Hi there, just doing some basic linear algebra for quantum computation / quantum information theory, and am wondering whether I'm changing the basis of an operator correctly.(adsbygoogle = window.adsbygoogle || []).push({});

If I have two orthogonal basis vectors of space C2 given by (~ = complex conjugate):

S1 = [|0>, |1>]

and S2 = [|u> = a|0> + b|1> and |v> = b~|0> - a~|1>]

(S2 is orthonormal given aa~+bb~=1, easy enough to prove (<u|v>=0))

and the operator, A, given in terms of the basis set S2:

A = |u><u| - |v><v|

(This is from the given fact that A has eigenvectors |u>,|v> with eigenvalues 1,-1 respectively)

To change A into the basis set S1, do I simply do:

A' = UA

where U is the unitary matrix |0><u|+|1><v|

This results in A' = |0><u| - |1><v|

So, if I want to find the probabiliy of a measurement of A on the state |0> I then do:

A'|0> = |0><u|o> - |1><v|0>

As <u|0> = a~ and <v|0> = b this gives

Therefore A'|0> = a~|0> - b|1>

So the probability of this measurement returning 1 is |b|^2

This also means the expectation value of the measurement is 0*p(0)+1*p(1) = 0*|a|^2 + 1*|b|^2 = |b|^2

Is this correct or have I made a mistake somewhere?

Cheers!

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# Operator change of basis (QM / QI)

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