Quantum information science has shown that models of information pro-cessing devices following quantum principles can execute algorithms that outperform the best known algorithms designed to be executed on mod- els following classical principles. The source of this extra-classical perfor-mance has been a perplexing problem as it has also been shown that some quantum models can be efficiently simulated classically. One such quan-tum model that can be simulated by a classical model is what we call the LOCC model, which makes use of a collection of two-level quantum systems (called qubits) and coordinated operations on individual qubits, a situation commonly called local operations and classical communication (LOCC). It is known that extra-classical performances can be achieved if multi-qubit unitary operations are added to the LOCC model, indicating that these operations must have certain properties that are essential for the gap in performance.

In general, there is a class of information processing tasks that cannot be accomplished when operations are restricted to LOCC. In this thesis, we refer to these tasks as global tasks, and the property of unitary operations that allow them to accomplish these tasks as globalness. The globalness of unitary operations is an important resource for realizing the extra-classical performance in quantum information processing, making it worthy of inves-tigation.

In this thesis, we analyze globalness of unitary operations that are partic-ularly relevant to global tasks dealing with unknown input states by studying three global tasks we call (i) two-piece delocalization, (ii) one-piece delo-calization, and (iii) entanglement-assisted LOCC implementation of global unitary operations.

We begin by explaining our notion of one piece of localized quantum information, which is a state of a d-level quantum system (or qudit) that is guaranteed to be representible by a certain d-dimensional state vector, but the choice of the vector is totally unknown. It is also argued in this chapter why we refer to this state as localized quantum information. Global tasks (i) and (ii) are then introduced. We observe that there are `degrees' of delocalization by considering four LOCC tasks we name one-piece LOCC relocalization, two-piece LOCC relocalization, one-piece LOCC relocation, and two-piece LOCC relocation. Roughly speaking, these LOCC tasks are used to measure how initially localized quantum information is moved out of its original Hilbert space. See Fig. 1 for a schematic representation of each task.

More precisely, we obtain the following three results when two-piece LOCC relocalization and one-piece LOCC relocalization are used. First, we prove that two-piece LOCC relocalization is possible after two-piece delocal-ization, if and only if the delocalizing unitary operation is a tensor product of two local unitary operations. Second, we prove that one-piece LOCC relocalization is possible after two-piece delocalization, if and only if the delocalizing unitary operation is local unitarily equivalent to a controlled-unitary operation. Third, we restrict the delocalizaing unitary operation to be a two-qubit unitary operation and prove that one-piece LOCC relocaliza-tion is possible after one-piece delocalization if the given unitary operation has at most two nonzero Kraus-Cirac coefficients.

As for two-piece LOCC relocation and one-piece LOCC relocation, we obtain the following three results. First, two-piece LOCC relocation is pos-sible after two-piece delocalization, if and only if the delocalizing unitary operation is local unitarily equivalent to the swap operation. Second, it is proven that if one-piece LOCC relocation is possible after one-piece delocal-ization, then the delocalizing unitary operation must be non-invertible under partial transposition. As a corollary, we show that local unitary equivalents of controlled-unitary operations do not give two pieces of delocalized quan-tum information that are one-piece LOCC relocateable. Finally, we give an example of a unitary operation that gives one piece of delocalized quantum information that is one-piece LOCC relocateable.

Before discussing how this analysis leads to a characterization of global-ness of unitary operations, we analyze the third global task, entanglement-assisted LOCC implementation of two-qubit global unitary operations, to evaluate the entanglement resources necessary to deterministically imple-ment any given controlled-unitary operation by LOCC. The minimal amount of entanglement measured in terms of LOCC monotones is obtained. It is proven that for any given two-qubit controlled-unitary operation, its de-terministic entanglement-assisted LOCC implementation requires at least 1 ebit when the resource state is two-qubit. We derive conditions that any LOCC protocol must satisfy when it implements the given controlled-unitary operation. These conditions are used to show that any such protocol can be transformed without loss of generality to a three-turn protocol for which the necessary entanglement resource must be a maximally entangled two-qubit state. This answers a decade long open question in entanglement theory.

Based on these analyses of the three global tasks (i)-(iii), we argue that they naturally lead to characterizations of globalness. The _rst characteri-zation is based on the delocalization power of global unitary operations. We present two ways to rank delocalization power of global unitary operation according to the degree of delocalization each unitary operation brings. The difference in these rankings lies in whether the degree of delocalization is measured by LOCC relocalization or LOCC relocation. The second charac-terization is based on the minimal entanglement cost required to perform a deterministic entanglement-assisted LOCC implementation for a given uni-tary operation. We study relations between these two characterizations.

Moreover, these characterizations, which are all based on gloal tasks on unknown input states, are compared against a characterization based on a global task on known input states. This characterization has been investigated elsewhere and is called entangling power, which is measured by the maximal amount of entanglement that a given unitary operation can generate. After proving a generic statement about entangling power, we find that entangling power is unrelated to the characterizations based on the global tasks on unknown input states. It is argued that the degree of globalness of unitary operations reects the fundamental difference between the known and unknown input states in the task used to characterize the globalness.

Figure 1: Schematics of delocalization, LOCC relocalization, and LOCC relocation. Each small colored box represents a qudit with one piece of quantum information (or, its corresponding Hilbert space, to be precise). Each un_lled small box represents a blank qudit. The dashed arrow A corresponds to two-piece delocalization, while B to one-piece delocalization. The solid arrows pointing down to each box on the bottom represent LOCC. Boxes 1, 2, 3, and 4 correspond to two-piece LOCC relocalization, one-piece LOCC relocalization, one-piece LOCC relocation, and two-piece LOCC relocation, respectively.

本論文は``Characterizing globalness of unitary operations for quantum information processing" (量子情報処理におけるユニタリ演算の非局所性の解析)と題し、5章からなる。近年、物理系が持つ量子性をうまく活用すると、より効率的な情報処理を行うことができる例が発見されつつある。量子性と情報処理能力の向上の関係は完全に理解されているわけではないが、量子的非局所性が情報処理における優位性を得るうえで必要であることは広く予想されている。本論文では量子的非局所性のなかでも、量子操作に内在する非局所性に着目し、「量子情報の非局在化」と「量子操作のエンタングルメントを使ったLOCC実装」と呼んでいる2種類の非局所的なタスク(=非局所的な量子操作なしでは達成できないタスク)を解析しており、これを元に量子操作が持つ非局所性を「非局在化力」および「非局所性のエンタングルメント・コスト」の観点より評価している。

第1章はイントロダクションであり、研究の背景および本論文の主目的について述べている。

第2章では未知量子状態を「量子情報」と定義し、d準位系1個に保存されている未知な純粋量子状態を「1ピース」の量子情報と定義した。また、d準位系が2個以上あるときに、量子情報1ピースが1つのd準位系に保存されているとき、量子情報が「局在化されている」としている。つぎに、d準位系2個に、2ピースまたは1ピースの局在化された量子情報が保存されている状況を考察し、この量子系に2体ユニタリ操作を施すことを「量子情報の非局在化」と定義している。そして、量子情報がユニタリ操作によってどれくらい「非局在化」されたかを評価するために、非局所的な量子操作を用いないタスクを2種類導入し、非局在化の度合いを評価する尺度を提案している。最後に、非局在化の度合いと非局在化に使ったユニタリ操作の関係を調べている。

非局所的な量子操作を用いずに実現できる量子操作のすべてはLOCCと呼ばれるが、第3章では、非局所的な量子操作をLOCCで実装する際に必要なエンタングルメントの量を解析し、2量子ビット上への制御ユニタリ操作は1ebitのエンタングルメントが必要なことを証明している。証明のため、まずLOCCプロトコルが、与えられた制御ユニタリ操作を実装するときに満たすべき条件を積算演算子を導入して求め、nターンの実装プロトコルは必ず3ターンのものに還元できることを示している。また、3ターンでLOCC実装するには必ず1ebit必要なことを証明した。

第4章は、前章までに得られた結果を元に、量子操作の非局所性を評価する2種類の方法を提案し、これまで知られていた評価法と比べている。まず、第2章の結果を用いて、与えられたユニタリ操作がどれくらい量子情報を非局所化できるかを解析し、非局在化に使ったユニタリ操作が持つ「量子情報の非局在化力」をランク付けした。次に、ユニタリ操作が持つ非局所性は一意ではないが、いかなる非局所性を情報処理に活用するにも、必ずユニタリ操作を実装する必要があることに着目し、ユニタリ操作を実装するためのコストをユニタリ操作の非局所性のコストとすることを提案している。そして、第3章で得られた結果を利用して、2量子ビット上の制御ユニタリ操作の非局所性に必要なエンタングルメント・コストは1ebitであることを示した。最後に、これら二つの評価値は、既知の状態を扱ったタスクを用いた非局所性の評価法であるエンタングルメント生成力と全く関係がないことも示されており、未知入力と既知入力の違いが量子操作の非局所性の評価の相違に反映されることが示された。

第5節では、本論文のまとめを行っている。

なお、第2、3、4章の研究は村尾美緒氏、第3章の研究はPeter S. Turner氏との共同研究であるが、論文提出者が主体となって分析および検証を行ったもので、論文提出者の寄与が十分であると判断する。

したがって、博士(理学)の学位を授与できるものと認める。