Friday, 6 September 2013

Postulates of Quantum Mechanics: State Measurement

Besides the unitary operators that could be applied to quantum states, a measurement on a quantum state can also be performed. If \(\left|\psi\right>\in\mathcal{H}\), where the dimension of the Hilbert space \(\mathcal{H}\) is \(N\), and \[\left|\psi\right>= \alpha_1\left|1\right>+ \alpha_2 \left|2\right>+ . . . + \alpha_N \left|N\right>,\]
then performing a measurement on \(\left|\psi\right>\) produces some basis state \(\left|i\right>\). Exactly what state \(\left|i\right>\) corresponds to can not usually be completely determined prior to the measurement being made. Instead, the state \(\left|i\right>\) will be observed with a probability given by \(|\alpha_i|^2\). If nothing is done to the state after the measurement has been made then the state will remain in the basis state \(\left|i\right>\). 
 Consider some state \(\left|\psi\right>\in \mathcal{H}_A\otimes \mathcal{H}_B\) of a bipartite system expressed as \[\left|\psi\right>=\displaystyle\sum\limits_{i, j}\alpha_{ij}\left|a_i\right>\left|b_i\right>,\] where the states \(\left|a_i\right>\in \mathcal{H}_A\) form a basis of \(\mathcal{H}_A\) and the states \(\left|b_i\right>\in \mathcal{H}_B\) form a basis of \(\mathcal{H}_B\). Suppose a measurement is performed only on part of the system represented by \(\mathcal{H}_B\) yielding some basis state \(\left|b_k\right>\). Then such a state will be observed with probability \[p(k)=\displaystyle\sum\limits_{lk}|\alpha_{lk}|^2\] leaving the state of the joint system  in the state \[\left|\psi'\right>=\displaystyle\sum\limits_{i}\frac{\alpha_{ik}}{p(k)}\left|a_i\right>\left|b_k\right>.\] Notice here, that the state has been renormalized with the factor \(p(x)\). Measurements, on either an entire quantum system or part of it, will be an essential step that needs to be performed at the end of quantum algorithms in order to extract information from the quantum state. Unlike the natural reversibility of the unitary operators, the act of measurement is an irreversible process, because there is not enough information in the measured quantum state alone to be able to tell exactly what state it was measured from.

 Up until now, there has not been much motivation for why we originally demanded quantum states to be of unit norm with the normalization constraint \[\displaystyle\sum\limits_{i=1}^N |\alpha_i|^2 = 1.\] However, given the probabilistic interpretation described here, this normalization constraint can now be seen to make the probabilities involved in the measurement process obey the standard properties of probability. Thus, if a state of unit norm is measured, we are guaranteed that at least some basis state will be observed after the measurement process.

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