Quantum Computing Measurement Postulate

Quantum computation and information is based on the postulates of Quantum Mechanics. They are:

  1. State: Each state is represented by a unit vector in a Hilbert space.
  2. Evolutions: The state evolves only by unitary matrices.
  3. Measurement Postulate: Quantum measurements are described by a collection {\{M_m\}} of measurement operators. These are operators acting on the state space of the system being measured.The measurement operators satisfy the completeness eqaution, i.e.,

    \displaystyle \sum_m M_m^\dagger M_m=I.

    The index {m} in {\{M_m\}} refers to the measurement outcome that may occur in the experiment. If {|\psi \rangle} is the state then the probability of outcome {m} is given by

    \displaystyle p(m)=\langle \psi|M_m^\dagger M_m|\psi\rangle.

    The state of after the measurement is given by

    \displaystyle \frac{M_m|\psi\rangle}{\sqrt{\langle \psi|M_m^\dagger M_m|\psi\rangle}}.

Let us take an example here. Let {|\psi \rangle} represent the following picture.

Handwritten 'S'

Now, we want to perfrom an experiment to determine what the picture represents. So, first we need to decide what measurement operators (i.e. to decide what type of output we want) to use. If we think that {|\psi \rangle} is a digit then we might want to use the measurement operators say, {\{N_n\}}, that gives an output from the following set: \{0, 1, 2, 3, … ,9\}. If we think that {|\psi \rangle} is an English alphabet then we use measurement operators say, {\{L_l\}}, that gives an output from the following set:\{a, b, c,…,z\}. The thing to note here is that we can two different set of operators to measure a quantum state, and based on the operators used the results might be different. If we use the former set of operators we are most likely to get the result {\mathbf{5}}. Instead, if we used the set of operators which produces the result in the alphabet we are most likely to the output {\mathbf{S}}. Until now everything looks good, we wanted to find out what the image represented and we can do it using the set of operators described above. This is exactly what an Optical Character Recognition (OCR) system would have done then what’s the difference?

Well, from here onwards the quantum system would behave differently, in the classical system the picture would remain the same but if the picture were really described by a quantum state, {|\psi\rangle}, it will change to some other state {|\phi\rangle}, say. Consider the set of operators {\{N_n\}}. If we think of 0, 1, .., 9 to be orthogonal vectors, then the operators can be the projectors on the respective spaces. In this case if we performed a measurement using {\{N_n\}} and the output was {\mathbf{5}} then resulting vector {|\phi\rangle} will lie in the subspace of {\mathbf{5}}.

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