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Chinese Physics Letters

Why Can We Copy Classical Information?

Yao Shen, Liang Hao, Gui‐Lu Long

Published Jan 1, 2011• 10 citations
Publisher

Abstract

It is pointed out that the noncloning theorem in quantum mechanics also holds for unknown state in linear classical physics. The apparent capability of copying of a classical state is essentially the capability of perfect measurement in classical physics. The difference in copying between quantum and classical physics is the difference in measurement between the two theories. A classical copying process is the combined action of measurement of an unknown state and the preparation of this state onto another system. Hence perfect measurability in classical physics enables the copying of a classical state.

DOI
10.1088/0256-307x/28/1/010306

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It is pointed out that the noncloning theorem in quantum mechanics also holds for unknown state in linear classical physics. The apparent capability of copying of a classical state is essentially the capability of perfect measurement in classical physics. The difference in copying between quantum and classical physics is the difference in measurement between the two theories. A classical copying process is the combined action of measurement of an unknown state and the preparation of this state onto another system. Hence perfect measurability in classical physics enables the copying of a classical state.
Copying the quantum states is contradictory to classical information processing since the fundamental difference between classical and quantum information is that while classical information can be copied perfectly, quantum information cannot. However, this statement does not rule out the risk of building a device that can reproduce a set of quantum states. This paper investigates the naturally arising question of how well or under what conditions one can copy and measure an arbitrary quantum superposition of states. The CNOT and XOR operation-based quantum circuit is presented that exhibits entanglement of states and allows for measuring the doubled qubits.
Quantum information theory is a wide area of investigation, which includes two main subjects: quantum computation and quantum communication. The success of quantum computation theory is due to a positive result: Shor's algorithm. The algorithm provides factorization of integers in a much more efficient way with respect to what is achieved not only by present-day computers, but even by the most powerful conceivable computers based on classical physics. On the other hand, the success of quantum communication is due to the intrinsic security of ideal quantum cryptographic protocols, and thus relies on a negative result, that is the impossibility of reading quantum information without scrambling it in an unrecoverable way. The no information without disturbance property spawns other no-go theorems, which represent some of the most famous and classic results in quantum information theory. Among these results one can certainly list the no-cloning theorem, stating the impossibility of copying quantum information. This impossibility is clearly a consequence of the impossibility of extracting information without disturbing it. Indeed, if we could copy exactly the state of a system, reproducing it on another system of the same type, then we would also be able to iterate this process, and produce as many copies as we want. We could then extract all the information we are interested in from the copies, keeping the original intact. This argument is so simple, but hides a very deep point about quantum information, and more generally about any type of information that can be purified. Suppose indeed that there exists a measurement allowing us to determine the state of a system, with absolute precision. We could then use such a measurement on the system, and once we know its exact state, we can in principle re-prepare the same state as many times as we want. If copying the state is forbidden within a theory, we must then conclude that, in that theory, measuring the state is forbidden, too! This is exactly the case in theories with purification, and in particular in quantum theory. In quantum theory, determining the state of a system is impossible. We can only have some information about what state our system is prepared in. The information may be accurate, but it will never be certain. This could be practically the case in classical theory, too.
A common way of stating the non-cloning theorem -- one of distinguishing characteristics of quantum theory -- is that one cannot make a copy of an arbitrary unknown quantum state. Even though this theorem is an important part of the ongoing discussion of the nature of a quantum state, the role of the theorem in the logical-algebraic approach to quantum theory has not yet been systematically studied. According to the standard point of view (which is in line with the logical tradition), quantum cloning amounts to two classical rules of inference, namely, monotonicity and idempotency of entailment. One can conclude then that the whole of quantum theory should be described through a logic wherein these rules do not hold, which is linear logic. However, in accordance with a supervaluational semantics (that allows one to retain all the theorems of classical logic while admitting `truth-value gaps'), quantum cloning necessitates the permanent loss of the truth values of experimental quantum propositions which violates the unalterability of the past. The present paper demonstrates this.