Consider two different preparations of a qubit

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · Quantum states, measurement and entanglement

Quantum Information and Technology, Ch Mixed states, generalised measurement and telling states apart

Problem

Consider two different preparations of a qubit. Ensemble

\mathcal{A}

: prepare

|0\rangle

with probability

\tfrac12

and

|1\rangle

with probability

\tfrac12

. Ensemble

\mathcal{B}

: prepare

|+\rangle

with probability

\tfrac12

and

|-\rangle

with probability

\tfrac12

. ((a)) For each ensemble form the two weighted vectors

\sqrt{p_i}|\psi_i\rangle

and assemble them, in the order given above, as the columns of a

2\times2

matrix

W_\mathcal{A}

and

W_\mathcal{B}

. Find a unitary matrix

u

such that

W_\mathcal{B}=W_\mathcal{A}u

. ((b)) State, in the form of the direction of the unitary freedom theorem needed here, why the existence of such a

u

guarantees

\rho_\mathcal{A}=\rho_\mathcal{B}

, and verify this directly by computing both density operators. ((c)) A third ensemble

\mathcal{C}

prepares

|{+i}\rangle

with probability

\tfrac12

and

|{-i}\rangle

with probability

\tfrac12

. Without repeating the calculation in full, say whether any finite number of measurements on qubits drawn from

\mathcal{A}

\mathcal{B}

and

\mathcal{C}

, in any bases chosen by the experimenter, could determine which ensemble was used, and why.

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