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CRC Press eBooks

Quantum Entanglement

Yanhua H. Shih

Published Dec 16, 2020• 0 citations
Publisher

Abstract

In quantum theory, a particle is allowed to be in a state of coherent superposition among a set of orthogonal states. A vivid picture of this concept is the “Schrödinger cat”: a cat is alive and dead simultaneously. This would be no surprise if one means a large number of an ensemble of cats with 50% alive and 50% dead. But, “Schrödinger cat” is a cat. People are talking about a cat being alive and dead simultaneously. In mathematics, the best word to characterize the relationship between states “alive” and “dead” is perhaps “orthogonal”. In quantum mechanics, the superposition of orthogonal states is used to describe the state of a quantum object, or a particle. The superposition principle is indeed a mystery compared to our everyday classical life experience. Beyond the superposition of orthogonal single-particle states, the superposition of orthogonal multi-particle states represents an even more troubling concept in classical theory.

DOI
10.1201/9781003130604-9

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In quantum theory, a particle is allowed to be in a state of coherent superposition among a set of orthogonal states. A vivid picture of this concept is the “Schrödinger cat”: a cat is alive and dead simultaneously. This would be no surprise if one means a large number of an ensemble of cats with 50% alive and 50% dead. But, “Schrödinger cat” is a cat. People are talking about a cat being alive and dead simultaneously. In mathematics, the best word to characterize the relationship between states “alive” and “dead” is perhaps “orthogonal”. In quantum mechanics, the superposition of orthogonal states is used to describe the state of a quantum object, or a particle. The superposition principle is indeed a mystery compared to our everyday classical life experience. Beyond the superposition of orthogonal single-particle states, the superposition of orthogonal multi-particle states represents an even more troubling concept in classical theory.
Abstract In this chapter, the notion of quantum superposition of states is introduced through the example of a polarized photon. This brings out the novel feature that the state of the system depends on how the experiment is set up. The paradoxical consequences of quantum superposition, such as that a cat can be simultaneously dead and alive, are also discussed. This is the essence of the famous Schrödinger’s cat paradox. This description motivates another important consequence of quantum mechanical description of multiple objects; namely, their ability to exist in an entangled state. The properties of the two objects can remain entangled no matter how far away they are from each other and thus have the ability to influence each other. After discussing these aspects of quantum mechanics, the application of quantum entanglement to novel phenomena of quantum teleportation and quantum computing are presented.
In this chapter, the notion of quantum superposition of states is introduced through the example of a polarized photon. This brings out the novel feature that the state of the system depends on how the experiment is set up. The paradoxical consequences of quantum superposition, such as a cat can be simultaneously dead and alive, are also discussed. This is the essence of the famous Schrödinger’s cat paradox. This description motivates another important consequence of quantum mechanical description of the multiple objects, namely, their ability to exist in an entangled state. The properties of the two objects can remain entangled no matter how far away they are from each other and thus have the ability to influence each other. After discussing these aspects of quantum mechanics, the application of quantum entanglement to novel phenomena of quantum teleportation and quantum swapping are presented.
Schrodinger's famous cat has long been misunderstood. According to quantum theory and experiments with entangled systems, an entangled state such as the Schrodinger's cat state is neither a superposition of states of either subsystem nor a superposition of compound states of the composite system, but rather a nonlocal superposition of correlations between pairs of states of the two subsystems. The entangled post-measurement state that results from an ideal measurement is not paradoxical, but is merely a coherent superposition of two statistical correlations at "zero phase angle," i.e. at 100% positive correlation. Thus the state of the radioactive nucleus and Schrodinger's cat is as follows: an undecayed nucleus is 100% positively correlated with an alive cat, and (i.e. superposed with) a decayed nucleus is 100% positively correlated with a dead cat. The superposition consists merely in the fact that both correlations are simultaneously true. Despite many published statements to the contrary, this superposition is not paradoxical. It is in fact what one expects intuitively.
Schrödinger's famous Gedankenexperiment has inspired multiple generations of physicists to think about apparent paradoxes that arise when the logic of quantum physics is applied to macroscopic objects. The development of quantum technologies enabled us to produce physical analogues of Schrödinger's cats, such as superpositions of macroscopically distinct states as well as entangled states of microscopic and macroscopic entities. Here we take one step further and prepare an optical state which, in Schrödinger's language, is equivalent to a superposition of two cats, one of which is dead and the other alive, but it is not known in which state each individual cat is. Specifically, the alive and dead states are, respectively, the displaced single photon and displaced vacuum (coherent state), with the magnitude of displacement being on a scale of 10810^8108 photons. These two states have significantly different photon statistics and are therefore macroscopically distinguishable.
This paper reviews and suggests a resolution of the problem of definite outcomes of measurement. This problem, also known as ‘Schrodinger’s cat’, has long posed an apparent paradox because the state resulting from a measurement appears to be a quantum superposition in which the detector is in two macroscopically distinct states (alive and dead in the case of the cat) simultaneously. Many alternative interpretations of the quantum mathematical formalism, and several alternative modifications of the theory, have been proposed to resolve this problem, but no consensus has formed supporting any one of them. Applying standard quantum theory to the measurement state, together with the analysis and results of decades of nonlocality experiments with pairs of entangled systems, this paper shows the entangled measurement state is not a paradoxical macroscopic superposition of states. It is instead a phase-dependent superposition of correlations between states of the subsystems. Thus Schrodinger’s cat is a non-paradoxical ‘macroscopic correlation’ in which one of the two correlated systems happens to be a detector. This insight resolves the problem of definite outcomes but it does not entirely resolve the measurement problem because the entangled state is still reversible.
Matrices, complex numbers, and tensor products are the holy trinity of quantum computing. During my quantum learning curve on the mathematical background, I wondered as you may: Where is the quantum magic in all this? How can a complex number or a matrix tell me about superposition or entanglement? Where is the spookiness? My immature understanding of superposition was that a system existed in multiple states at the same time: Like the cat in the box, it is dead and alive at once. However that is incorrect, the cat cannot be dead and alive at the same time just like a quantum system cannot exist in two simultaneous states.