論文の公開元へEntanglement theory in distributed quantum information processing
概要
Quantum entanglement is a type of correlation characteristic of quantum mechanics, exhibited in quantum states of a composite quantum system. In quantum information theory, information processing tasks exploiting such quantum mechanical properties are analyzed, so as to quantita- tively characterize consequences of the quantum mechanical properties beyond classical mechanics. This thesis investigates properties of quantum entanglement from the viewpoint of distributed quantum information processing over networks. Recent advances in quantum technology facilitate quantum information processing using quantum devices capable of coherently keeping quantum states of a quantum system inside and of performing low-noise operations for transforming these quantum states. There exists, however, technical difficulty in increasing the number of low-noise qubits built in one quantum device, and hence, the quantum system size of such a quantum device may be limited on small and intermediate scales of up to several dozens of qubits at least in the near future. To achieve large-scale quantum information processing, larger quantum system sizes than those in such small- and intermediate-scale quantum devices are required. For scaling up quantum information processing, distributed quantum information processing is considered to be a promising platform, where larger-scale information processing is achieved using multiple quantum devices connected by a network of quantum channels for communicating quantum information, that is, quantum communication. In contrast to quantum information processing performed by arbitrarily transforming quantum states of a quantum system, the quantum devices in distributed quantum information processing share a composite quantum system whose subsystems are located in each device, and each device is allowed to perform state transformations only on the subsystem in the device. Nonlocal state transformations over different quantum devices are performed by combining these local state transformations in each device with quantum communication.
In distributed quantum information processing, quantum entanglement among the multiple quantum devices serves as a resource for achieving nonlocal information processing tasks using local operations and classical communication (LOCC). If a quantum state has entanglement, the state is called an entangled state, and otherwise called a separable state. To generate arbitrary en- tangled states shared among multiple devices from a separable state, quantum communication for transferring quantum states between the devices is necessary in addition to LOCC, and sufficiently much use of quantum communication allows the quantum devices to perform arbitrary nonlocal transformations of quantum states of the shared composite system. Conversely, if two devices ini- tially share a particular type of bipartite entangled state, quantum communication between these devices can be simulated by a protocol, quantum teleportation, using LOCC assisted by the shared bipartite entanglement. Hence, entanglement serves as a resource assisting LOCC, for achieving nonlocal state transformations over spatially separated quantum systems. While classical commu- nication can be reliably performed using current technology, quantum communication for sharing entanglement is more challenging and costly. In this regard, it is natural to investigate efficient use of entanglement when cost of LOCC is negligible. This approach of regarding entanglement as resources is a fundamental starting point of entanglement theory, which has been successful in establishing operational understanding of bipartite entanglement, i.e., entanglement between two quantum systems. Among bipartite entangled states, convertibility between these entangled states under LOCC establishes partial order of the states in terms of their usability as a resource. This partial order yields quantifications of entanglement in terms of its value as a resource, where it is required that these quantifications are monotonically nonincreasing under LOCC.
Multipartite entanglement, i.e., entanglement among more than two quantum systems, also serves as a resource for multiparty tasks relevant to distributed quantum information processing and ubiquitously appears in many-body quantum systems in condensed matter physics and quan- tum gravity. However, straightforward applications of the above bipartite analysis is not sufficient to characterize multipartite entanglement, because mathematical structures of multipartite entan- gled states are not as simple as bipartite entanglement. Especially, in case of multi-qudit systems whose subsystems are of equal dimension, almost no LOCC transformation among given quantum states of the system is possible, and hence, the paradigm based on the partial order of bipartite entanglement under LOCC does not generalize to multipartite entanglement.
This thesis aims to characterize properties of multipartite entanglement through an operational approach, not only using the framework of LOCC, but using quantum communication networks in addition to LOCC, motivated by distributed quantum information processing. In distributed quantum information processing, a nonlocal transformation of a quantum state shared between two quantum devices can be performed by first transferring a part of the state of one of the devices from this device to the other device, and then performing the transformation locally on the latter device, followed by transferring the state back. While this strategy for performing a nonlocal state transformation is not always the most efficient in terms of a communication cost, this strategy exactly and deterministically achieves the transformation. When two quantum devices, namely, A and B, share a partially unknown quantum state represented by a mixed state ψAB, the communication task of transferring A’s part of this shared state to B is called quantum state merging. In state merging, the other quantum devices than A and B are introduced and are collectively denoted by R. This third party R is called reference, on which A and B cannot perform any operation. More formally, A, B, and R in state merging initially share a pure state ψ RAB representing the partially unknown shared state as a purified state, where A and B know classical description of ψ RAB . The goal of state merging is to transfer A’s part of ψ RAB from A to B and obtain ψ RBJ B , where BJ is B’s system corresponding to A, and coherence between B and R has to be kept. To achieve state merging, A and B performs LOCC assisted by additionally shared bipartite entanglement between A and B for simulating quantum communication by means of quantum teleportation. In quantum state merging, B’s part of the shared state, if correlated with A’s, is called quantum side information. This quantum side information may be used for reducing the required cost of the initially shared entangled resource state for achieving quantum state merging, which is called entanglement cost. Entanglement cost can be regarded as the required amount of quantum communication when the cost of LOCC is negligible.
The former half of this thesis aims to reduce the entanglement cost in quantum state merging. Aiming at transferring quantum information on small and intermediate scales relevant to dis- tributed quantum information processing, this thesis considers one-shot scenarios of state merging, where only a single copy of the shared state is given. While the original formulation of quantum state merging and their successive works are mainly targeted at quantum communication on large scales, protocols aimed at efficient distributed quantum information processing over a network should be designed to be suitable for arbitrarily-small-dimensional quantum systems, especially, on the small and intermediate scales. In contrast to existing protocols achieving nearly optimal one-shot approximate quantum state merging on a large scale, this thesis constructs protocols for one-shot exact quantum state merging so that these protocols are applicable to any given state of an arbitrarily-small-dimensional system. The protocols retain the essential feature of state merging; that is, entanglement cost can be reduced depending on a structure of the given state derived from the Koashi-Imoto decomposition of ψ RAB . This feature arises because the Koashi-Imoto decompo- sition of the given state shows the classical part, the quantum part, and the redundant part of the state, and entanglement can be gained from the redundant part by entanglement distillation, while the classical part can be merged at zero entanglement cost by a measurement followed by classical communication of the measurement outcome. In these protocols, it is crucial to coherently combine different subprocesses, namely, entanglement distillation from the redundant part and quantum teleportation of the quantum part, using controlled measurements and controlled isometries. Also, an improved converse bound for exact quantum state merging is presented, which is proven to be achievable for qubits but not necessarily achievable in general. As for approximate quantum state merging, protocols and improved converse bounds are obtained by applying smoothing to those for exact state merging.
Moreover, to make the most of the quantum side information in one-shot state merging, this thesis proves that B’s preprocessing regarding quantum side information and backward classical communication from B to A can strictly reduce entanglement cost of the protocols. This contrasts with the asymptotic scenario of state merging, where one-way classical communication from A to B suffices to achieve the asymptotically minimal entanglement cost, and these preprocessing and backward communication leading to a two-way LOCC protocol do not contribute to reducing this cost. In the proof, it is crucial to interconnect state merging with another task, local state discrimination, and combine their proof techniques to demonstrate an asymptotically non-surviving separation between one-way communication and two-way communication in terms of entanglement cost in state merging, by showing an instance.
Consequently, the obtained results on state merging complement existing protocols achieving nearly optimal one-shot state merging on a large scale, opening the way to another direction for future research on transferring quantum information on small and intermediate scales.
The latter half of this thesis analyzes multipartite entanglement in the framework of distributed quantum information processing using networks, quantitatively characterizing requirements of en- tanglement cost and quantum system sizes for achieving distributed quantum information pro-cessing. Encoding and decoding quantum information in a multipartite quantum system are indis- pensable quantum state transformations for quantum error correction and also play crucial roles in multiparty tasks in distributed quantum information processing such as quantum secret sharing. To quantitatively characterize nonlocal properties of multipartite state transformations for encoding and decoding, entanglement costs of encoding and decoding quantum information in a multipar- tite quantum system are analyzed, where the multipartite system is distributed among spatially separated parties connected by a network. This analysis generalizes previous studies of entan- glement costs for preparing bipartite and multipartite quantum states and implementing bipartite quantum state transformations by entanglement-assisted LOCC. The analysis identifies conditions for the parties being able to encode or decode quantum information in the distributed quantum system deterministically and exactly, when inter-party quantum communication is restricted to a tree-topology network. In this analysis, the multiparty tasks of implementing the encoding and decoding are reduced to sequential applications of one-shot exact quantum state merging and its inverse task, quantum state splitting, for two parties. While encoding and decoding are inverse tasks of each other, these results suggest that a quantitative difference in entanglement cost between encoding and decoding arises due to the difference between quantum state merging and splitting. In addition, advantage of using multipartite entanglement over bipartite entanglement is estab- lished. Any multipartite entangled state can be generated from appropriately distributed bipartite entangled states by LOCC, and in this sense, any distributed process based on shared multipar- tite entanglement and LOCC is simulatable by using only bipartite entangled states and LOCC. However, it is shown that this reduction scenario does not necessarily hold when there exists a limitation on the size of the local quantum systems. Under such a limitation, it is proven that there exists a set of multipartite quantum states such that these states in the set cannot be prepared from any distribution of bipartite entanglement, while the states can be prepared from a common resource state exhibiting multipartite entanglement. It is also shown that temporal uses of bipar- tite quantum communication resources on a network within a limitation of local system sizes are sufficient for preparing this common resource state exhibiting multipartite entanglement, yet there also exist other states exhibiting multipartite entanglement which cannot be prepared even in this setting. Hence, when the local quantum system sizes are limited, multipartite entanglement is an indispensable resource without which certain processes still cannot be accomplished.
These analyses clarify fundamental limitations and potential applications of distributed quan- tum information processing to characterize properties of quantum entanglement in the small- and intermediate-scale settings and multipartite settings relevant to distributed quantum information processing, providing a paradigm for investigating multipartite entanglement in distributed quan- tum information processing over networks beyond the state convertibility introducing the partial order under LOCC.
