Cat state

In quantum computing, the cat state, named after Schrödinger's cat,[1] is the special pure 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,[2] such as the possibilities that a cat be alive and dead at the same time. This is sometimes connected to the many worlds hypothesis by proponents of the many worlds interpretation of quantum mechanics. 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.[3] 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.[4]

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 referred 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

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

Even and odd coherent states were first introduced by Dodonov, Malkin, and Man'ko in 1974.[5]

Linear superposition of coherent states

A simple example of a cat state is a linear superposition of coherent states with opposite phases, when each state has the same weight:

$|\mathrm{c}\rangle = \frac{1}{\sqrt{2(1+e^{-2|\alpha|^2})}}(|\alpha\rangle+|{-}\alpha\rangle)$
$|\mathrm{c}\rangle = \frac{1}{\sqrt{2(1-e^{-2|\alpha|^2})}}(|\alpha\rangle-|{-}\alpha\rangle)$