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arXiv:quant-ph/0409157 · AIP conference proceedings

Entanglement in Random Subspaces

Patrick M. Hayden

Published Jan 1, 2004• 6 citations
arXivHTMLPDF

Abstract

The selection of random subspaces plays a role in quantum information theory analogous to the role of random strings in classical information theory. Recent applications have included protocols achieving the quantum channel capacity and methods for extending superdense coding from bits to qubits. In addition, random subspaces have proved useful for studying the structure of bipartite and multipartite entanglement.

DOI
10.1063/1.1834421

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The selection of random subspaces plays a role in quantum information theory analogous to the role of random strings in classical information theory. Recent applications have included protocols achieving the quantum channel capacity and methods for extending superdense coding from bits to qubits. In addition, random subspaces have proved useful for studying the structure of bipartite and multipartite entanglement.
We introduce the notion of entanglement of subspaces as a measure that quantifies the entanglement of bipartite states in a randomly selected subspace. We discuss its properties and in particular we show that for maximally entangled subspaces it is additive. Furthermore, we show that maximally entangled subspaces can play an important role in the study of quantum error-correction codes. We discuss both degenerate and nondegenerate codes and show that the subspace spanned by the logical code words of a nondegenerate code is a kkk-totally- (maximally) entangled subspace. As for non-degenerate codes, we provide a mathematical definition in terms of subspaces and, as an example, we analyze Shor's 9-qubit code in terms of 22 mutually orthogonal subspaces.
Abstract Superdense coding uses entanglement as a resource to communicate classical information efficiently through quantum channels. A superdense coding method is optimal when its capacity reaches Holevo bound. We show that for optimality, maximal entanglement is a necessity across the bipartition of Alice and Bob, but neither absolute nor genuine multipartite entanglement is required. Unlike the previous schemes, which can transmit either even or odd bits of information, we demonstrate a generalized dense coding protocol using the genuine multipartite entangled GHZ state to send arbitrary information bits. Expressed in the eigenbasis of different Pauli operators, GHZ state is characterized by a unique parity pattern which enables us to formulate a security checking technique to ensure absolute security of the protocol. We show this method is better applicable in a scenario, where the initial information is distributed among spatially separated parties. Finally, optimizing the number of qubit(s) sent to Bob, we construct a distributed dense coding method, which completely depicts absolutely secure quantum communication between many to one party.
The purpose of this review is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review and of more detailed examples—coming mainly from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels.
Abstract Superdense coding promises increased classical capacity and communication security but this advantage may be undermined by noise in the quantum channel. We present a numerical study of how forward error correction (FEC) applied to the encoded classical message can be used to mitigate against quantum channel noise. By studying the bit error rate under different FEC codes, we identify the unique role that burst errors play in superdense coding, and we show how these can be mitigated against by interleaving the FEC codewords prior to transmission. We conclude that classical FEC with interleaving is a useful method to improve the performance in near-term demonstrations of superdense coding.
Quantum entanglement stands as one of the most intriguing phenomena in quantum mechanics, defying classical intuition and paving the way for revolutionary advancements in information processing. the fundamentals of quantum entanglement and explores its applications in quantum information processing. Beginning with a concise overview of entanglement theory, we delve into various protocols and techniques that leverage entangled states to perform tasks such as quantum teleportation, superdense coding, and quantum cryptography. Furthermore, we discuss recent developments in the field, including entanglement-based quantum key distribution schemes and entanglement swapping protocols. Through a comprehensive examination of both theoretical principles and experimental implementations, this paper elucidates the pivotal role of quantum entanglement in shaping the landscape of quantum information processing, with implications for secure communication, computational speedup, and future quantum technologies.