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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Operator change of basis (QM / QI)

**Physics Forums | Science Articles, Homework Help, Discussion**