the asymptotic setting where we encode into many copies of the resource state, our bounds yield an operational interpretation of resource monotones such as the relative entropy of coherence and its corresponding relative entropy variance.

We consider sequential hypothesis testing between two quantum states using adaptive and non-adaptive strategies. In this setting, samples of an unknown state are requested sequentially and a decision to either continue or to accept one of the two hypotheses is made after each test. Under the constraint that the number of samples is bounded, either in expectation or with high probability, we exhibit adaptive strategies that minimize both types of misidentification errors. Namely, we show that these errors decrease exponentially (in the stopping time) with decay rates given by the measured relative entropies between the two states. Moreover, if we allow joint measurements on multiple samples, the rates are increased to the respective quantum relative entropies. We also fully characterize the achievable error exponents for non-adaptive strategies and provide numerical evidence showing that adaptive measurements are necessary to achieve our bounds.

We introduce an axiomatic approach to entropies and relative entropies that relies only on minimal information-theoretic axioms, namely monotonicity under mixing and data-processing as well as additivity for product distributions. We find that these axioms induce sufficient structure to establish continuity in the interior of the probability simplex and meaningful upper and lower bounds, e.g., we find that every relative entropy satisfying these axioms must lie between the Rényi divergences of order 0 and ∞. We further show simple conditions for positive definiteness of such relative entropies and a characterisation in terms of a variant of relative trumping. Our main result is a one-to-one correspondence between entropies and relative entropies.

Smooth entropies are a tool for quantifying resource trade-offs in (quantum) information theory and cryptography. In typical bi- and multi-partite problems, however, some of the sub-systems are often left unchanged and this is not reflected by the standard smoothing of information measures over a ball of close states. We propose to smooth instead only over a ball of close states which also have some of the reduced states on the relevant sub-systems fixed. This partial smoothing of information measures naturally allows to give more refined characterizations of various information-theoretic problems in the one-shot setting. In particular, we immediately get asymptotic second-order characterizations for tasks such as privacy amplification against classical side information or classical state splitting. For quantum problems like state merging the general resource trade-off is tightly characterized by partially smoothed information measures as well.

We study the general framework of quantum channel simulation, that is, the ability of a quantum channel to simulate another one using different classes of codes. First, we show that the minimum error of simulation and the one-shot quantum simulation cost under no-signalling assisted codes are given by semidefinite programs. Second, we introduce the channel's smooth max-information, which can be seen as a one-shot generalization of the mutual information of a quantum channel. We provide an exact operational interpretation of the channel's smooth max-information as the one-shot quantum simulation cost under no-signalling assisted codes, which significantly simplifies the study of channel simulation and provides insights and bounds for the case under entanglement-assisted codes. Third, we derive the asymptotic equipartition property of the channel's smooth max-information; i.e., it converges to the quantum mutual information of the channel in the independent and identically distributed asymptotic limit. This implies the quantum reverse Shannon theorem in the presence of no-signalling correlations. Finally, we explore the simulation cost of various quantum channels.

We propose a branch-and-bound algorithm for minimizing a bilinear functional of the form f (X, Y) = tr((X ⊗ Y) Q) +tr(AX) +tr(BY), of pairs of Hermitian matrices (X, Y) restricted by joint semidefinite programming constraints. The functional is parametrized by self-adjoint matrices Q, A and B. This problem generalizes that of a bilinear program, where X and Y belong to polyhedra. The algorithm converges to a global optimum and yields upper and lower bounds on its value in every step. Various problems in quantum information theory can be expressed in this form. As an example application, we compute Dobrushin curves of quantum channels, giving upper bounds on classical coding with energy constraints.

Prior work has shown that there exists a relation problem which can be solved with certainty by a constant-depth quantum circuit composed of geometrically local gates in two dimensions, but cannot be solved with high probability by any classical constant depth circuit composed of bounded fan-in gates. Here we provide two extensions of this result. Firstly, we show that a separation in computational power persists even when the constant-depth quantum circuit is restricted to geometrically local gates in one dimension. The corresponding quantum algorithm is the simplest we know of which achieves a quantum advantage of this type. Our second, main result, is that a separation persists even if the shallow quantum circuit is corrupted by noise. We construct a relation problem which can be solved with near certainty using a noisy constant-depth quantum circuit composed of geometrically local gates in three dimensions, provided the noise rate is below a certain constant threshold value. On the other hand, the problem cannot be solved with high probability by a noise-free classical circuit of constant depth. A key component of the proof is a quantum error-correcting code which admits constant-depth logical Clifford gates and single-shot logical state preparation. We show that the surface code meets these criteria.

Smooth entropies are a tool for quantifying resource trade-offs in information theory and cryptography. However, in typical multi-partite problems some of the sub-systems are often left unchanged and this is not reflected by the standard smoothing of information measures over a ball of close states. We propose to smooth instead only over a ball of close states which also have some of the reduced states on the relevant sub-systems fixed. This partial smoothing of information measures naturally allows to give more refined characterizations of various information-theoretic problems in the one-shot setting. As a consequence, we can derive asymptotic second-order characterizations for tasks such as privacy amplification against classical side information or classical state splitting. For quantum problems like state merging the general resource trade-off is tightly characterized by partially smoothed information measures as well.

We consider the problem of interconverting a finite amount of resources within all theories whose single-shot transformation rules are based on a majorisation relation, e.g. the resource theories of entanglement and coherence (for pure state transformations), as well as thermodynamics (for energy-incoherent transformations). When only finite resources are available we expect to see a non-trivial trade-off between the rate rnat which n copies of a resource state ρ can be transformed into nrncopies of another resource state σ, and the error level εnof the interconversion process, as a function of n. In this work we derive the optimal trade-off in the so-called moderate deviation regime, where the rate of interconversion rnapproaches its optimum in the asymptotic limit of unbounded resources (n → ∞), while the error εnvanishes in the same limit. We find that the moderate deviation analysis exhibits a resonance behaviour which implies that certain pairs of resource states can be interconverted at the asymptotically optimal rate with negligible error, even in the finite n regime.

We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in general classical-quantum channels. Second, we establish a finite blocklength sphere-packing bound for classical-quantum channels, which significantly improves Dalai's prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o(logn/n), indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.

Entanglement distillation, an essential quantum information processing task, refers to the conversion from multiple copies of noisy entangled states to a smaller number of highly entangled states. In this paper, we study the non-asymptotic fundamental limits for entanglement distillation. We investigate the optimal tradeoff between the distillation rate, the number of prepared states, and the error tolerance. First, we derive the one-shot distillable entanglement under completely positive partial transpose preserving operations as a semidefinite program and demonstrate an exact characterization via the quantum hypothesis testing relative entropy. Second, we establish efficiently computable second-order estimations of the distillation rate for general quantum states. In particular, we provide explicit as well as approximate evaluations for various quantum states of practical interest, including pure states, mixture of Bell states, maximally correlated states, and isotropic states.

We explore several new converse bounds for classical communication over quantum channels in both the one-shot and asymptotic regimes. First, we show that the Matthews-Wehner meta-converse bound for entanglementassisted classical communication can be achieved by activated, no-signaling assisted codes, suitably generalizing a result for classical channels. Second, we derive a new efficiently computable meta-converse on the amount of classical information unassisted codes can transmit over a single use of a quantum channel. As applications, we provide a finite resource analysis of classical communication over quantum erasure channels, including the second-order and moderate deviation asymptotics. Third, we explore the asymptotic analogue of our new meta-converse, the Υ-information of the channel. We show that its regularization is an upper bound on the classical capacity, which is generally tighter than the entanglement-assisted capacity and other known efficiently computable strong converse bounds. For covariant channels, we show that the Υ-information is a strong converse bound.

We study the general framework of quantum channel simulation, that is, the ability of a quantum channel to simulate another one using different classes of codes. Our main results are as follows. First, we show that the minimum error of simulation under non-signalling assisted codes is efficiently computable via semidefinite programming. The cost of simulating a channel via noiseless quantum channels under non-signalling assisted codes can also be characterized as a semidefinite program. Second, we introduce the channel's smooth max-information, which can be seen as a one-shot generalization of the channel's mutual information. We show that the one-shot quantum simulation cost under non-signalling assisted codes is exactly equal to the channel's smooth max-information. Due to the quantum reverse Shannon theorem, the channel's smooth max-information converges to the channel's mutual information in the independent and identically distributed asymptotic limit. Together with earlier findings on the (activated) non-signalling assisted one-shot capacity of channels [Wang et al., arXiv:1709.05258], this suggest that the operational min- and max-type one-shot analogues of the channel's mutual information are the channel's hypothesis testing relative entropy and the channel's smooth max-information, respectively.

We explore several new converse bounds for classical communication over quantum channels in the finite blocklength regime. First, we show that the Matthews-Wehner meta-converse bound for entanglement-assisted classical communication can be achieved by activated, no-signalling assisted codes, suitably generalizing a result for classical channels. Second, we derive a new meta-converse on the amount of information unassisted codes can transmit over a single use of a quantum channel. We further show that this meta-converse can be evaluated via semidefinite programming. As an application, we provide a second-order analysis of classical communication over quantum erasure channels.

We revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis. We introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms. In this case, we find the threshold rate, the optimal error and strong converse exponents (at large deviations from the threshold), and the second order asymptotics (at small deviations from the threshold). We apply our results to find the operational interpretations of various Rényi information measures. In case the alternative hypothesis is comprised of bipartite product distributions, we find that the optimal error and strong converse exponents are determined by the variations of Rényi mutual information. In case the alternative hypothesis consists of tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by the variations of Rényi conditional mutual information. In either case, the relevant notion of Rényi mutual information depends on the precise choice of the alternative hypothesis. As such, this paper also strengthens the view that different definitions of Rényi mutual information, conditional entropy, and conditional mutual information are adequate depending on the context in which the measures are used.

“To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award.” We provide a sphere-packing lower bound for the optimal error probability in finite blocklengths when coding over a symmetric classical-quantum channel. Our result shows that the pre-factor can be significantly improved from the order of the subexponential to the polynomial, This established pre-factor is arguably optimal because it matches the best known random coding upper bound in the classical case. Our approaches rely on a sharp concentration inequality in strong large deviation theory and crucial properties of the error-exponent function.

We analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases. To characterise the fundamental tradeoff between decoding error, code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by Altŭg and Wagner as well as Polyanskiy and Verdú. We derive such a tradeoff for classical-quantum (as well as image-additive) channels in terms of the channel capacity and the channel dispersion, giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data. To derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime. Due to the central importance of the latter task, we expect that our techniques will find further applications in the analysis of other quantum information processing tasks.

A Markov chain is a tripartite quantum state ρABCwhere there exists a recovery map RB→BCsuch that ρABC= RB→BC(ρAB). More generally, an approximate Markov chain ρABCis a state whose distance to the closest recovered state RB→BC(ρAB) is small. Recently it has been shown that this distance can be bounded from above by the conditional mutual information I(A : C|B)ρof the state. We improve on this connection by deriving the first bound that is tight in the commutative case and features an explicit recovery map that only depends on the reduced state pBC. The key tool in our proof is a multivariate extension of the Golden-Thompson inequality, which allows us to extend logarithmic trace inequalities from two to arbitrarily many matrices.

We establish a converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here we use this approach along with a “privacy test” to establish a general meta-converse bound for private communication.

We study lower bounds on the optimal error probability in channel coding at rates below capacity, commonly termed sphere-packing bounds. In this work, we establish a sphere-packing bound for classical-quantum channels, which significantly improves previous prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o(log n/n), indicating our sphere-packing bound is almost exact in the high rate regime. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.

We revisit a fundamental open problem in quantum information theory, namely, whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here, we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication. For any sequence of codes with rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical postprocessing on the transmitted quantum data. As an application of this result, for generalized dephasing channels, we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. Thus, we conclusively settle the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.

This paper establishes several converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here, we use this approach along with a “privacy test” to establish a general meta-converse bound for private communication, which has a number of applications. The meta-converse allows for proving that any quantum channel's relative entropy of entanglement is a strong converse rate for private communication. For covariant channels, the meta-converse also leads to second-order expansions of relative entropy of entanglement bounds for private communication rates. For such channels, the bounds also apply to the private communication setting in which the sender and the receiver are assisted by unlimited public classical communication, and as such, they are relevant for establishing various converse bounds for quantum key distribution protocols conducted over these channels. We find precise characterizations for several channels of interest and apply the methods to establish converse bounds on the private transmission capabilities of all phase-insensitive bosonic channels.

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.

The quantum relative entropy between two states satisfies a monotonicity property, meaning that applying the same quantum channel to both states can never increase their relative entropy. It is known that this inequality is only tight when there is a “recovery map” that exactly reverses the effects of the quantum channel on both states. In this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version. The recovery map is a convex combination of rotated Petz recovery maps and perfectly reverses the quantum channel on the second state. As a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved in [Fawzi and Renner, Commun. Math. Phys., 2015]. Our proof only relies on elementary properties of pinching maps and the operator logarithm.

We revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis. We introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms. In this case we find the threshold rate, the optimal error and strong converse exponents (at large deviations from the threshold) and the second-order asymptotics (at small deviations from the threshold). We apply our results to find operational interpretations of Rényi information measures. In particular, in case the alternative hypothesis consists of certain tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by the Rényi conditional mutual information.

Fawzi and Renner recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states ABC in terms of the fidelity of recovery (FoR), i.e., the maximal fidelity of the state ABC with a state reconstructed from its marginal BC by acting only on the C system. The FoR measures quantum correlations by the local recoverability of global states and has many properties similar to the CQMI. Here, we generalize the FoR and show that the resulting measure is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brandão et al. of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions.

The quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy. It is known that this inequality is only tight when there is a recovery map that exactly reverses the effects of the quantum channel on both states. In this paper, we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version. The recovery map is a convex combination of rotated Petz recovery maps and perfectly reverses the quantum channel on the second state. As a special case, we reproduce recent lower bounds on the conditional mutual information, such as the one proved by Fawzi and Renner. Our proof only relies on the elementary properties of pinching maps and the operator logarithm.

The entanglement-assisted capacity of a quantum channel is known to provide the formal quantum generalization of Shannon's classical channel capacity theorem, in the sense that it admits a single-letter characterization in terms of the quantum mutual information and does not increase in the presence of a noiseless quantum feedback channel from receiver to sender. In this work, we investigate second-order asymptotics of the entanglement-assisted communication task. That is, we consider how quickly the rates of entanglement-assisted codes converge to the entanglement-assisted capacity of a channel as a function of the number of channel uses and the error tolerance. We define a quantum generalization of the mutual information variance of a channel in the entanglement-assisted setting. For covariant channels, we show that this quantity is equal to the channel dispersion, and characterizes the convergence towards the entanglement-assisted capacity when the number of channel uses increases. More generally, we prove that the Gaussian approximation for a second-order coding rate is achievable for all quantum channels.

Recently, a variety of new measures of quantum Rényi mutual information and quantum Rényi conditional entropy have been proposed, and some of their mathematical properties explored. Here, we show that the Rényi mutual information attains operational significance in the context of composite hypothesis testing, when the null hypothesis is a fixed bipartite state and the alternate hypothesis consists of all product states that share one marginal with the null hypothesis. This hypothesis testing problem occurs naturally in channel coding, where it corresponds to testing whether a state is the output of a given quantum channel or of a “useless” channel whose output is independent of the channel input and environment. Similarly, we establish an operational interpretation of Rényi conditional entropy by choosing an alternative hypothesis that consists of product states that are maximally mixed on one system. Specialized to classical probability distributions, our results also establish an operational interpretation of Rényi mutual information and Rényi conditional entropy.

We revisit a fundamental open problem in quantum information theory, namely whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication: For any code with a rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical post-processing on the transmitted quantum data. Our result has several applications. Most importantly, for generalized dephasing channels we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. This for the first time conclusively settles the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.

This paper shows that, under the average error probability formalism, the third-order term in the normal approximation for the additive white Gaussian noise channel with a maximal or equal power constraint is at least (1/2) log n + O(1). This improves on the lower bound by Polyanskiy-Poor-Verdú (2010) and matches the upper bound proved by the same authors.

This paper shows that, under the average error probability formalism, the third-order term in the normal approximation for the additive white Gaussian noise channel with a maximal or equal power constraint is at least 1 over 2 log n+O(1). This improves on the lower bound by Polyanskiy-Poor-Verdú (2010) and matches the upper bound proved by the same authors.

Recently a new quantum generalization of the Rényi divergence and the corresponding conditional Rényi entropies was proposed. Here we report on a surprising relation between conditional Rényi entropies based on this new generalization and conditional Rényi entropies based on the quantum relative Rényi entropy that was used in previous literature. This generalizes the well-known duality relation H(A|B)+H(A|C) = 0 for tripartite pure states to Rényi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Rényi entropies.

The security of quantum key distribution protocols is guaranteed by the laws of quantum mechanics. However, a precise analysis of the security properties requires tools from both classical cryptography and information theory. Here, we employ recent results in non-asymptotic classical information theory to show that information reconciliation imposes fundamental limitations on the amount of secret key that can be extracted in the finite key regime. In particular, we find that an often used approximation for the information leakage during one-way information reconciliation is flawed and we propose an improved estimate.

We study the non-asymptotic fundamental limits for transmitting classical information over memoryless quantum channels, i.e. we investigate the amount of information that can be transmitted when the channel is used a finite number of times and a finite average decoding error is permissible. We show that, if we restrict the encoder to use ensembles of separable states, the non-asymptotic fundamental limit admits a Gaussian approximation that illustrates the speed at which the rate of optimal codes converges to the Holevo capacity as the number of channel uses tends to infinity. To do so, several important properties of quantum information quantities, such as the capacity-achieving output state, the divergence radius, and the channel dispersion, are generalized from their classical counterparts. Further, we exploit a close relation between classical-quantum channel coding and quantum binary hypothesis testing and rely on recent progress in the non-asymptotic characterization of quantum hypothesis testing and its Gaussian approximation.

Most coding theorems in quantum Shannon theory can be proven using the decoupling technique. To send data through a channel, one guarantees that the environment gets no information about it. Uhlmann's theorem then ensures that the receiver must be able to decode. While a wide range of problems can be solved this way, one of the most basic coding problems remains impervious to a direct application of this method, sending classical information through a quantum channel. We will show that this problem can, in fact, be solved using decoupling ideas, specifically by proving a dequantizing theorem, which ensures that the environment is only classically correlated with the sent data. Our techniques naturally yield a generalization of the Holevo-Schumacher-Westmoreland theorem to the one-shot scenario, where a quantum channel can be applied only once.

We study the performance limits of state-dependent discrete memoryless channels with a discrete state available at both the encoder and the decoder. We establish the ε-capacity as well as necessary and sufficient conditions for the strong converse property for such channels when the sequence of channel states is not necessarily stationary, memoryless, or ergodic. We then seek a finer characterization of these capacities in terms of second-order coding rates. The general results are supplemented by several examples including independent identically distributed and Markov states and mixed channels.

This paper shows that the logarithm of the ε-error capacity (average error probability) for n uses of a discrete memoryless channel with positive conditional information variance at every capacity-achieving input distribution is upper bounded by the normal approximation plus a term that does not exceed 1/2 log n + O(1).

We consider state-dependent memoryless channels with general state available at both encoder and decoder. We establish the ε-capacity and the optimistic ε-capacity. This allows us to prove a necessary and sufficient condition for the strong converse to hold. We also provide a simpler sufficient condition on the first- and second-order statistics of the state process that ensures that the strong converse holds.

We investigate two-party cryptographic protocols that are secure under assumptions motivated by physics, namely special relativity and quantum mechanics. In particular, we discuss the security of bit commitment in the so-called split models, i.e., models in which at least one of the parties is not allowed to communicate during certain phases of the protocol. We find the minimal splits that are necessary to evade the Mayers-Lo-Chau no-go argument and present protocols that achieve security in these split models. Furthermore, we introduce the notion of local versus global command, a subtle issue that arises when the split committer is required to delegate noncommunicating agents to open the commitment. We argue that classical protocols are insecure under global command in the split model we consider. On the other hand, we provide a rigorous security proof in the global command model for Kent's quantum protocol. The proof employs two fundamental principles of modern physics, the no-signaling property of relativity and the uncertainty principle of quantum mechanics.

We consider two fundamental tasks in quantum information theory, data compression with quantum side information, as well as randomness extraction against quantum side information. We characterize these tasks for general sources using so-called one-shot entropies. These characterizations-in contrast to earlier results-enable us to derive tight second-order asymptotics for these tasks in the i.i.d. limit. More generally, our derivation establishes a hierarchy of information quantities that can be used to investigate information theoretic tasks in the quantum domain: The one-shot entropies most accurately describe an operational quantity, yet they tend to be difficult to calculate for large systems. We show that they asymptotically agree (up to logarithmic terms) with entropies related to the quantum and classical information spectrum, which are easier to calculate in the i.i.d. limit. Our technique also naturally yields bounds on operational quantities for finite block lengths.

This paper shows that the logarithm of the ε-error capacity (average error probability) for n uses of a discrete memoryless channel (DMC) is upper bounded by the normal approximation plus a third-order term that does not exceed [ 1/ 2] logn +O(1) if the ε-dispersion of the channel is positive. This matches a lower bound by Y. Polyanskiy (2010) for DMCs with positive reverse dispersion. If the ε-dispersion vanishes, the logarithm of the ε-error capacity is upper bounded by n times the capacity plus a constant term except for a small class of DMCs and ε ≥ [ 1/ 2].

The chain rule for the Shannon and von Neumann entropy, which relates the total entropy of a system to the entropies of its parts, is of central importance to information theory. Here, we consider the chain rule for the more general smooth min- and max-entropies, used in one-shot information theory. For these entropy measures, the chain rule no longer holds as an equality. However, the standard chain rule for the von Neumann entropy is retrieved asymptotically when evaluating the smooth entropies for many identical and independently distributed states.

The Leftover Hash Lemma states that the output of a two-universal hash function applied to an input with sufficiently high entropy is almost uniformly random. In its standard formulation, the lemma refers to a notion of randomness that is (usually implicitly) defined with respect to classical side information. Here, a strictly more general version of the Leftover Hash Lemma that is valid even if side information is represented by the state of a quantum system is shown. Our result applies to almost two-universal families of hash functions. The generalized Leftover Hash Lemma has applications in cryptography, e.g., for key agreement in the presence of an adversary who is not restricted to classical information processing.

The Leftover Hash Lemma states that the output of a two-universal hash function applied to an input with sufficiently high entropy is almost uniformly random. In its standard formulation, the lemma refers to a notion of randomness that is (usually implicitly) defined with respect to classical side information. Here, we prove a (strictly) more general version of the Leftover Hash Lemma that is valid even if side information is represented by the state of a quantum system. Furthermore, our result applies to arbitrary δ-almost two-universal families of hash functions. The generalized Leftover Hash Lemma has applications in cryptography, e.g., for key agreement in the presence of an adversary who is not restricted to classical information processing.

In classical and quantum information theory, operational quantities such as the amount of randomness that can be extracted from a given source or the amount of space needed to store given data are normally characterized by one of two entropy measures, called smooth min-entropy and smooth max-entropy, respectively. While both entropies are equal to the von Neumann entropy in certain special cases (e.g., asymptotically, for many independent repetitions of the given data), their values can differ arbitrarily in the general case. In this paper, a recently discovered duality relation between (nonsmooth) min- and max-entropies is extended to the smooth case. More precisely, it is shown that the smooth min-entropy of a systemAconditioned on a systemBequals the negative of the smooth max-entropy ofAconditioned on a purifying systemC. This result immediately implies that certain operational quantities (such as the amount of compression and the amount of randomness that can be extracted from given data) are related. We explain how such relations have applications in cryptographic security proofs.

The classical asymptotic equipartition property is the statement that, in the limit of a large number of identical repetitions of a random experiment, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. In this paper, a fully quantum generalization of this property is shown, where both the output of the experiment and side information are quantum. An explicit bound on the convergence is given, which is independent of the dimensionality of the side information. This naturally leads to a family of REacutenyi-like quantum conditional entropies, for which the von Neumann entropy emerges as a special case.