We introduce and analyze a task in which a tripartite quantum state is transformed to an approximately recoverable state by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of n copies of a tripartite state ρABC, and is transformed by a random unitary operation on Anto another state, which is approximately recoverable from its reduced state on AnBn(Case 1) or BnCn (Case 2). We analyze the minimum cost of randomness per copy required for the task in an asymptotic limit of infinite copies and vanishingly small error of recovery, mainly focusing on the case of pure states. We prove that the minimum cost in Case 1 is equal to the Markovianizing cost of the state, for which a single-letter formula is known. With an additional requirement on the convergence speed of the recovery error, we prove that the minimum cost in Case 2 is also equal to the Markovianizing cost. Our results have an application for distributed quantum computation.
We introduce and analyze a task in which a tripartite quantum state is transformed to an approximately recoverable state by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of n copies of a tripartite state ρABC, and is transformed by a random unitary operation on Anto another state, which is approximately recoverable from its reduced state on AnBn(Case 1) or BnCn (Case 2). We analyze the minimum cost of randomness per copy required for the task in an asymptotic limit of infinite copies and vanishingly small error of recovery, mainly focusing on the case of pure states. We prove that the minimum cost in Case 1 is equal to the Markovianizing cost of the state, for which a single-letter formula is known. With an additional requirement on the convergence speed of the recovery error, we prove that the minimum cost in Case 2 is also equal to the Markovianizing cost. Our results have an application for distributed quantum computation.
We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of n copies of a tripartite state ρABC, and is transformed to a quantum Markov chain conditioned by Bnwith a small error, by a random unitary operation on An. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a single-letter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is an approximate quantum Markov chain. Our results have an application for distributed quantum computation.
We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is the tensor product of n copies of a tripartite state ρ ABC, and is transformed to a quantum Markov chain conditioned by Bn with a small error, using a random unitary operation on An. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a singleletter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is close to a quantum Markov chain. Our results have an application in analyzing the cost of resources for simulating a bipartite unitary gate by local operations and classical communication.
We introduce and analyze a task that we call symmetrization, in which a state of a quantum system, associated with a symmetry group, is transformed by a random unitary operation to a symmetric state. Each element of the unitary ensemble is required to be symmetry preserving, in the sense that it keeps the set of symmetric states invariant. We consider an asymptotic limit of infinitely many copies and vanishingly small error, and analyze the symmetrizing cost, that is the minimum cost of randomness per copy required for symmetrization. We prove that the symmetrizing cost of an arbitrary quantum state is equal to the relative entropy of frameness, thereby providing it with a direct operational meaning.
Here we address a task denoted as deimaginarity, which is to transform a state into a real state with the aid of random covariant-free unitary operations. We consider the minimum cost of randomness required for deimaginarity in the scenario of infinite copies and suiciently small error, and we prove that the deimaginarity cost of a state ρ is equal to its regularized relative entropy of imaginarity, which can be seen as an operational interpretation of the relative entropy of imaginarity.
Quantum state cloning is the task of approximately producing additional copies of an unknown quantum state from a finite number of input copies. The optimal cloning fidelity is known exactly for pure states, but no comparable characterization is known for general mixed states. We determine the optimal asymptotic copy complexity for d-dimensional states of rank at most r: producing M additional copies with worst-case fidelity at least 1−ε requires and is achievable with N=Θ(Mrd/ε) input copies. Remarkably, the lower bound already holds for a family of states with a fixed flat spectrum, while the matching upper bound is achieved by random purification followed by optimal pure-state cloning. For M=1, we further show that high-fidelity tomography can be coherently converted into cloning with comparable error, revealing an operational origin of the matching cloning and tomography complexities.
Distributed quantum computation is a task in which many distant parties perform a large quantum computation in collaboration with each other, by using classical communication, quantum communication or shared entanglement as resources.In this thesis, we consider one of the simplest nontrivial task of distributed quantum computation, that is, implementation of bipartite unitary by LOCC (local operations and classical communication) assisted by shared entanglement.Here, two input states are distributed over two distant parties.Contrary to previous researches, we consider an asymptotic scenario in which the two parties perform the same bipartite unitary on infinitely many independent input pairs.We analyze the minimum amount of resources required for the task by exploiting concepts and techniques developed in the field of quantum Shannon theory, which has originally been developed to analyze quantum communication.As a tool for analyzing distributed quantum computation, we first introduce a task that we call Markovianization.Markovianization is a task in which a tripartite pure state is transformed to a state called quantum Markov chain, by a randomizing operation on one of the three subsystems.We derive the minimum cost of randomness required for Markovianization (Markovianizing cost) by exploiting a novel decomposition of a Hilbert space introduced in the context of quantum cryptography.We then apply the obtained result to distributed quantum computation.We mainly consider implementation of bipartite unitaries by entanglement-assisted LOCC protocols consisting of three steps, called one-round protocols.This is the first nontrivial case because it is not possible to implement bipartite unitaries by protocols consisting of only one or two steps.Our main result is that we derive the minimum costs of entanglement and classical communication.We show that the minimum costs are given by the Markovianizing cost of a tripartite state associated with the unitary.The result indicates that information about input states of computation is divided into three components: one of which only information of amplitude in a basis is required for computation (information of phase is not), another of which both amplitude and phase information are required, and the other of which no information is involved in computation.Our construction of a protocol that requires the smallest amount of resources is based on this decomposition.We also propose a method to compute the Markovianizing cost of a state associated with the unitary.We also consider two-round protocols, that is, protocols consisting of concatenation of two one-round protocols, for two-qubit controlled unitaries.We show that, for a particular class of two-qubit controlled unitaries, the minimum entanglement cost in two-round protocols is strictly smaller than that in one-round protocols.This result is the first reported case that indicates a property of a trade-off relation between entanglement requirement and the number of turns in entanglement-assisted LOCC tasks.The research in this thesis opens a possibility to build a new bridge between quantum computation theory and quantum communication theory, which are the two largest subfield of quantum information theory, by analyzing a problem in the former using concepts and techniques developed in the latter.