Cat state

In quantum computing, the cat state, named after Schrödinger's cat, is the special quantum state where the qubits are in an equal superposition of all being |0> and all being |1>, i.e. (in Bra-ket notation): |00...0> +|11...1>.

In other quantum mechanics contexts, according to The New York Times for example, physicists view the cat state as composed of two diametrically opposed conditions at the same time, such as the possibilities that a cat be alive and dead at the same time. This is sometimes called the many worlds hypothesis. More prosaically, a cat state might be the possibilities that six atoms be spin up and spin down, as published by a team at NIST, December 1, 2005. This spin up/down formulation was proposed by David Bohm, who conceived of spin as an observable in a version of thought experiments formulated in the 1935 EPR paradox. David Bohm and Yakir Aharonov later attained recognition for the Aharonov-Bohm effect, 1959.

Erwin Schrödinger formulated this concept by positing a condition in which one intrinsically does not know something, and that only by performing an observation could one know it. He dramatized the situation by positing a moral choice (bestowing life or death on a cat). Later formulations have de-emphasized the life/death choice by showing the possibility of a cat state which is independent of human action.

In quantum optics

In quantum optics, a cat state is defined as the coherent superposition of two coherent states with opposite phase:

$|\mathrm{cat}_e\rangle \propto|\alpha\rangle+|{-}\alpha\rangle$,

where

$|\alpha\rangle =e^{-{|\alpha|^2\over2}}\sum_{n=0}^{\infty}{\alpha^n\over\sqrt{n!}}|n\rangle$,

and

$|{-}\alpha\rangle =e^{-{|{-}\alpha|^2\over2}}\sum_{n=0}^{\infty}{({-}\alpha)^n\over\sqrt{n!}}|n\rangle$,

are coherent states defined in the number (Fock) basis. Notice that if we add the two states together, the resulting cat state only contains even Fock state terms

$|\mathrm{cat}_e\rangle \propto 2e^{-{|\alpha|^2\over2}}\left({\alpha^0\over\sqrt{0!}}|0\rangle+{\alpha^2\over\sqrt{2!}}|2\rangle+{\alpha^4\over\sqrt{4!}}|4\rangle+\dots\right)$.

As a result of this property, the above cat state is often refereed to as an even cat state. Alternatively, we can define an odd cat state as

$|\mathrm{cat}_o\rangle \propto|\alpha\rangle-|{-}\alpha\rangle$,

which only contains odd Fock states

$|\mathrm{cat}_o\rangle \propto 2e^{-{|\alpha|^2\over2}}\left({\alpha^1\over\sqrt{1!}}|1\rangle+{\alpha^3\over\sqrt{3!}}|3\rangle+{\alpha^5\over\sqrt{5!}}|5\rangle+\dots\right)$.

See also

The reading list contained in Incompleteness of quantum physics.