Random unitaries
As a first step toward understanding the quantum description of an evolving open system, consider a “two-qubit universe” in which we observe only one of the qubits.
Let’s revisit the controlled-\texttt{NOT} gate, in which two qubits undergo the unitary transformation
U
= |0\rangle\langle 0|\otimes\mathbf{1}+|1\rangle\langle 1|\otimes X
= \begin{bmatrix}\mathbf{1}&0\\0&X\end{bmatrix}
but we’re going to focus on the transformation of the target qubit alone.
We know that it depends on the state of the control qubit:
- if the input state of the control qubit is |0\rangle, the target qubit evolves (unitarily) according to the identity operator \mathbf{1};
- if the input state of the control qubit is |1\rangle, the target qubit evolves (unitarily) according to the bit-flip operator X;
- … but for input states of the control that are superpositions of |0\rangle and |1\rangle the evolution of the target qubit is not unitary.
To justify this last point, note that, if the control qubit is in the state \alpha_0|0\rangle+\alpha_1|1\rangle and the target qubit is in some state |\psi\rangle, then the output state can be written as
\alpha_0|0\rangle\otimes\mathbf{1}|\psi\rangle + \alpha_1|1\rangle\otimes X|\psi\rangle
which shows that the control and the target become entangled.
The target qubit alone ends up in the statistical mixture of states |\psi\rangle with probability |\alpha_0|^2 and X|\psi\rangle with probability |\alpha_1|^2.
We can verify this by expressing the above output state of the two qubits as the density matrix
\begin{aligned}
|\alpha_0|^2|0\rangle\langle 0|\otimes \mathbf{1}|\psi\rangle\langle\psi|\mathbf{1}
\quad &+\quad |\alpha_1|^2|1\rangle\langle 1|\otimes X|\psi\rangle\langle\psi|X
\\+\, \alpha_0\alpha_1^\star|0\rangle\langle 1| \otimes \mathbf{1}|\psi\rangle\langle\psi|X
\quad &+\quad \alpha_0^\star\alpha_1|1\rangle\langle 0| \otimes X|\psi\rangle\langle\psi|\mathbf{1}
\end{aligned}
and then tracing over the control qubit, which gives
|\alpha_0|^2 \mathbf{1}|\psi\rangle\langle\psi|\mathbf{1}+ |\alpha_1|^2 X|\psi\rangle\langle\psi| X.
Then we can say that the input state of the target qubit evolves either according to the identity operator (with probability |\alpha_0|^2) or according to the X operator (with probability |\alpha_1|^2).
This argument works even if the target qubit is initially in a mixed state: we are dealing with a linear transformation, and any mixed state can be expressed as a statistical ensemble of pure states (via the convex decomposition \rho=\sum_i p_i|\psi_i\rangle\langle\psi_i| of a density matrix).
So, in general, we can express the evolution of the target qubit as
\rho\longmapsto
\rho'= |\alpha_0|^2 \mathbf{1}\rho\mathbf{1}+ |\alpha_1|^2 X\rho X
where \rho and \rho' are the input and the output states, respectively.
We may think about this input-output relation as a mathematical representation of a quantum communication channel in which an input qubit is bit-flipped (via the operator X) with some prescribed probability |\alpha_1|^2.
But we may also take a more “global” view and see the action of the channel as arising from a unitary evolution on a larger (dilated) system, here composed of two qubits (namely the target and the control).
Our discussion can easily be extended beyond two qubits to cover any conditional dynamics of the type
U
= \sum_i |i\rangle\langle i|\otimes U_i
= \begin{bmatrix}
U_1 & 0 & 0 & \ldots
\\0 & U_2 & 0 & \ldots
\\0 & 0 & U_3 & \ldots
\\\vdots & \vdots &\vdots & \ddots
\end{bmatrix}
where the vectors |i\rangle form an orthonormal basis in the Hilbert space associated with a control system, and the U_i are the corresponding unitary operations performed on a target system.
If the control system is prepared in state \sum_i\alpha_i|i\rangle and the target in state |\psi\rangle, then the final state of the two systems is
\sum_i \alpha_i|i\rangle\otimes U_i|\psi\rangle
and, by the same sequence of arguments as before, we obtain the evolution of the target system alone, and express it as
\rho\longmapsto
\rho' = \sum_{i=1} |\alpha_i|^2 U_i \rho U^\dagger_i.
That is, the state of the target system is modified by the unitary U_i chosen randomly with probability p_i=|\alpha_i|^2.
The reason we are paying particular attention to random unitaries is that each unitary is invertible, and, as such, offers a sliver of hope for being able to reverse the overall action of the channel.
Indeed, if we can learn, post factum, which particular unitary operation U_i was chosen, then we can simply apply the inverse U_i^{-1}=U_i^\dagger of that unitary and recover the original state.
For example, if we can measure the control system in the |i\rangle basis, then measuring the outcome to be k tells us that we have to apply U_k^\dagger to the target to recover its input state.
However, if we do not have access to the control system, then there is very little we can do: we cannot figure out which particular unitary was applied by inspecting the target system alone.
In this case the best we can do is to apply the inverse of the most likely unitary, which will then recover the input state, but only with some probability of success.
In order to do better than that we have to look at slightly different channels.
First though, a fundamental example of a random unitary evolution:
A single-qubit Pauli channel applies one of the Pauli operators, X, Y or Z, chosen randomly with some prescribed probabilities p_x, p_y and p_z, giving
\rho\longmapsto
p_0 \mathbf{1}\rho\mathbf{1}+ p_x X\rho X+ p_y Y\rho Y+ p_z Z\rho Z.
The Pauli operators represent quantum errors: bit-flip X, phase-flip Z, and the composition of the two Y=iXZ.