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mtak0114
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1. Show that an arbitrary density operator for a mixed state qubit may be written as
2. \rho = \frac{I+r^i\sigma_i}{2}, where ||r||<1
(Nielsen and Chuang pg 105)
3. So my attempt was as follows
Given that a \rho is hermitian it may be written as a linear combination of the pauli matrices and the identity.
\rho = aI+r^i\sigma_i where r^i is arbitrary
the trace condition tr(\rho)=1 implies a=1/2 and
the positivity condition \langle\varphi|\rho|\varphi\rangle \geq 0
implies that 2cos\theta |r||\langle\varphi|\sigma^i|\varphi\rangle| \geq -1 which implies that
|r|\leq 1/2, fnally redefing r above gives the result QED
is this correct or am i assuming too much?
cheers
Mark
2. \rho = \frac{I+r^i\sigma_i}{2}, where ||r||<1
(Nielsen and Chuang pg 105)
3. So my attempt was as follows
Given that a \rho is hermitian it may be written as a linear combination of the pauli matrices and the identity.
\rho = aI+r^i\sigma_i where r^i is arbitrary
the trace condition tr(\rho)=1 implies a=1/2 and
the positivity condition \langle\varphi|\rho|\varphi\rangle \geq 0
implies that 2cos\theta |r||\langle\varphi|\sigma^i|\varphi\rangle| \geq -1 which implies that
|r|\leq 1/2, fnally redefing r above gives the result QED
is this correct or am i assuming too much?
cheers
Mark
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