Authors: C S Sudheer Kumar
Consider two ensembles of N qubits each: E1 and E2. The ratio Na1/N (Na2/N) is approximately 1/2 (1/2), where Na1 (Na2) is the total number of qubits in E1 (E2) which are in the state |a1> (|a2>), a1=0,1 (a2=+,-). |0>, |1> are eigenkets of Z (Pauli-z matrix), and |+>=(|0>+|1>)/sqrt(2), |->=(|0>-|1>)/sqrt(2). If we cannot address and control each of the N qubits in the ensemble separately (i.e., no local control, but only global control), like in nuclear magnetic resonance (NMR) spin ensembles, then both E1 and E2 are said to be maximally mixed, and hence it is not possible to discriminate between them. This is because, both E1 and E2 give same expectation value of an arbitrary observable. As number of measurements increases, variance of sample mean (of measurement outcomes) decreases, and hence sample mean approaches expectation value of the observable being measured. We are going to show that, if we have local control (like in experiments with single photons), then selectively rotating about x-axis (on Bloch sphere) each of the N qubits in the ensemble by a random angle, reduces variance of sample mean in E1 (this is due to sort of convoluting two independent probability distributions). As random x-rotations does nothing (up to an insignificant global phase) to the states |+>, |->, variance of sample mean remains unaltered in E2. Without random x-rotations, both E1 and E2 give same variance of sample mean. Hence we can discriminate between E1 and E2, via variance of sample mean. We also show that, numerical simulation results support theoretical predictions.
Comments: 17 Pages.
[v1] 2017-02-06 14:22:10
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