Entangling power and operator entanglement of nonunitary quantum evolutions

We propose a method to calculate the operator entanglement and entangling power of a noisy nonunitary operation in terms of linear entropy. By decomposing the Kraus operators of noisy evolution as the sum of products of Pauli matrices, we derive the analytical expression of the operator entanglement for a general nonunitary operation. The definition of entangling power is extended from the ideal unitary operation case to the nonunitary case via a Kraus operator representation and the analytical expression of the entangling power for a general nonunitary operation is derived. To demonstrate the effectiveness of the above method, we investigate the properties of operator entanglement and entangling power of nonunitary operations caused by phase damping noise. Our findings imply that the pure phase damping noise has its own operator entanglement and entangling power, which increase exponentially with time and asymptotically approach their respective upper bounds. In addition, when the phase damping noise is added to an ideal operation, such as an iswap operation or a controlled-$Z$ operation, it can make the operation's entangling power grow exponentially with the strength of noise, but leave its operator entanglement invariant. In this sense, we can conclude that, for a general operation, operator entanglement is a more intrinsic property than entangling power.

Figure 1

(a) Operator entanglement of phase damping noise versus $\tau$. (b) Entangling power of phase damping noise versus $\tau$. Here $\tau =\sqrt{\mathrm{\Gamma }}t$.

Figure 2

(a) Entangling power of the evolution for realizing an iswap gate versus $Jt$ without noise. (b) Entangling power of the evolution for realizing an iswap gate versus $\tau$ in the presence of phase damping noise. Here $\tau =\sqrt{\mathrm{\Gamma }}t$.

Figure 3

(a) Entanglement of the evolution for realizing an iswap gate versus $Jt$ without noise. (b) Entanglement of the evolution for realizing an iswap gate versus $\tau$ in the presence of phase damping noise. Here $\tau =\sqrt{\mathrm{\Gamma }}t$.

Figure 4

(a) Entangling power of the evolution for realizing a controlled-$Z$ gate versus $gt$ without noise. (b) Entangling power of the evolution for realizing a controlled-$Z$ gate versus $\tau$ in the presence of phase damping noise. Here $\tau =\sqrt{\mathrm{\Gamma }}t$.

Figure 5

(a) Entanglement of the evolution for realizing a controlled-$Z$ gate versus $gt$ without noise. (b) Entanglement of the evolution for realizing a controlled-$Z$ gate versus $\tau$ in the presence of phase damping noise. Here $\tau =\sqrt{\mathrm{\Gamma }}t$.