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The existence of quantum LDPC codes with minimal distance scaling linearly in the number of qubits is a central open problem in quantum information. Despite years of research good quantum LDPC codes are not known to exist, but at the very least it is known they cannot be defined on very regular topologies, like low-dimensional grids. In this work we establish a complementary result, showing that good quantum CSS codes which are sparsely generated require “structure” in the local terms that constrain the code space so as not to be “too-random” in a well-defined sense. To show this, we prove a weak converse to a theorem of Krasikov and Litsyn on weight distributions of classical codes due to which may be of independent interest: subspaces for which the distribution of weights in the dual space is approximately binomial have very few codewords of low weight, tantamount to having a non-negligible “approximate” minimal distance. While they may not have a large minimal non-zero weight, they still have very few words of low Hamming weight. [ABSTRACT FROM AUTHOR]

Shannon's entropy power inequality (EPI) can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule: f (√a X + √1 - a Y) ≥ a f (X) + (1 - a) f (Y) ∀ a ∊ [0,1]. Here, X and Y are continuous random variables and the function f is either the differential entropy or the entropy power. König and Smith [IEEE Trans. Inf. Theory 60(3), 1536-1548 (2014)] and De Palma, Mari, and Giovannetti [Nat. Photonics 8(12), 958-964 (2014)] obtained quantum analogues of these inequalities for continuous-variable quantum systems, where X and Y are replaced by bosonic fields and the addition rule is the action of a beam splitter with transmissivity a on those fields. In this paper, we similarly establish a class of EPI analogues for d-level quantum systems (i.e., qudits). The underlying addition rule for which these inequalities hold is given by a quantum channel that depends on the parameter a ∊ [0,1] and acts like a finite-dimensional analogue of a beam splitter with transmissivity a, converting a two-qudit product state into a single qudit state. We refer to this channel as a partial swap channel because of the particular way its output interpolates between the states of the two qudits in the input as a is changed from zero to one. We obtain analogues of Shannon's EPI, not only for the von Neumann entropy and the entropy power for the output of such channels, but also for a much larger class of functions. This class includes the Rényi entropies and the subentropy. We also prove a qudit analogue of the entropy photon number inequality (EPnI). Finally, for the subclass of partial swap channels for which one of the qudit states in the input is fixed, our EPIs and EPnI yield lower bounds on the minimum output entropy and upper bounds on the Holevo capacity. [ABSTRACT FROM AUTHOR]

Entanglement is a fundamental resource for quantum information processing. In its pure form, it allows quantum teleportation and sharing classical secrets. Realistic quantum states are noisy and their usefulness is only partially understood. Bound-entangled states are central to this question-they have no distillable entanglement, yet sometimes still have a private classical key. We present a construction of bound-entangled states with a private key based on classical probability distributions. From this emerge states possessing a new classical analogue of bound entanglement, distinct from the long-sought bound information. We also find states of smaller dimensions and higher key rates than previously known. Our construction has implications for classical cryptography: we show that existing protocols are insufficient for extracting private key from our distributions due to their "bound-entangled" nature. We propose a simple extension of existing protocols that can extract a key from them. [ABSTRACT FROM AUTHOR]

Local operations with classical communication (LOCC) and separable operations are two classes of quantum operations that play key roles in the study of quantum entanglement. Separable operations are strictly more powerful than LOCC, but no simple explanation of this phenomenon is known. We show that, in the case of von Neumann measurements, the ability to interpolate measurements is an operational principle that sets apart LOCC and separable operations. [ABSTRACT FROM AUTHOR]

Limited quantum memory is one of the most important constraints for near-term quantum devices. Understanding whether a small quantum computer can simulate a larger quantum system, or execute an algorithm requiring more qubits than available, is both of theoretical and practical importance. In this Letter, we introduce cluster parameters K and d of a quantum circuit. The tensor network of such a circuit can be decomposed into clusters of size at most d with at most K qubits of inter-cluster quantum communication. We propose a cluster simulation scheme that can simulate any (K,d)-clustered quantum circuit on a d-qubit machine in time roughly 2O(K), with further speedups possible when taking more fine-grained circuit structure into account. We show how our scheme can be used to simulate clustered quantum systems--such as large molecules--that can be partitioned into multiple significantly smaller clusters with weak interactions among them. By using a suitable clustered ansatz, we also experimentally demonstrate that a quantum variational eigensolver can still achieve the desired performance for estimating the energy of the BeH2 molecule while running on a physical quantum device with half the number of required qubits. [ABSTRACT FROM AUTHOR]