Quantum Entanglement and Information

Quantum entanglement is a physical resource, like energy, associated with the peculiar nonclassical correlations that are possible between separated quantum systems. Entanglement can be measured, transformed, and purified. A pair of quantum systems in an entangled state can be used as a quantum information channel to perform computational and cryptographic tasks that are impossible for classical systems. The general study of the information-processing capabilities of quantum systems is the subject of quantum information.

* 1. Quantum Entanglement

* 2. Exploiting Entanglement: Quantum Teleportation

* 3. Quantum Information

* 4. Quantum Cryptography

* 5. Quantum Computation

* 6. Interpretative Remarks

* Bibliography

* Other Internet Resources

* Related Entries

1. Quantum Entanglement

In 1935 and 1936, Schrцdinger published a two-part article in the Proceedings of the Cambridge Philosophical Society in which he discussed and extended a remarkable argument by Einstein, Podolsky, and Rosen. The Einstein-Podolsky-Rosen (EPR) argument was, in many ways, the culmination of Einstein's critique of the orthodox Copenhagen interpretation of quantum mechanics, and was designed to show that the theory is incomplete. (See the entries on the EPR argument and on the Copenhagen interpretation.) In classical mechanics the state of a system is essentially a list of the system's properties -- more precisely, it is the specification of a set of parameters from which the list of properties can be reconstructed: the positions and momenta of all the particles comprising the system (or similar parameters in the case of fields). The dynamics of the theory specifies how properties change in terms of a law of evolution for the state. Pauli characterized this mode of description of physical systems as a 'detached observer' idealization. See Pauli's letter to Born in The Born-Einstein Letters (Born, 1992; p. 218). On the Copenhagen interpretation, such a description is not possible for quantum systems. Instead, the quantum state of a system should be understood as a catalogue of what an observer has done to the system and what has been observed, and the import of the state then lies in the probabilities that can be inferred (in terms of the theory) for the outcomes of possible future observations on the system. Einstein rejected this view and proposed a series of arguments to show that the quantum state is simply an incomplete characterization of the system. The missing parameters are sometimes referred to as 'hidden parameters' or 'hidden variables' (although Einstein did not use this terminology, presumably because he did not want to endorse any particular 'hidden variable' theory).

It should not be supposed that Einstein's definition of a complete theory included the requirement that it be deterministic. Rather, he required certain conditions of separability and locality for composite systems consisting of separated component systems: each component system separately should be characterized by its own properties (even if these properties manifest themselves stochastically), and it should be impossible to alter the properties of a distant system instantaneously (or the probabilities of these properties) by acting on a local system. In later analyses -- notably in Bell's extension of the EPR argument -- it became apparent that these conditions, suitably formulated as probability constraints, are equivalent to the requirement that statistical correlations between separated systems should be reducible to probability distributions over common causes (deterministic or stochastic) in the sense of Reichenbach. (See the entries on Bell's theorem and on Reichenbach's common cause principle.)

In the original EPR article, two particles are prepared from a source in a certain quantum state and then move apart. There are 'matching' correlations between both the positions of the two particles and their momenta: a measurement of either position or momentum on a particular particle will allow the prediction, with certainty, of the outcome of a position measurement or momentum measurement, respectively, on the other particle. These measurements are mutually exclusive: either a position measurement can be performed, or a momentum measurement, but not both simultaneously. Either correlation can be observed, but the subsequent measurement of momentum, say, after establishing a position correlation, will no longer yield any correlation in the momenta of the two particles. It is as if the position measurement disturbs the correlation between the momentum values. The puzzle is that the quantum state of the particle pair is inconsistent with the assignment of labels to the particles separately that could be associated with appropriately correlated values for the outcomes of position and momentum measurements. These labels would be the common causes of the correlations, and would provide an explanation of the correlations in terms of the initial correlations between the properties of the two systems at their source. EPR concluded that the quantum state was incomplete.

Here is how Schrцdinger put the puzzle in the first part of his two-part article (Schrцdinger, 1935; p. 559):

Yet since I can predict either x1 or p1 without interfering with the system No. 1 and since system No. 1, like a scholar in an examination, cannot possibly know which of the two questions I am going to ask first: it so seems that our scholar is prepared to give the right answer to the first question he is asked, anyhow. Therefore he must know both answers; which is an amazing knowledge; quite irrespective of the fact that after having given his first answer our scholar is invariably so disconcerted or tired out, that all the following answers are 'wrong.'

What Schrцdinger showed was that if two particles are prepared in a quantum state such that there is a matching correlation between two 'canonically conjugate' dynamical quantities -- quantities like position and momentum whose values suffice to specify all the properties of a classical system -- then there are infinitely many dynamical quantities of the two particles for which there exist similar matching correlations: every function of the canonically conjugate pair of the first particle matches with the same function of the canonically

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