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Characterizing the minimal communication needed for the quantum channel simulation is a fundamental task in the quantum information theory. In this paper, we show that, under the purified distance, 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. Using the bounds, we also recover the quantum reverse Shannon theorem in a much simpler way.

Weak coin flipping is a cryptographic primitive in which two mutually distrustful parties generate a shared random bit to agree on a winner via remote communication. While a stand-alone secure weak coin flipping protocol can be constructed from noiseless communication channels, its composability has not been explored. In this work, we demonstrate that no weak coin flipping protocol can be abstracted into a black box resource with composable security. Despite this, we also establish the overall stand-alone security of weak coin flipping protocols under sequential composition.

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., Comment: 39 pages, 3 figures

We address the problem of exact and approximate transformation of quantum dichotomies in the asymptotic regime, i.e., the existence of a quantum channel $\mathcal E$ mapping $\rho_1^{\otimes n}$ into $\rho_2^{\otimes R_nn}$ with an error $\epsilon_n$ (measured by trace distance) and $\sigma_1^{\otimes n}$ into $\sigma_2^{\otimes R_n n}$ exactly, for a large number $n$. We derive second-order asymptotic expressions for the optimal transformation rate $R_n$ in the small, moderate, and large deviation error regimes, as well as the zero-error regime, for an arbitrary pair $(\rho_1,\sigma_1)$ of initial states and a commuting pair $(\rho_2,\sigma_2)$ of final states. We also prove that for $\sigma_1$ and $\sigma_2$ given by thermal Gibbs states, the derived optimal transformation rates in the first three regimes can be attained by thermal operations. This allows us, for the first time, to study the second-order asymptotics of thermodynamic state interconversion with fully general initial states that may have coherence between different energy eigenspaces. Thus, we discuss the optimal performance of thermodynamic protocols with coherent inputs and describe three novel resonance phenomena allowing one to significantly reduce transformation errors induced by finite-size effects. What is more, our result on quantum dichotomies can also be used to obtain, up to second-order asymptotic terms, optimal conversion rates between pure bipartite entangled states under local operations and classical communication., Comment: 53 pages, 6 figures, comments welcome

We introduce and analyse approximate quantum secret sharing in a formal cryptographic setting, wherein a dealer encodes and distributes a quantum secret to players such that authorized structures (sets of subsets of players) can approximately reconstruct the quantum secret and omnipotent adversarial agents controlling non-authorized subsets of players are approximately denied the quantum secret. In particular, viewing the map encoding the quantum secret to shares for players in an authorized structure as a quantum channel, we show that approximate reconstructability of the quantum secret by these players is possible if and only if the information leakage, given in terms of a certain entanglement-assisted capacity of the complementary quantum channel to the players outside the structure and the environment, is small., Comment: 6 pages, 1 figure

We study a recommender system for quantum data using the linear contextual bandit framework. In each round, a learner receives an observable (the context) and has to recommend from a finite set of unknown quantum states (the actions) which one to measure. The learner has the goal of maximizing the reward in each round, that is the outcome of the measurement on the unknown state. Using this model we formulate the low energy quantum state recommendation problem where the context is a Hamiltonian and the goal is to recommend the state with the lowest energy. For this task, we study two families of contexts: the Ising model and a generalized cluster model. We observe that if we interpret the actions as different phases of the models then the recommendation is done by classifying the correct phase of the given Hamiltonian and the strategy can be interpreted as an online quantum phase classifier., Comment: 15 pages, 9 figures

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 Mu et al, we show that if certain monotones - namely multivariate extensions of R\'{e}nyi divergences - are strictly ordered between the two tuples, then for sufficiently large $n$, there exists a stochastic matrix taking the $n$-fold Kronecker power of each input distribution to the $n$-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., Comment: 59 pages, 3 figures. Comparing to the earlier version, some typos and terminology were fixed and a further corollary (Corollary 46) was added

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 Renyi $\alpha$-entropies. However, due to the lack of a traditional chain rule for Renyi entropies the picture remained incomplete. In this work we establish the missing link by providing Renyi chain rules connecting different definitions of Renyi entropies by Hayashi and Arimoto. This allows us to provide new information combining bounds for the Arimoto Renyi 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 $\alpha=2$ we give the first optimal information combining bounds with quantum side information., Comment: 14 pages

Smooth Csisz\'ar $f$-divergences can be expressed as integrals over so-called hockey stick divergences. This motivates a natural quantum generalization in terms of quantum Hockey stick divergences, which we explore here. Using this recipe, the Kullback-Leibler divergence generalises to the Umegaki relative entropy, in the integral form recently found by Frenkel. We find that the R\'enyi divergences defined via our new quantum $f$-divergences are not additive in general, but that their regularisations surprisingly yield the Petz R\'enyi divergence for $\alpha < 1$ and the sandwiched R\'enyi divergence for $\alpha > 1$, unifying these two important families of quantum R\'enyi divergences. Moreover, we find that the contraction coefficients for the new quantum $f$ divergences collapse for all $f$ that are operator convex, mimicking the classical behaviour and resolving some long-standing conjectures by Lesniewski and Ruskai. We derive various inequalities, including new reverse Pinsker inequalites with applications in differential privacy and also explore various other applications of the new divergences., Comment: 44 pages. v2: improved results on reverse Pinsker inequalities + minor clarifications

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.

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 show that the proof of the generalised quantum Stein's lemma [Brandão & Plenio, Commun. Math. Phys. 295, 791 (2010)] is not correct due to a gap in the argument leading to Lemma III.9. Hence, the main achievability result of Brandão & Plenio is not known to hold. This puts into question a number of established results in the literature, in particular the reversibility of quantum entanglement [Brandão & Plenio, Commun. Math. Phys. 295, 829 (2010); Nat. Phys. 4, 873 (2008)] and of general quantum resources [Brandão & Gour, Phys. Rev. Lett. 115, 070503 (2015)] under asymptotically resource non-generating operations. We discuss potential ways to recover variants of the newly unsettled results using other approaches.

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

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 both types of error probabilities decrease to zero exponentially fast and, when using adaptive strategies, the rates are characterized by the measured relative entropy between two quantum channels, yielding a strictly larger region than that achievable by non-adaptive strategies. Allowing for quantum memory, we see that the optimal rates are given by the regularized channel relative entropy. Finally, we discuss achievable rates when allowing for repeated measurements via quantum instruments and conjecture that the achievable rate region is not larger than that achievable with POVMs by connecting the result to the strong converse for the quantum channel Stein's Lemma.

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 [1] and later, more explicitly, by Polyanskiy-Poor-Verd\'u [2]. However, the proofs outlined in both works contain gaps that are nontrivial to close. This comment provides an alternative, complete proof of this property., Comment: Submitted to IEEE Transactions on Information Theory on Oct. 27th, 2022. Revised on March 7th, 2023. Accepted on April 30th 2023

Quantum homomorphic encryption, which allows computation by a server directly on encrypted data, is a fundamental primitive out of which more complex quantum cryptography protocols can be built. For such constructions to be possible, quantum homomorphic encryption must satisfy two privacy properties: data privacy which ensures that the input data is private from the server, and circuit privacy which ensures that the ciphertext after the computation does not reveal any additional information about the circuit used to perform it, beyond the output of the computation itself. While circuit privacy is well-studied in classical cryptography and many homomorphic encryption schemes can be equipped with it, its quantum analogue has received little attention. Here we establish a definition of circuit privacy for quantum homomorphic encryption with information-theoretic security. Furthermore, we reduce quantum oblivious transfer to quantum homomorphic encryption. By using this reduction, our work unravels fundamental trade-offs between circuit privacy, data privacy and correctness for a broad family of quantum homomorphic encryption protocols, including schemes that allow only the computation of Clifford circuits.

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 R\'enyi divergence chain rules for channels in the quantum setting. Our results simplify and in some cases generalise previous derivations in the literature., Comment: v2: 6 pages, technical note, will appear at IEEE International Symposium on Information Theory 2022, final version with updated references

Quantum state transfer involves two parties who use pre-shared entanglement and noiseless communication in order to transfer parts of a quantum state. In this work, we quantity the communication cost of one-shot state splitting in terms of the partially smoothed max-information. We then give an analysis of state splitting in the moderate deviation regime, where the error in the protocol goes sub-exponentially fast to zero as a function of the number of i.i.d. copies. The main technical tool we derive is a tight relation between the partially smoothed max-information and the hypothesis testing relative entropy, which allows us to obtain the expansion of the partially smoothed max-information for i.i.d. states in the moderate deviation regime. This then also establishes the moderate deviation analysis for other variants of state transfer such as state merging and source coding.

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., Comment: 32 pages

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 under some additional assumptions., Comment: 31 pages, 4 figures

We develop a framework to extend resource measures from one domain to a larger one. We find that all extensions of resource measures are bounded between two quantities that we call the minimal and maximal extensions. We discuss various applications of our framework. We show that any relative entropy (i.e. an additive function on pairs of quantum states that satisfies the data processing inequality) must be bounded by the min and max relative entropies. We prove that the generalized trace distance, the generalized fidelity, and the purified distance are optimal extensions. And in entanglement theory we introduce a new technique to extend pure state entanglement measures to mixed bipartite states., Comment: 6 pages (main text) + 16 pages (supplemental material), v3, added references and a closed formula for the sandwiched or minimal quantum Renyi divergence with alpha in [0,1/2)

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 must lie between the R\'enyi divergences of order $0$ and $\infty$. We further show simple conditions for positive definiteness of such relative entropies and a characterisation in term of a variant of relative trumping. Our main result is a one-to-one correspondence between entropies and relative entropies.

Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence., Comment: v2: published version, v3: license changed, v4: missing reference fixed

Encoding schemes and error-correcting codes are widely used in information technology to improve the reliability of data transmission over real-world communication channels. Quantum information protocols can further enhance the performance in data transmission by encoding a message in quantum states; however, most proposals to date have focused on the regime of a large number of uses of the noisy channel, which is unfeasible with current quantum technology. We experimentally demonstrate quantum enhanced communication over an amplitude damping noisy channel with only two uses of the channel per bit and a single entangling gate at the decoder. By simulating the channel using a photonic interferometric setup, we experimentally increase the reliability of transmitting a data bit by greater than 20% for a certain damping range over classically sending the message twice. We show how our methodology can be extended to larger systems by simulating the transmission of a single bit with up to eight uses of the channel and a two-bit message with three uses of the channel, predicting a quantum enhancement in all cases., Australian Research Council (Future Fellowship Project FT130101744), Australian Research Council. Centre of Excellence for Engineered Quantum Systems (Project CE110001013)

Quantum hypothesis testing is one of the most basic tasks in quantum information theory and has fundamental links with quantum communication and estimation theory. In this paper, we establish a formula that characterizes the decay rate of the minimal type-II error probability in a quantum hypothesis test of two Gaussian states given a fixed constraint on the type-I error probability. This formula is a direct function of the mean vectors and covariance matrices of the quantum Gaussian states in question. We give an application to quantum illumination, which is the task of determining whether there is a low-reflectivity object embedded in a target region with a bright thermal-noise bath. For the asymmetric-error setting, we find that a quantum illumination transmitter can achieve an error probability exponent stronger than a coherent-state transmitter of the same mean photon number, and furthermore, that it requires far fewer trials to do so. This occurs when the background thermal noise is either low or bright, which means that a quantum advantage is even easier to witness than in the symmetric-error setting because it occurs for a larger range of parameters. Going forward from here, we expect our formula to have applications in settings well beyond those considered in this paper, especially to quantum communication tasks involving quantum Gaussian channels.

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 $r_n$ at which $n$ copies of a resource state $\rho$ can be transformed into $nr_n$ copies of another resource state $\sigma$, and the error level $\varepsilon_n$ of 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 $r_n$ approaches its optimum in the asymptotic limit of unbounded resources ($n\to\infty$), while the error $\epsilon_n$ vanishes 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., Comment: 16 pages, 1 figure. Comments welcome

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., Comment: v2: slightly improved achievability bound for classical state splitting + fixed converse proof for privacy amplification, 25 pages

Heisenberg's uncertainty principle forms a fundamental element of quantum mechanics. Uncertainty relations in terms of entropies were initially proposed to deal with conceptual shortcomings in the original formulation of the uncertainty principle and, hence, play an important role in quantum foundations. More recently, entropic uncertainty relations have emerged as the central ingredient in the security analysis of almost all quantum cryptographic protocols, such as quantum key distribution and two-party quantum cryptography. This review surveys entropic uncertainty relations that capture Heisenberg's idea that the results of incompatible measurements are impossible to predict, covering both finite- and infinite-dimensional measurements. These ideas are then extended to incorporate quantum correlations between the observed object and its environment, allowing for a variety of recent, more general formulations of the uncertainty principle. Finally, various applications are discussed, ranging from entanglement witnessing to wave-particle duality to quantum cryptography., Quantum Internet Division, Quantum Information and Software, QuTech

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.

Heisenberg's uncertainty principle forms a fundamental element of quantum mechanics. Uncertainty relations in terms of entropies were initially proposed to deal with conceptual shortcomings in the original formulation of the uncertainty principle and, hence, play an important role in quantum foundations. More recently, entropic uncertainty relations have emerged as the central ingredient in the security analysis of almost all quantum cryptographic protocols, such as quantum key distribution and two-party quantum cryptography. This review surveys entropic uncertainty relations that capture Heisenberg's idea that the results of incompatible measurements are impossible to predict, covering both finite- and infinite-dimensional measurements. These ideas are then extended to incorporate quantum correlations between the observed object and its environment, allowing for a variety of recent, more general formulations of the uncertainty principle. Finally, various applications are discussed, ranging from entanglement witnessing to wave-particle duality to quantum cryptography., Quantum Internet Division, Quantum Information and Software

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 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 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 $\Upsilon$-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 $\Upsilon$-information is a strong converse bound., Comment: v3: published version; v2: 18 pages, presentation and results improved

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 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 $o(\log n / n)$, 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.

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 Altug and Wagner as well as Polyanskiy and Verdu. 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., Comment: 24 pages, 1 figure. Published version. See also concurrent work by Cheng and Hsieh, arXiv:1701.03195

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. The established pre-factor is essentially 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., Comment: submitted to ISIT 2017

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.

Quantum hypothesis testing is one of the most basic tasks in quantum information theory and has fundamental links with quantum communication and estimation theory. In this paper, we establish a formula that characterizes the decay rate of the minimal Type-II error probability in a quantum hypothesis test of two Gaussian states given a fixed constraint on the Type-I error probability. This formula is a direct function of the mean vectors and covariance matrices of the quantum Gaussian states in question. We give an application to quantum illumination, which is the task of determining whether there is a low-reflectivity object embedded in a target region with a bright thermal-noise bath. For the asymmetric-error setting, we find that a quantum illumination transmitter can achieve an error probability exponent stronger than a coherent-state transmitter of the same mean photon number, and furthermore, that it requires far fewer trials to do so. This occurs when the background thermal noise is either low or bright, which means that a quantum advantage is even easier to witness than in the symmetric-error setting because it occurs for a larger range of parameters. Going forward from here, we expect our formula to have applications in settings well beyond those considered in this paper, especially to quantum communication tasks involving quantum Gaussian channels., Comment: v2: 13 pages, 1 figure, final version published in Physical Review Letters

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

We prove several trace inequalities that extend the Golden-Thompson and the Araki-Lieb-Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb's triple matrix inequality. As an example application of our four matrix extension of the Golden-Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities., Comment: v3: 21 pages, 2 figures, minor changes, published version; v2: 21 pages, 2 figures, minor changes; v1: 20 pages, 2 figures

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 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 several converse bounds on the private transmission capabilities of all phase-insensitive bosonic channels., Comment: v3: 53 pages, 3 figures, final version accepted for publication in IEEE Transactions on Information Theory

Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback-Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki's quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz' conclusion remains true if we allow general positive operator valued measures. Second, we extend the result to Renyi relative entropies and show that for non-commuting states the sandwiched Renyi relative entropy is strictly larger than the measured Renyi relative entropy for $\alpha \in (\frac12, \infty)$, and strictly smaller for $\alpha \in [0,\frac12)$. The latter statement provides counterexamples for the data-processing inequality of the sandwiched Renyi relative entropy for $\alpha < \frac12$. Our main tool is a new variational expression for the measured Renyi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive., Comment: v2: final published version

In the study of Markovian processes, one of the principal achievements is the equivalence between the $\Phi$-Sobolev inequalities and an exponential decrease of the $\Phi$-entropies. In this work, we develop a framework of Markov semigroups on matrix-valued functions and generalize the above equivalence to the exponential decay of matrix $\Phi$-entropies. This result also specializes to spectral gap inequalities and modified logarithmic Sobolev inequalities in the random matrix setting. To establish the main result, we define a non-commutative generalization of the carr\'e du champ operator, and prove a de Bruijn's identity for matrix-valued functions. The proposed Markov semigroups acting on matrix-valued functions have immediate applications in the characterization of the dynamical evolution of quantum ensembles. We consider two special cases of quantum unital channels, namely, the depolarizing channel and the phase-damping channel. In the former, since there exists a unique equilibrium state, we show that the matrix $\Phi$-entropy of the resulting quantum ensemble decays exponentially as time goes on. Consequently, we obtain a stronger notion of monotonicity of the Holevo quantity - the Holevo quantity of the quantum ensemble decays exponentially in time and the convergence rate is determined by the modified log-Sobolev inequalities. However, in the latter, the matrix $\Phi$-entropy of the quantum ensemble that undergoes the phase-damping Markovian evolution generally will not decay exponentially. This is because there are multiple equilibrium states for such a channel. Finally, we also consider examples of statistical mixing of Markov semigroups on matrix-valued functions. We can explicitly calculate the convergence rate of a Markovian jump process defined on Boolean hypercubes, and provide upper bounds of the mixing time on these types of examples.

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 various Renyi 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 variations of Renyi mutual information. In case the alternative hypothesis consists of tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by variations of Renyi conditional mutual information. In either case the relevant notion of Renyi mutual information depends on the precise choice of the alternative hypothesis. As such, our work also strengthens the view that different definitions of Renyi mutual information, conditional entropy and conditional mutual information are adequate depending on the context in which the measures are used., Comment: published version

The quantum capacity of a memoryless channel is often used as a single figure of merit to characterize its ability to transmit quantum information coherently. The capacity determines the maximal rate at which we can code reliably over asymptotically many uses of the channel. We argue that this asymptotic treatment is insufficient to the point of being irrelevant in the quantum setting where decoherence severely limits our ability to manipulate large quantum systems in the encoder and decoder. For all practical purposes we should instead focus on the trade-off between three parameters: the rate of the code, the number of coherent uses of the channel, and the fidelity of the transmission. The aim is then to specify the region determined by allowed combinations of these parameters. Towards this goal, we find approximate and exact characterizations of the region of allowed triplets for the qubit dephasing channel and for the erasure channel with classical post-processing assistance. In each case the region is parametrized by a second channel parameter, the quantum channel dispersion. In the process we also develop several general inner (achievable) and outer (converse) bounds on the coding region that are valid for all finite-dimensional quantum channels and can be computed efficiently. Applied to the depolarizing channel, this allows us to determine a lower bound on the number of coherent uses of the channel necessary to witness super-additivity of the coherent information., Comment: v2: 24 pages, 6 figures, new author, new title, merged with arXiv:1504.05376v1

One of the predominant challenges when engineering future quantum information processors is that large quantum systems are notoriously hard to maintain and control accurately. It is therefore of immediate practical relevance to investigate quantum information processing with limited physical resources, for example to ask: How well can we perform information processing tasks if we only have access to a small quantum device? Can we beat fundamental limits imposed on information processing with classical resources? This book will introduce the reader to the mathematical framework required to answer such questions. A strong emphasis is given to information measures that are essential for the study of devices of finite size, including R\'enyi entropies and smooth entropies. The presentation is self-contained and includes rigorous and concise proofs of the most important properties of these measures. The first chapters will introduce the formalism of quantum mechanics, with particular emphasis on norms and metrics for quantum states. This is necessary to explore quantum generalizations of R\'enyi divergence and conditional entropy, information measures that lie at the core of information theory. The smooth entropy framework is discussed next and provides a natural means to lift many arguments from information theory to the quantum setting. Finally selected applications of the theory to statistics and cryptography are discussed., Comment: 135 pages, partly based on arXiv:1203.2142; v3: published version; v4: typos removed, previous Lemma 3.3 removed; v5: typos removed, new proof for stronger triangle inequality of purified distance

Fawzi and Renner [Commun. Math. Phys. 340(2):575, 2015] 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 Brandao et al. [Phys. Rev. Lett. 115(5):050501, 2015] 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., Comment: v2: 9 pages, published version

A prominent formulation of the uncertainty principle identifies the fundamental quantum feature that no particle may be prepared with certain outcomes for both position and momentum measurements. Often the statistical uncertainties are thereby measured in terms of entropies providing a clear operational interpretation in information theory and cryptography. Recently, entropic uncertainty relations have been used to show that the uncertainty can be reduced in the presence of entanglement and to prove security of quantum cryptographic tasks. However, much of this recent progress has been focused on observables with only a finite number of outcomes not including Heisenberg’s original setting of position and momentum observables. Here, we show entropic uncertainty relations for general observables with discrete but infinite or continuous spectrum that take into account the power of an entangled observer. As an illustration, we evaluate the uncertainty relations for position and momentum measurements, which is operationally significant in that it implies security of a quantum key distribution scheme based on homodyne detection of squeezed Gaussian states.

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 one-way 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 information reconciliation is not generally valid. We propose an improved approximation that takes into account finite key effects and numerically test it against codes for two probability distributions, that we call binary-binary and binary-Gaussian, that typically appear in quantum key distribution protocols.

A variety of new measures of quantum Renyi mutual information and quantum Renyi conditional entropy have recently been proposed, and some of their mathematical properties explored. Here, we show that the Renyi mutual information attains operational meaning 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 decoupled from the environment. Similarly, we establish an operational interpretation of Renyi 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 Renyi mutual information and Renyi conditional entropy., Comment: v3: various fixes, v4: changed proof of Lemma 5, v5: published version

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 post-processing 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., Comment: v4: 13 pages, accepted for publication in IEEE Transactions on Information Theory