Characterizing Quantum Dynamics with Initial System-Environment Correlations

We fully characterize the reduced dynamics of an open quantum system initially correlated with its environment. Using a photonic qubit coupled to a simulated environment, we tomographically reconstruct a superchannel—a generalized channel that treats preparation procedures as inputs—from measurement of the system alone. We introduce novel quantitative measures for determining the strength of initial correlations, and to allow an experiment to be optimized in regard to its environment.

Figure 1

System dynamics in the presence of an environment. (a) With no initial SE correlations, the reduced dynamics of the system, which interacts unitarily ($U$) with an environment, can be completely reconstructed from tomographically complete sets of input states $\{{\rho }_i\}$ (resulting from preparation procedures $\{{\mathcal{P}}_i\}$) and measurements $\{M_j\}$. (b) The joint SE state may be initially correlated before the state preparation procedure. The superchannel approach encompasses this situation by treating the preparation procedure $P_{ki}$ as the input state to a more general description of the reduced system dynamics.

Figure 2

Experimental setup. System and environment photons are created in the state ${\rho }_{\mathrm{SE}}$ with controllable degree of entanglement, using the source of Ref. [32]. Arbitary preparations ${\mathcal{P}}_{ij}$ on the system and measurements $\{M_j\}$ are implemented by means of polarizers (PBS), quarter- and half-wave plates (QWP, HWP), and single-photon detectors (APD). The joint SE evolution $\mathcal{U}$ is implemented as a CZ gate between a set of HWPs and QWPs. In a case with no initial correlations, this setup implements the target system evolution $U_{\mathrm{S}}$. The CZ gate is based on nonclassical interference at a partially polarizing beam splitter (PPBS) with reflectivities of $r_H=0$ ($r_V=2/3$) for horizontally (vertically) polarized light [40].

Figure 3

(a) Real parts of ${\mathrm{\Lambda }}_{\mathcal{M}}$ for $U_{\mathrm{S}}=H$ in the ideal, uncorrelated case and experimental results for increasingly strong initial correlations. The matrices ${\mathrm{\Lambda }}_{\mathcal{M}}$ are shown in a polarization-Pauli basis, with the elements from left to right corresponding to $\{|H⟩,|V⟩\}\otimes \{\mathcal{I},X,Y,Z\}$ and from front to back corresponding to $\{⟨H|,⟨V|\}\otimes \{\mathcal{I},X,Y,Z\}$. The emergence of a peak corresponding to the identity operation (shown in yellow) is characteristic for the simulated increased tendency of the single-qubit operation $U_{\mathrm{S}}$ (shown in green) to fail in the presence of stronger initial correlations. The negligible imaginary parts are not shown. (b) Real parts of the Choi matrices (shown in the Pauli basis) for $U_{\mathrm{S}}$ obtained via QPT for different choices of preparation procedure in the case of low initial correlation $\tau =0.136$. Cases (i) and (ii) correspond to a fixed ${\rho }_k$ in Fig. 1, (iii) corresponds to ${\rho }_k={\rho }_i$, and (iv) is the case where $1\le k\le 4$. The information contained in the superchannel $\mathcal{M}$ can be used to identify the optimal preparation procedure.

Figure 4

Initial correlation norm $\parallel \mathcal{M}{\parallel }_{\mathrm{IC}}$ vs correlation strength $\tau$ of ${\rho }_{\mathrm{SE}}$ for $U_{\mathrm{S}}=Z$ (blue circles), $U_{\mathrm{S}}=H$ (yellow squares), and $U_{\mathrm{S}}={ZR}_{\mathrm{Y}}$ (green diamonds). The values of $\tau$ were obtained from state tomography of ${\rho }_{\mathrm{SE}}$ for each experiment. The measured real parts of the states with weakest and strongest initial correlations are shown in the respective insets. The solid line corresponds to the IC norm in the ideal case. Error bars from Poissonian counting statistics are on the order of the symbol size.

Figure 5

Optimization of the preparation procedure. The average preparation fidelity $F_{\text{prep}}(\mathcal{M},{\rho }_1,Z)$ for (a) $U_{\mathrm{S}}=Z$ and (b) $U_{\mathrm{S}}={ZR}_{\mathrm{Y}}$ is shown as a density plot on the surface of the Bloch sphere of the initial-projection state ${\rho }_1$. In both cases, we chose the lowest strength of initial correlation realized in the experiment to visualize the effect even for very weak SE correlations.