Operator structures and quantum one-way LOCC conditions

Abstract

Abstract. We conduct the first detailed analysis in quantum information of recently derived operator relations from the study of quantum one-way local operations and classical communications (LOCC). We show how operator structures such as operator systems, operator algebras, and Hilbert C * -modules all naturally arise in this setting, and we make use of these structures to derive new results and new derivations of some established results in the study of LOCC. We also show that perfect distinguishability under oneway LOCC and under arbitrary operations is equivalent for several families of operators that appear jointly in matrix and operator theory and quantum information theory.

Optimal resource states for local state discrimination

“…We note that this same result can be shown using the operator system methods in the recent work of Kribs, et al [52].…”

Operator and Graph Theoretic Techniques for Distinguishing Quantum States via One-Way LOCC

“…The communication paradigm called local (quantum) operations and classical communication, usually denoted by its acronym LOCC, is fundamental to quantum information theory, and includes many central topics such as quantum teleportation, data hiding, and many of their derivations [1][2][3]. The somewhat more restricted version called one-way LOCC, in which communicating parties must perform their measurements in a prescribed order, has received expanded attention due it being more tractable mathematically while still capturing many of the more important communication scenarios [4][5][6][7][8][9][10][11][12][13]. A particularly important subclass of problems in this field involves the determination of when sets of known quantum states can be distinguished using only LOCC operations or some subset thereof.…”

“…Our work in the theory of LOCC [10][11][12][13] has for the first time brought techniques and tools from operator theory, operator algebras, and graph theory to the basic theory of quantum state distinguishability in one-way LOCC. Given the overlapping nature of some of our results and applications, including improvements on some results as our work progressed, we felt a review paper bringing together a selection of main features from our work could be a useful contribution to the literature.…”

“…In Section 2, we give necessary preliminaries, including the mathematical description of one-way LOCC in terms of operator relations. Section 3 includes a brief introduction to the relevant operator structures in our analysis, and then we present some of our main results and applications from [10][11][12]. We finish in Section 4 by first giving a brief introduction to our necessary notions from graph theory, then we present one of our main results from [13] and some examples, and we derive a new graph-theoretic description for the case of a single-qubit sender.…”

Non-commutative graphs and quantum error correction for a two-mode quantum oscillator

“…The information is encoded in the density matrix ρ of the system. Given a quantum channel Φ (i.e., a completely positive trace preserving map) which describes information transmission from sender to receiver and includes noise and other destructive for information transmission effects, an error correction code is a set of states in H which can be exactly distinguished after a transmission via the channel Φ. Quantum error correcting codes are actively studied theoretically [7][8][9][10][11][12] and experimentally [13,14]. They can rely on encoding the information in finite-dimensional states of a quantum system, as was originally considered for example in [2], or in infinite-dimensional states as for example using states of a quantum oscillator [7,12].…”