Experimental Demonstration of Instrument-Specific Quantum Memory Effects and Non-Markovian Process Recovery for Common-Cause Processes

The duration, strength, and structure of memory effects are crucial properties of physical evolution. Because of the invasive nature of quantum measurement, such properties must be defined with respect to the probing instruments employed. Here, using a photonic platform, we experimentally demonstrate this necessity via two paradigmatic processes: future-history correlations in the first process can be erased by an intermediate quantum measurement; for the second process, a noisy classical measurement blocks the effect of history. We then apply memory truncation techniques to recover an efficient description that approximates expectation values for multitime observables. Our proof-of-principle analysis paves the way for experiments concerning more general non-Markovian quantum processes and highlights where standard open systems techniques break down.

Figure 1

Process schematic. Parts of an initial state ${\gamma }_{\mathrm{ABC}}$ are sent to Alice, Bob, and Charlie in a time-ordered fashion. Alice and Bob’s outputs are discarded, corresponding to an identity matrix in the Choi representation, where each time step has two Hilbert spaces (input and output). The resulting process tensor is ${\mathrm{\Gamma }}_{\mathrm{ABC}}={\gamma }_{{\mathrm{A}}^i{\mathrm{B}}^i{\mathrm{C}}^i}\otimes {\mathbb{1}}_{{\mathrm{A}}^o{\mathrm{B}}^o}$.

Figure 2

Experimental setup. An entangled photon pair is generated via spontaneous parametric down-conversion (SPDC) at ppKTP1 in a Sagnac interferometer pumped by a 404 nm laser. For Process 1, the polarization acts as history and memory and path qubits as future and herald. For Process 2, an additional SPDC at ppKTP2 constructs a third level. Conditional Alice-Charlie correlations for each of Bob’s outcomes encode memory effects. Inset: Bob’s measurement apparatus [58, 59]. Abbreviations: ODL—optical delay line; HWP—half-wave plate; QWP—quarter-wave plate; RM/BS—reflection mirror/beam splitter; PBS—polarizing beam splitter; FC—fiber coupler; SPD—single photon detector; and ppKTP—periodically poled potassium titanyl phosphate.

Figure 3

Non-Markovianity and memory strength of (a) Process 1 and (b) Process 2. Experimental results for the non-Markovianity $\mathcal{N}$ (black, left), the (nonvanishing) memory strength for (a) the computational-basis measurement ${\mathcal{Z}}_{\mathrm{B}}=\{{\mathcal{Z}}_{\mathrm{B}}^{(x)}\}$, and (b) POVM ${\mathrm{\Pi }}_{\mathrm{B}}=\{{\mathrm{\Pi }}_{\mathrm{B}}^{(x)}\}$ (red, middle) [Eq. (4)], and the (vanishing) memory strength for (a) the POVM ${\mathrm{\Theta }}_{\mathrm{B}}=\{{\mathrm{\Theta }}_{\mathrm{B}}^{(x)}\}$ [Eq. (3)] and (b) noisy measurement ${\mathrm{\Xi }}_{\mathrm{B}}=\{{\mathrm{\Xi }}_{\mathrm{B}}^{(x)}\}$ (yellow, right). Each bar shows the mutual information between Alice and Charlie, conditioned on Bob’s event. Gray edges show theoretical predictions.

Figure 4

Multitime expectations for (a) Process 1 and (b) Process 2. Alice and Charlie project onto orthogonal states $\{|\varphi ⟩=\cos {\theta }_1|0⟩+e^{i\varphi }\sin {\theta }_1|1⟩,|\varphi {⟩}^{\perp }=\sin {\theta }_1|0⟩-e^{-i\varphi }\cos {\theta }_1|1⟩\}$ and $\{|\psi ⟩=\cos {\theta }_2|0⟩+e^{i\psi }\sin {\theta }_2|1⟩,|\psi {⟩}^{\perp }=\sin {\theta }_2|0⟩-e^{-i\psi }\cos {\theta }_2|1⟩\}$, respectively, and Bob performs any nonselective measurement. The difference of expectation values via the recovered and true process is plotted for fixed phase (a) $\varphi =0$ and $\psi =0$ and (b) $\varphi =1.920\pi$ and $\psi =\pi$.