An Isaac Newton Institute Workshop

Quantum Statistics - quantum measurements, estimation and related topics

State estimation on a circle

Authors: Samuel L. Braunstein (The University of York), Sibasish Ghosh (The University of York)


Gisin and Popescu [PRL, 83, 432 (1999)] showed that more information about the direction of the Bloch vector of a pure qubit |\psi(\theta, \phi)> = cos(\theta/2)|0> + exp[i\phi] sin(\theta/2)|1> can be obtained from anti-parallel states |{\Psi}_{11}(\theta, \phi)> = |\psi(\theta, \phi)> X |\psi(\pi-\theta, \pi+\phi)>, compared to parallel states |{\Psi}_{20}(\theta, \phi)> = |\psi(\theta, \phi)> X |\psi(\theta, \phi)>, where (\theta, \phi) is distributed uniformly over [0, \pi] x [0, 2\pi]. As a cause behind this difference, they pointed out the difference between the dimensions of the subspaces spanned by parallel and anti-parallel states separately. When \theta = \pi/2, there is no difference in the amount of information as in that case (and only in that case) exact spin-flip is possible. For any fixed \theta, the dimension of the space spanned by N no. of same and M no. of its orthogonal qubits is (N+M+1). We found that whenever we fix \theta, anti-parallel states always give more information if \theta is different from 0 or \pi/2 or \pi, in the case when we estimate the direction of the Bloch vector of the qubit. We generalized this to the case of N no. of same and M no. of its orthogonal qubits. Here the measurement basis for optimal estimation strategy always turns out to be a quantum Fourier transform. But in the case of estimating the direction of the Bloch vector of the qubit |\psi(\theta, \phi)> = cos(\theta/2)|0> + exp[i\phi]sin(\theta/2)|1>, we found that both the sets S_P(\theta) = {|{\Psi}_{20}(\theta, \phi)> : \phi \in [0, 2\pi]} U {|{\Psi}_{20}(\pi-\theta, \pi+\phi)> : \phi \in [0, 2\pi]} and S_A(\theta) = {|{\Psi}_{11}(\theta, \phi)> : \phi \in [0, 2\pi]} U {|{\Psi}_{11}(\pi-\theta, \pi+\phi)> : \phi \in [0, 2\pi] give the same information.