Standard forms of noisy quantum operations via depolarization

We consider completely positive maps that describe noisy, multiparticle unitary operations. We show that by random single-particle operations the completely positive maps can be depolarized to a standard form with a reduced number of parameters describing the noise process in such a way that the noiseless (unitary) part of the evolution is not altered. A further reduction of the parameters, in many cases even to a single one (i.e., global white noise), is possible by tailoring the decoherence process and increasing the amount of noise. We generalize these results to the dynamical case where the noisy evolution is described by a master equation of Lindblad form, and the noiseless evolution is specified by an interaction Hamiltonian. The resulting standard forms may be used to compute lower bounds on channel capacities, to simplify quantum process tomography or to derive error thresholds for entanglement purification and quantum computation.

Figure 1

In order to obtain the state $E$ the CPM $\mathcal{E}$ is applied to system $\overline{A}$ of the joint system of $A$ and $\overline{A}$, which is prepared in the maximally entangled state $P_{\Phi }^{\overline{A}A}$.

Figure 2

(Color online) Given the state $E$ on the composite system $A^{\prime }$ and $A$, the CPM $\mathcal{E}$ is evaluated for an arbitrary input state $\rho$ by taking $\rho$ as an input at system $\overline{A}$. Then the joint system $A\overline{A}$ is measured in a Bell basis containing the maximally entangled state $P_{\Phi }^{A\overline{A}}$. With probability $1∕d_A^2$ the desired output state $\mathcal{E}\left(\rho \right)$ is then obtained at system $A^{\prime }$.

Figure 3

(Color online) In order to obtain the state $E$ the CPM $\mathcal{E}$ is applied to the systems ${\overline{A}}_i$ of the joint system $\overline{A}=({\overline{A}}_1,\dots ,{\overline{A}}_N)$, which are (locally) prepared in the maximally entangled states $P_{\Phi }^{{\overline{A}}_i{A}_i}$.

Figure 4

(Color online) Given the state $E$ on the composite system $A^{\prime }=(A_1^{\prime },\dots ,A_N^{\prime })$ and $A=(A_1,\dots ,A_N)$, the CPM $\mathcal{E}$ is evaluated for an arbitrary multipartite input state $\rho$ by taking $\rho$ as an input at system $\overline{A}=({\overline{A}}_1,\dots ,{\overline{A}}_N)$. Then the joint systems $A_i{\overline{A}}_i$ are (locally) measured in a Bell basis containing the maximally entangled state $P_{\Phi }^{A_i{\overline{A}}_i}$. With probability $1∕d_A^2$ the desired output state $\mathcal{E}\left(\rho \right)$ is then obtained at system $A^{\prime }$.