Quantum Information Processing

Inner-Product Evaluation
The product of a logical bra and a logical ket is evaluated according to the identities
= 1
= 0 ,
= 0 , and
= 1.
It follows that for logical states, if a bra multiplies a ket, the result cancels unless the
states match, in which case the answer is 1. Applying inner-product evaluation to
Equation (15) results in
i3i4i3i i
5 5 5 0
4
5 1
4
+ = + = .
5
To simplify the notation, we can omit one of the two vertical bars in products such as
〈a||b〉 and write 〈a|b〉.
To understand inner-product evaluation, think of the expressions as products of row
and column vectors. For example,
(1 0) ⎛ 0⎞
〈|〉 ↔ ⎜ ⎟ = 0 .
⎝ 1⎠
That is, as vectors, the two states |〉 and |〉 are orthogonal. In general, if |φ〉 and
|ψ〉 are states, then 〈φ|ψ〉 is the inner product, or “overlap,” of the two states. In the
expression for the overlap, we compute 〈φ| from |φ〉 = α|〉 + β|〉 by conjugating
the coefficients and converting the logical kets to bras: 〈φ| = α〈| + β〈|. In the vector
representation, this is the conjugate transpose of the column vector for |φ〉, so the inner
product agrees with the usual one. Two states are orthogonal if their overlap is zero.
We write |φ〉 † = 〈φ| and 〈φ| † = |φ〉.
Every linear operator on states can be expressed with the bra-ket notation. For example,
the bra-ket expression for the noop gate is noop = |〉〈| + |〉〈|. To apply noop to
a qubit, you multiply its state on the left by the bra-ket expression
( ) = ( + ) ( + )
= ( α + β ) + ( α + β )
noop α + β α β
= α + β + α + β
= α 1 + β 0 + α 0 + β 1
= α + β
One way to think about an operator such as |a〉〈b| is to notice that, when it is used to
operate on a ket expression, the 〈b| picks out the matching kets in the state, which are
.
Number 27 2002 Los Alamos Science 11
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Quantum Information Processing