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)

See also

The reading list contained in Incompleteness of quantum physics.

Notes

  1. ↑ John Gribbin (1984), In Search of Schrödinger's Cat, ISBN 0-552-12555-5, 22 February 1985, Transworld Publishers, Ltd, 318 pages.
  2. ↑ Dennis Overbye, "Quantum Trickery: Testing Einstein's Strangest Theory". New York Times Tuesday (Science Times), December 27, 2005 pages D1,D4.
  3. ↑ D. Leibfried, E. Knill, S. Seidelin, J. Britton, R.B. Blakestad, J. Chiaverini, D. Hume, W.M. Itano, J.D. Jost, C. Langer, R. Ozeri, R. Reichle, and D.J. Wineland. "Creation of a six atom 'Schrödinger cat' state". Nature. Dec. 1, 2005, 639–642.
  4. ↑ Amir D. Aczel (2001), Entanglement: the unlikely story of how scientists, mathematicians, and philosophers proved Einstein's spookiest theory. ISBN 0-452-28457-0 Penguin: paperback, 284 pages, index.
  5. ↑ V.V. Dodonov, I.A. Malkin, V.I. Man'ko. Physica, Volume 72, Issue 3, 15 March 1974, Pages 597–615.