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AIP conference proceedings

A principle behind no-cloning and no-imprinting theorems

Masato Koashi

Published Jan 1, 2001• 0 citations
Publisher

Abstract

Consider a situation in which a quantum system is secretly prepared in a state chosen from the known set of n states. We present a principle that gives a definite distinction between the operations that preserve the marginal states of the system and those that do not. Many types of no-cloning and no-imprinting conditions for mixed states and entangled states can easily be derived from this principle. The principle also gives a unified view on how various schemes of quantum cryptography work.

DOI
10.1063/1.1388687

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Consider a situation in which a quantum system is secretly prepared in a state chosen from the known set of n states. We present a principle that gives a definite distinction between the operations that preserve the marginal states of the system and those that do not. Many types of no-cloning and no-imprinting conditions for mixed states and entangled states can easily be derived from this principle. The principle also gives a unified view on how various schemes of quantum cryptography work.
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 no-cloning theorem is a cornerstone of quantum cryptography. Here we generalize and rederive in a unified framework various upper bounds on the maximum achievable fidelity of probabilistic and deterministic cloning machines. Building on ideas by Gisin [Phys. Lett. A 242, 1 (1998)], our result follows from the fact that remote state preparation is possible and the no-signaling principle holds. We apply our general theorem to several subsets of states that are of interest in quantum cryptography.
The no-cloning principle tells us that nonorthogonal quantum states cannot be cloned, but it does not tell us that orthogonal states can always be cloned. We suggest a situation where the cloning transformations are restricted, leading to a novel type of no-cloning principle. In the case of a composite system made of two subsystems: if the subsystems are only available one after the other then there are various cases when orthogonal states cannot be cloned. Surprising examples are given, which give a radically better insight regarding the basic concepts of quantum cryptography.
Modern quantum physics now enables control of quantum systems at the level of individual trajectories, opening a new frontier that links quantum information theory, quantum many-body physics, and quantum thermodynamics, and uncovers novel non-equilibrium phenomena such as deep thermalization and measurement-induced entanglement. However, a central challenge remains: their characterization relies on measuring nonlinear properties of individual quantum states, a task tantamount to fine-grained cloning of a quantum ensemble. Here, the fundamental laws governing the cloning of quantum ensembles are investigated. First, a general no-cloning theorem for arbitrary ensembles is established from an information-theoretic perspective, even assuming multiple copies of the ensemble's purification. It is then shown that this barrier can be unexpectedly circumvented for physical ensembles generated by finite-time evolutions. Nevertheless, these tasks are proven to remain computationally intractable, even when the full circuit description of state preparation is known. This stands in sharp contrast to the conventional no-cloning theorem, which relies on the state being unknown. Together, these results establish new fundamental principles of quantum mechanics, reveal intrinsic trade-offs among sample complexity, computational complexity, and quantum measurements, and highlight the necessity of problem-specific strategies for probing measurement-induced quantum phenomena.