13.1 Decoherence simplified

Consider the following interaction between a qubit and its environment: \begin{aligned} |0\rangle|e\rangle &\longmapsto |0\rangle|e_{00}\rangle \\|1\rangle|e\rangle &\longmapsto |1\rangle|e_{11}\rangle \end{aligned} where |e\rangle, |e_{00}\rangle, and |e_{11}\rangle are the states of the environment, which not need to be orthogonal.255 Now assume that the qubit is initially in some general state |\psi\rangle = \alpha|0\rangle + \beta|1\rangle. The resulting qubit-environment interaction is essentially the environment trying to measure the qubit and, as the result, entangling the two together: \Big( \alpha|0\rangle + \beta|1\rangle \Big) |e\rangle \longmapsto \alpha |0\rangle|e_{00}\rangle + \beta |1\rangle |e_{11}\rangle.

Now we can also write this evolution as \begin{aligned} \Big( \alpha|0\rangle + \beta|1\rangle \Big) |e\rangle \longmapsto & \Big( \alpha|0\rangle + \beta|1\rangle \Big) \frac{|e_{00}\rangle+|e_{11}\rangle}{2} \\+& \Big( \alpha|0\rangle - \beta|1\rangle \Big) \frac{|e_{00}\rangle-|e_{11}\rangle}{2}. \\=& \mathbf{1}|\psi\rangle|e_{\mathbf{1}}\rangle + Z|\psi\rangle|e_Z\rangle, \end{aligned} where |e_{\mathbf{1}}\rangle = \frac{1}{2}(|e_{00}\rangle + |e_{11}\rangle) and |e_Z\rangle = \frac{1}{2}(|e_{00}\rangle - |e_{11}\rangle). We can roughly interpret this expression as saying that two things can happen to the qubit: nothing \mathbf{1} (first term), or phase-flip Z (second term).

This, however, should not be taken literally unless the states of the environment, |e_{\mathbf{1}}\rangle and |e_Z\rangle, are orthogonal.256

This process is what we refer to as decoherence.

  1. The reason we use two indices in |e_{00}\rangle and |e_{11}\rangle will become clear in a moment, when we consider more general interaction with the environment.↩︎

  2. Exercise. Why not?↩︎