Quantum State Merging for Arbitrarily Small-Dimensional Systems

Abstract

Recent advances in quantum technology facilitate the realization of information processing using quantum computers at least on the small and intermediate scales of up to several dozens of qubits. We investigate entanglement cost required for one-shot quantum state merging, aiming at quantum state transformation on these scales. In contrast to existing coding algorithms achieving nearly optimal approximate quantum state merging on a large scale, we construct algorithms for exact quantum state merging so that the algorithms are applicable to any given state of an arbitrarily-small-dimensional system. In the algorithms, entanglement cost can be reduced depending on a structure of the given state derived from the Koashi-Imoto decomposition. We also provide improved converse bounds for exact quantum state merging achievable for qubits but not necessarily achievable in general. As for approximate quantum state merging, we obtain algorithms and improved converse bounds by applying smoothing to those for exact state merging. Our results are applicable to distributed quantum information processing and multipartite entanglement transformation on small and intermediate scales.Index Terms-Quantum state merging, multipartite entanglement transformation, small and intermediate scale. I. INTRODUCTIONT HE ERA of small-and intermediate-scale quantum computers of up to several dozens of qubits is approaching due to advances in quantum technology. There exists, however, technical difficulty in increasing the number of lownoise qubits built in one quantum device [1]. For further scaling up, distributed quantum information processing using multiple quantum devices connected by a network for quantum communication is considered to be promising [2], [3]. Aimed at efficient quantum information processing, coding algorithms for quantum communication tasks in such a distributed setting should be designed to be suitable for transferring quantum states on these small and intermediate scales.Quantum state merging [4], [5] is a task playing crucial roles in distributed quantum information processing [6]-[8] and multipartite entanglement transformations [9]-[13]. Originally, state merging, or state redistribution [14], [15] as a generalized task including state merging, was introduced in the context of quantum Shannon theory, and it has applied to the analyses of

Hierarchy of quantum operations in manipulating coherence and entanglement

“…An asymptotically non-surviving separation has been recently discovered in Ref. [74] in a communication task of quantum state merging [29,30,72]. Reference [74] discloses this phenomena by proving the difference in the required amount of entanglement resources in achieving one-shot quantum state merging under one-way and two-way LOCC, while this difference ceases to exist in the conventional asymptotic quantum state merging.…”

Spread quantum information in one-shot quantum state merging

“…In this letter, we show that the minimal entanglement cost in state merging under one-way LOCC and that under two-way LOCC can be different in a one-shot scenario, whereas they coincide in the asymptotic scenario [5,6]. State merging and its generalized task, state redistribution [43,44], have also been defined and analyzed in various one-shot scenarios [4,[45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60], as well as other derivatives [61][62][63][64][65][66][67]. In the one-shot scenarios, state merging can be achieved using techniques of one-shot decoupling [49], the convex-split lemma [55], or the Koashi-Imoto decomposition [4].…”

“…State merging and its generalized task, state redistribution [43,44], have also been defined and analyzed in various one-shot scenarios [4,[45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60], as well as other derivatives [61][62][63][64][65][66][67]. In the one-shot scenarios, state merging can be achieved using techniques of one-shot decoupling [49], the convex-split lemma [55], or the Koashi-Imoto decomposition [4]. However, these existing techniques employ only one-way communication similarly to the asymptotic scenario.…”

“…A task of transferring A's part of D−1 l=0 α l |ψ l AB to B without destroying coherence is equivalent to quantum state merging [4] , and a protocol transfers A's part of |ψ RAB to B, keeping coherence between R and B [5,6]. The final state in state merging can be written as |ψ RB B , where B is B's system corresponding to A.…”

One-shot quantum error correction of classical and quantum information

“…In contrast, recent studies develop analyses without taking the asymptotic limit, which is called the one-shot scenarios [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Since one-shot analyses overcome the difficulties of the asymptotic results [40], they are of great importance not only theoretically but also experimentally.…”