Show that this dephasing operation is Markovian

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The discussion centers on demonstrating that the given dephasing operation is Markovian, defined by the condition E_g E_h = E_{g+h}. The operation is expressed mathematically, and initial calculations show the composition of the operations leads to a complex expression involving the density matrix ρ and the operator Z. There is a suggestion to simplify the algebra using hyperbolic trigonometric identities, which may clarify the relationship between the operations. A key point raised is the challenge of assuming ρ commutes with Z, as this may not hold true in general cases. The conversation emphasizes the need for careful manipulation of the mathematical expressions to establish the Markovian property.
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Show that this dephasing operation is Markovian
$$E_t(\rho)=\left(\frac{1+e^{-t}}{2}\right)\rho + \left(\frac{1-e^{-t}}{2}\right)Z\rho Z$$

The operation is supposed to be Markovian when
$$E_g E_h = E_{g+h}$$

So this is what I get when I apply this multiplication
$$
E_g E_h=\frac{1+e^{-g}}{2}\frac{1+e^{-h}}{2} \rho^2 + \frac{1-e^{-g}}{2} \frac{1-e^{h}}{2}Z\rho ZZ\rho Z + \frac{1+e^{-g}}{2} \frac{1-e^{-h}}{2} = \rho Z \rho Z + \frac{1+e^{-g}}{2} \frac{1-e^{-h}}{2} Z \rho Z \rho
$$
We can lose ZZ because
$$ZZ=I$$
but I can't see what else can be done. ρ after all only commutes with Z if it's diagonal but we can't assume that for ρ in general?
 
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You have misunderstood the question. ##E_gE_h## means the composition of those functions.
The algebra might be a bit simpler using hyperbolic trig - if you know the formulae for ##\cosh(x+y)## etc.
 
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