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Physical Review A

Operations that do not disturb partially known quantum states

Masato Koashi, Nobuyuki Imoto

Published Aug 30, 2002• 70 citations
Publisher

Abstract

Consider a situation in which a quantum system is secretly prepared in a state chosen from the known set of states. We present a principle that gives a definite distinction between the operations that preserve the states of the system and those that disturb the states. The principle is derived by alternately applying a fundamental property of classical signals and a fundamental property of quantum ones. The principle can be cast into a simple form by using a decomposition of the relevant Hilbert space, which is uniquely determined by the set of possible states. The decomposition implies the classification of the degrees of freedom of the system into three parts depending on how they store the information on the initially chosen state: one storing it classically, one storing it nonclassically, and the other one storing no information. Then the principle states that the nonclassical part is inaccessible and the classical part is read-only if we are to preserve the state of the system. From this principle, many types of no-cloning, no-broadcasting, and no-imprinting conditions can easily be derived in general forms including mixed states. It also gives a unified view on how various schemes of quantum cryptography work. The principle helps one to derive optimum amount of resources (bits, qubits, and ebits) required in data compression or in quantum teleportation of mixed-state ensembles.

DOI
10.1103/physreva.66.022318

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Consider a situation in which a quantum system is secretly prepared in a state chosen from the known set of states. We present a principle that gives a definite distinction between the operations that preserve the states of the system and those that disturb the states. The principle is derived by alternately applying a fundamental property of classical signals and a fundamental property of quantum ones. The principle can be cast into a simple form by using a decomposition of the relevant Hilbert space, which is uniquely determined by the set of possible states. The decomposition implies the classification of the degrees of freedom of the system into three parts depending on how they store the information on the initially chosen state: one storing it classically, one storing it nonclassically, and the other one storing no information. Then the principle states that the nonclassical part is inaccessible and the classical part is read-only if we are to preserve the state of the system. From this principle, many types of no-cloning, no-broadcasting, and no-imprinting conditions can easily be derived in general forms including mixed states. It also gives a unified view on how various schemes of quantum cryptography work. The principle helps one to derive optimum amount of resources (bits, qubits, and ebits) required in data compression or in quantum teleportation of mixed-state ensembles.
Consider a situation in which a quantum system is secretly prepared in a state chosen from the known set of states. We present a principle that gives a definite distinction between the operations that preserve the states of the system and those that disturb the states. The principle is derived by alternately applying a fundamental property of classical signals and a fundamental property of quantum ones. The principle can be cast into a simple form by using a decomposition of the relevant Hilbert space, which is uniquely determined by the set of possible states. The decomposition implies the classification of the degrees of freedom of the system into three parts depending on how they store the information on the initially chosen state: one storing it classically, one storing it nonclassically, and the other one storing no information. Then the principle states that the nonclassical part is inaccessible and the classical part is read-only if we are to preserve the state of the system. From this principle, many types of no-cloning, no-broadcasting, and no-imprinting conditions can easily be derived in general forms including mixed states. It also gives a unified view on how various schemes of quantum cryptography work. The principle helps to derive optimum amount of resources (bits, qubits, and ebits) required in data compression or in quantum teleportation of mixed-state ensembles.
The engine that powers quantum cryptography is the principle that there are no physical means for gathering information about the identity of a quantum system's state (when it is known to be prepared in one of a set of nonorthogonal states) without disturbing the system in a statistically detectable way. This situation is often mistakenly described as a consequence of the ``Heisenberg uncertainty principle.'' A more accurate account is that it is a unique feature of quantum phenomena that rests ultimately on the Hilbert space structure of the theory along with the fact that time evolutions for isolated systems are unitary. In this paper I explore several aspects of the ``information / disturbance principle'' in an attempt to make it firmly quantitative for both pure and mixed states. The final section briefly explores the extent to which such a principle can be taken as a foundation for unitary dynamics rather than as a consequence.
Abstract—Remote state preparation is the variant of quantum state teleportation in which the sender knows the quantum state to be communicated. The original paper introducing teleportation established minimal requirements for classical communication and entanglement but the corresponding limits for remote state preparation have remained unknown until now: previous work has shown, however, that it not only requires less classical communication but also gives rise to a tradeoff between these two resources in the appropriate setting. We discuss this problem from first principles, including the various choices one may follow in the definitions of the actual resources. Our main result is a general method of remote state preparation for arbitrary states of many qubits, at a cost of 1 bit of classical communication and 1 bit of entanglement per qubit sent. In this “universal ” formulation, these ebit and cbit requirements are shown to be simultaneously optimal by exhibiting a dichotomy. Our protocol then yields the exact tradeoff curve for memoryless sources of pure states (including the case of incomplete knowledge of the ensemble probabilities), based on the recently established quantum-classical tradeoff for visible quantum data compression. A variation of that method allows us to solve the even more general problem of preparing entangled states between sender and receiver (i.e., purifications of mixed state ensembles). The paper includes an extensive discussion of our results, including the impact of the choice of model on the resources, the topic of obliviousness, and an application to private quantum channels and quantum data hiding. Index Terms—Cryptography, entanglement, large deviations, teleportation, tradeoff. I.
The well-known duality relating entangled states and noisy quantum channels is expressed in terms of a channel ket, a pure state on a suitable tripartite system, which functions as a pre-probability allowing the calculation of statistical correlations between, for example, the entrance and exit of a channel, once a framework has been chosen so as to allow a consistent set of probabilities. In each framework the standard notions of ordinary (classical) information theory apply, and it makes sense to ask whether information of a particular sort about one system is or is not present in another system. Quantum effects arise when a single pre-probability is used to compute statistical correlations in different incompatible frameworks, and various constraints on the presence and absence of different kinds of information are expressed in a set of all-or-nothing theorems which generalize or give a precise meaning to the concept of ``no-cloning.'' These theorems are used to discuss the location of information in quantum channels modeled using a mixed-state environment, the classical-quantum channels introduced by Holevo, and the location of information in the physical carriers of a quantum code. It is proposed that both channel and entanglement problems be classified in terms of pure states (functioning as pre-probabilities) on systems of $p\ensuremath{\geqslant}2$ parts, with mixed bipartite entanglement and simple noisy channels belonging to the category p=3p=3p=3, a five-qubit code to the category p=6p=6p=6, etc., then by the dimensions of the Hilbert spaces of the component parts, along with other criteria yet to be determined.
The engine that powers quantum cryptography is the principle that there are no physical means for gathering information about the identity of a quantum system's state (when it is known to be prepared in one of a set of nonorthogonal states) without disturbing the system in a statistically detectable way. This situation is often mistakenly described as a consequence of the “Heisenberg uncertainty principle.” A more accurate account is that it is a unique feature of quantum phenomena that rests ultimately on the Hilbert space structure of the theory along with the fact that time evolutions for isolated systems are unitary. In this paper we shall explore several aspects of the information-disturbance principle in an attempt to make it firmly quantitative and flesh out its significance for quantum theory as a whole.