The effect of quantum memory on quantum speed limit time for CP-(in)divisible channels

Abstract

Quantum speed limit time defines the limit on the minimum time required for a quantum system to evolve between two states. Investigation of bounds on speed limit time of quantum system under non-unitary evolution is of fundamental interest, as it reveals interesting connections to quantum (non-)Markovianity. Here, we discuss the characteristics of quantum speed limit time as a function of quantum memory, quantified as the deviation from temporal self-similarity of quantum dynamical maps for CP-divisible as well as indivisible maps, and show that the presence of quantum memory can speed up quantum evolution. This demonstrates the enhancement of the speed of quantum evolution in the presence of quantum memory for a wider class of channels than indicated by the CP-indivisibility criterion.

Appendix

An MT type bound, in which speed limit is derived in terms of variance of the generator, was obtained as a function of quantum Fisher information for non-unitary evolution [25]. A bound on ${\mathcal {B}}(\rho _{0},\rho _{\tau })$ can be obtained in terms of the integral of the quantum Fisher information $F_{Q}(t)$ along the evolution path. The Bures fidelity ${\mathcal {F}}$ (Eq. 16) is connected to the quantum Fisher information $F_{Q}(t)$ [25],

$$\begin{aligned} {\mathcal {F}}(t,t+dt)=1-(dt)^2 F_{Q}(t)/4+{\mathcal {O}}(dt)^3. \end{aligned}$$

(A.1)

Quantum Fisher information is defined by $F_{Q}(t)=\text {tr}[\rho (t)L^2(t)]$. Here, the Hermitian operator ${\hat{L}}(t)$ is known as the symmetric logarithimic derivative (SLD) operator. It is defined as $d{\hat{\rho }}(t)/dt=({\hat{\rho }}(t){\hat{L}}(t)+{\hat{L}}(t)\hat{\rho (t)})/2$.

Fig. 5

Quantum speed limit $V_{QSL}$ is plotted as a function of the measure of non-Markovianity $\zeta$ for OUN, RTN and NMAD channels. Quantum speed limit $V_{QSL}$ is estimated for the initial states $\frac{1}{\sqrt{2}}[\vert 0\rangle +\vert 1\rangle ]$ for OUN and RTN channels, and $\vert 1\rangle \langle 1\vert$ for NMAD.The channel parameter $\Gamma =0.1\mu$, $\Gamma =\frac{1}{4}\mu$ and $\frac{a}{\mu }=1$ for NMAD, OUN and RTN, respectively, for an actual driving time $\tau =1$

The instantaneous speed of evolution between two time intervals is proportional to the square root of quantum Fisher information. The upper bound on Bures angle is,

$$\begin{aligned} {\mathcal {B}}(\rho _{0},\rho _{\tau })=\arccos (\sqrt{{\mathcal {F}}(\rho _{0},\rho _{\tau })})\le \frac{1}{2}\int _{0}^{\tau }\sqrt{F_{Q}(t)} dt, \end{aligned}$$

(A.2)

and the quantum speed limit can be identified as,

$$\begin{aligned} V_{QSL}=\frac{1}{2}\sqrt{F_{Q}(t)}. \end{aligned}$$

(A.3)

This bound is attained only if the evolution occurs on a geodesic, a condition for the MT bound for unitary evolution under a time-independent Hamiltonian.

We estimate the quantum speed limit $V_{QSL}$ in terms of quantum Fisher information for CP-divisible and CP-indivisible maps, for various initial pure states; $\frac{1}{\sqrt{2}}(\vert 0\rangle +\vert 1\rangle )$ for OUN and RTN channels, and $\vert 1\rangle \langle 1\vert$ for the non-Markovian amplitude damping channel. In Fig. 5, the quantum speed limit is seen to increase as non-Markovianity increases for both CP-divisible and indivisible channels. Thus, the evolution speed of quantum states is seen to increase with the strength of non-Markovianity.