It is known that continuous variable quantum information cannot be protected against naturally occurring noise using Gaussian states and operations only. Noh et al. (PRL 125:080503, 2020) proposed bosonic oscillator-to-oscillator codes relying on non-Gaussian resource states as an alternative, and showed that these encodings can lead to a reduction of the effective error strength at the logical level as measured by the variance of the classical displacement noise channel. An oscillator-to-oscillator code embeds K logical bosonic modes (in an arbitrary state) into N physical modes by means of a Gaussian N-mode unitary and N-K auxiliary one-mode Gottesman-Kitaev-Preskill-states. Here we ask if - in analogy to qubit error-correcting codes - there are families of oscillator-to-oscillator codes with the following threshold property: They allow to convert physical displacement noise with variance below some threshold value to logical noise with variance upper bounded by any (arbitrary) constant. We find that this is not the case if encoding unitaries involving a constant amount of squeezing and maximum likelihood error decoding are used. We show a general lower bound on the logical error probability which is only a function of the amount of squeezing and independent of the number of modes. As a consequence, any physically implementable family of oscillator-to-oscillator codes combined with maximum likelihood error decoding does not admit a threshold.

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.

We correct an intermediate step in the derivation of our main statements in the above-named work. Specifically, inequality (63) in the mentioned paper, intended to give an upper bound on the entropy of certain Gaussian states, is incorrect. In that paper, we used inequality (63) to derive the asymptotic (large-time) scaling of the entropy under the quantum version of the heat equation. We provide alternative derivations of this result, which sidestep the bound (63). The main conclusions of the paper therefore remain unaffected. We thank Giacomo de Palma, Andrea Mari and Vittorio Giovannetti for pointing out this issue, and simultaneously providing us with a resolution in a very detailed communication. We also thank them for their permission to include their discussion in this erratum. We note that one of the alternative derivations of the asymptotic scaling presented here has previously appeared in their publication.

When two independent analog signals, X and Y are added together giving Z=X+Y, the entropy of Z, H(Z), is not a simple function of the entropies H(X) and H(Y), but rather depends on the details of X and Y's distributions. Nevertheless, the entropy power inequality (EPI), which states that e2H(Z)≥ e2H(X)+e2H(Y), gives a very tight restriction on the entropy of Z. This inequality has found many applications in information theory and statistics. The quantum analogue of adding two random variables is the combination of two independent bosonic modes at a beam splitter. The purpose of this paper is to give a detailed outline of the proof of two separate generalizations of the EPI to the quantum regime. Our proofs are similar in spirit to the standard classical proofs of the EPI, but some new quantities and ideas are needed in the quantum setting. In particular, we find a new quantum de Bruijin identity relating entropy production under diffusion to a divergence-based quantum Fisher information. Furthermore, this Fisher information exhibits certain convexity properties in the context of beam splitters.

Let$X_1, \ldots, X_n$be a sequence of$n$classical random variables and consider a sample$X_{s_1}, \ldots, X_{s_r}$of$r \leq n$positions selected at random. Then, except with (exponentially in$r$) small probability, the min-entropy$H_{\min}(X_{s_1} \cdots X_{s_r})$of the sample is not smaller than, roughly, a fraction${r\over n}$of the overall entropy$H_{\min}(X_1 \cdots X_n)$, which is optimal. Here, we show that this statement, originally proved in [S. Vadhan, LNCS 2729, Springer, 2003] for the purely classical case, is still true if the min-entropy$H_{\min}$is measured relative to a quantum system. Because min-entropy quantifies the amount of randomness that can be extracted from a given random variable, our result can be used to prove the soundness of locally computable extractors in a context where side information might be quantum-mechanical. In particular, it implies that key agreement in the bounded-storage model—using a standard sample-and-hash protocol—is fully secure against quantum adversaries, thus solving a long-standing open problem.

In this paper, we show that the conditional min-entropyHmin(A|B) of a bipartite staterhoABis directly related to the maximum achievable overlap with a maximally entangled state if only local actions on theB-part ofrhoABare allowed. In the special case whereAis classical, this overlap corresponds to the probability of guessingAgivenB. In a similar vein, we connect the conditional max-entropyHmax(A|B) to the maximum fidelity ofrhoABwith a product state that is completely mixed onA. In the case whereAis classical, this corresponds to the security ofAwhen used as a secret key in the presence of an adversary holdingB. Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessingAgivenBis a lower bound on the number of uniform secret bits that can be extracted fromArelative to an adversary holdingB.

We address the question whether quantum memory is more powerful than classical memory. In particular, we consider a setting where information about a random n-bit string X is stored in s classical or quantum bits, for s

We introduce a new class of realistic sources of randomness and give concrete procedures for deterministic extraction of almost uniform random bits from these sources. Moreover, we show how randomness can be extracted from general Markov sources. This extends the types of sources for which explicit deterministic randomness extractors are known.

We introduce the concept of selectable knowledge, which models the information stored in an arbitrary (e.g., quantum mechanical) device. We then analyze a situation where an entity A holds selectable knowledge about some random variable X and quantify the information A has about the output H(X) of a randomly chosen function H applied to X. This generalizes the setting of privacy amplification by universal hashing. In particular, our result can be used to prove that privacy amplification remains secure even if the enemy possesses quantum instead of classical information.