A qubit is coupled unitarily to a second qubit (an environment qubit initially…

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · Open quantum systems, decoherence and error correction

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

Problem

A qubit

A

is coupled unitarily to a second qubit

E

(an environment qubit initially prepared in

|0\rangle

) by the two-qubit unitary

U

defined by

U|0\rangle_A|0\rangle_E=|0\rangle_A|0\rangle_E,\qquad U|1\rangle_A|0\rangle_E=|1\rangle_A\left(\cos\tfrac{\theta}{2}|0\rangle_E+\sin\tfrac{\theta}{2}|1\rangle_E\right),

with

\theta=\pi/2

: if

A

|0\rangle

E

is left alone; if

A

|1\rangle

E

is rotated away from

|0\rangle

by angle

\theta

. Immediately afterwards

E

is discarded (traced out). ((a)) Using

K_k=\langle e_k|U|e_0\rangle

with

\{|e_k\rangle\}=\{|0\rangle_E,|1\rangle_E\}

and

|e_0\rangle=|0\rangle_E

, write the two operators

K_0

K_1

for

\theta=\pi/2

2\times2

matrices in the

\{|0\rangle,|1\rangle\}

basis of

A

, and check

K_0^\dagger K_0+K_1^\dagger K_1=I

. ((b)) Qubit

A

starts in

|+\rangle

. Compute

\rho_A'=K_0\rho_A K_0^\dagger+K_1\rho_A K_1^\dagger

with

\rho_A=|+\rangle\langle+|

, and give the resulting Bloch vector. ((c)) State in one sentence what has happened to the coherence of

A

, and compare with the CNOT-onto-environment case, where the off-diagonal entries are erased completely.

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