Witnessing non-Markovianity with Gaussian quantum steering in a collision model

The nonincreasing feature of temporal quantum steering under a completely positive trace-preserving (CPTP) map, as proposed by Chen et al. [Phys. Rev. Lett. 116, 020503 (2016)], has been considered as a practical measure of non-Markovianity. In this paper we utilize an all-optical scheme to simulate a non-Markovian collision model and to examine how Gaussian steering can be used as a tool for quantifying the non-Markovianity of a structured continuous-variable (CV) Gaussian channel. By modifying the reflectivity of the beam splitters (BSs), we are able to tune the degree of non-Markovianity of the channel. After analyzing the non-Markovian degree of the dissipative channel within two steering scenarios, we discovered that the Gaussian-steering-based non-Markovian measure depends the specific scenario because of the asymmetry of Gaussian steering. We also compared the Gaussian-steering-based non-Markovianity to the one based on violation of divisibility of a CPTP map.

Figure 1

(a) Schematic diagram showing the block-interaction routes of collision model. The collision model consists of the ancilla An, the system $S$, and the environment blocks $E_j$, $j\in [1,L]$. The system is initialized to be entangled with the ancilla in the form of a two-mode squeezed vacuum (TMSV) state. Each collision route contains two separate steps in sequence: Step I describes the interaction that occurs between the system and the $j\mathrm{th}$ environment block; Step II shows the internal interacting process of the environment blocks $E_j$ and $E_{j+1}$. (b) A pictorial illustration of the all-optical scheme for simulating the collisions is shown in (a), which contains BSs for introducing the mixing of optical modes.

Figure 2

The Markovian and non-Markovian regions which are quantified by the Gaussian steering ${\mathcal{N}}_{\text{GS}}\left(L\right)$ in the $r_1\text{−}{r}_2$ plane. The environmental modes are vacuum states. In both panels the dissipation channel acts on the system mode, but for (a) the system can steer the ancilla, and for (b) the system will be steered by the ancilla. The phase shift is always $\phi =0$. The number of collisions in each case is $L=250$.

Figure 3

The Gaussian steerabilities ${\mathcal{G}}^{\to }$ in Eqs. (12) and (13) as functions of the number of steps $L$ in the different scenarios: the first and second rows correspond to the cases that the system can steer the ancilla, ${\mathcal{G}}^{S\to \text{An}}$, and the system can be steered by the ancilla, ${\mathcal{G}}^{\text{An}\to S}$, respectively. We have chosen six sets of parameters, denoted as $P_{\beta }=(r_1,r_2)$, with $\beta =a,b,c,d,e$, and $f$ for each case. The specific parameter settings are $P_a=(0.4,0.3)$, $P_b=(0.5,0.5)$, $P_c=(0.4,0.8)$, $P_d=(0.5,0.9)$, $P_e=(0.8,0.9)$, and $P_f=(0.9,0.9)$, and these parameter points are also have been labeled in Fig. 2.

Figure 4

Variation of the matrix element $|C_{2,2}{|}^2$ with the number of collisions $L$. Each panel (a), (b), and (c) corresponds to the parameter points $P_b$, $P_e$, and $P_f$ of the ${\mathcal{G}}^{S\to \text{An}}$ case in Fig. 3, respectively.

Figure 5

Gaussian steerability ${\mathcal{G}}^{\to }$ as a function of collision time for $r_1=0.4$, $r_2=0.3$.

Figure 6

Gaussian steerability ${\mathcal{G}}^{S\to \text{An}}$ and $|C_{2,2}{|}^2$ (inset) as a function of collision time. The parameters are $r_1=0.75$, $r_2=0.15$, and $\phi =\pi$.

Figure 7

(a) The Markovian and non-Markovian regions in the $r_1-r_2$ plane, which is measured by the violation of the CPTP map divisibility ${\mathcal{N}}_{\text{CPTP}}\left(L\right)$ in Eq. (20). The white dashed line corresponds to the boundary of the Markovian and non-Markovian regions in Fig. 2. The number of collisions is $L=250$. (b.1)–(b.3) The time evolution of Gaussian steering ${\mathcal{G}}^{\text{An}\to S}$ in the different intervals of the number of collisions. (c.1)–(c.3) The variation of ${\nu }_{j,-}$ with the different intervals of the numbers of collision. The environmental optical modes are initialized to be the vacuum states, and the other parameters are specified as $r_1=r_2=0.2$ in (b) and (c).