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Sunday, 8 September 2013

Measurements in the Bell Basis

The state of a quantum system is generally represented in terms of the classical states that would result after the system is measured. These classical states have the property that they are  classically distinguishable from one another. The computational basis conventionally serves to index these classical states. The postulates of quantum mechanics allows non-classical states to be defined, and these states can be used to construct an arbitrary basis to describe the states in the system. Therefore, there ought to be a way to make measurements in an arbitrary basis. Abstractly speaking, this can always be performed. In practice however, these basis states must be related to the computational basis states in order to make better physical sense of the measurements.

 In the previous section an explicit circuit was constructed that output the Bell states given the computational basis states as input. This circuit can be thought of as performing the unitary operation \(U=(H\otimes I)(c-X)\). Then the inverse operation of \(U\) given by \(U^\dagger=(c-X)^\dagger(H\otimes I)^\dagger\) is the operation that takes as input some Bell state \(\left| \beta_{ij}\right>\) and outputs the corresponding classical basis state \(\left|ij\right>\). Therefore, in order to perform a measurement in the Bell basis it suffices to apply \(U^\dagger\) to some Bell state \(\left| \beta_{ij}\right>\), measure the system, and use information obtained from the observed state \(\left|ij\right>\) to infer \(\left| \beta_{ij}\right>\). This procedure can be generalized to make measurements in an arbitrary basis provided with a circuit that can implement the transformation of the classical basis to the desired one.

 In the figure below, a circuit for performing a measurement in the Bell basis is given along with the Bell measurement circuit to be interpreted as an equivalent circuit. Notice that this circuit before the measurements are performed is precisely the circuit that results from reversing the circuit for the construction of Bell sates. To construct the actual inverse of an arbitrary circuit not containing any measurements, simply reverse the order of the gates and replace each with its conjugate transpose. In the case of the circuit for measuring in the Bell basis, since \(H^\dagger=H\) and \(c-X^\dagger=c-X\), the circuit also happens to be the same circuit that results from merely reversing the order of the gates.

(A circuit for measuring in the Bell basis. This circuit takes some Bell state \(\left|\beta_{ij}\right>\) as input and applies a controlled\(-NOT\) gate \(c-X\) followed and a Hadamard gate \(H\) followed by a measurement to each each register to yield the corresponding computational basis state \(\left|ij\right>\). Shown below is the Bell measurement gate, which will be used as a shorthand symbol that represents the circuit shown above.)

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