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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We propose a branch-and-bound algorithm for minimizing a bilinear functional of the form $f(X,Y) = \mathrm {tr}((X\otimes Y)Q)+ \mathrm {tr}(AX)+ \mathrm {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. [ABSTRACT FROM AUTHOR]

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 $o(\log n / n)$ , 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 refid="ref1"/. The proof employs two fundamental principles of modern physics, the no-signaling property of relativity and the uncertainty principle of quantum mechanics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]