Quantum-to-Classical Transition via Single-Shot Generalized Measurements

Abstract

The quantum-to-classical transition in finite-dimensional systems is generally believed to occur asymptotically, yet this process has not been systematically examined and the mechanism governing the intermediate regime remains unclear. We address this question using an operational framework that bridges discrete generalized measurements and continuous isotropic depolarizing channels. We show that a single round of such a generalized measurement can eliminate quasiprobability negativity, that is, nonclassicality, in quantum phase space for finite-dimensional systems. From the perspective of decoherence, we find that although the overall process is continuous, the loss of quasiprobability negativity occurs abruptly, defying the conventional expectation of asymptotic decay. Additionally, we propose quantum-circuit implementations and analyze their feasibility on current quantum platforms, such as superconducting systems. Our results provide new insight into the intermediate regime of the quantum-to-classical transition in finite-dimensional systems and enable the simulation of open system dynamics through operational generalized measurements.

Figures (3)

• Figure 1: Quantum-to-classical transition via a single round POVM. Top: The Wigner quasiprobability distribution $W_{\rho_n}^{(0)}$ for a qubit after $n$ rounds of two-level coherent-state POVMs. The coherent state is given by $|\Omega\rangle = \cos (\theta/2)|0\rangle + \sin (\theta/2)\,\mathrm{e}^{\mathrm{i}\phi}|1\rangle$. Bottom: The equivalence between the coherent-state POVM and the isotropic depolarizing channel. Both results show that a single round of POVM is sufficient to drive the transition from $W_{\rho_0}^{(0)} < 0$ to $W_{\rho_1}^{(0)} > 0$.
• Figure 2: Sudden disappearance of the negative-quasiprobability volume. The volume of the negative Wigner quasiprobability function, $\mathcal{P}(\rho_{n(t)})$, is plotted against time $t$ or the number of POVM rounds $n$, related by $t = \tfrac{2n}{\gamma N}\ln(1 + N)$ [Eq. (\ref{['Dengjia']})]. Solid curves $\mathcal{P}(\rho_t)$ correspond to $N = 2, 5, 10, 10^{2}, 10^{3}$ (dark to light blue) and depict the evolution of the negative-quasiprobability volume over time, showing a sudden disappearance at $t_c (n_c = 1/2)$. Markers at $n = 0$ denote the initial negative-probability volume, while those at $n = 1$ (dashed vertical line) indicate that a single coherent-state POVM completely eliminates the negativity.
• Figure 3: Schematic quantum circuits for implementing $N$-level coherent-state POVMs. (a) Direct measurement protocol via application of the inverse unitary $U_N^{-1}$, i.e., $\operatorname{Tr}[U_N^{-1} \rho U_N |0\rangle_N \langle 0|] = \operatorname{Tr}[\rho E(\Omega)] / N$. (b) Ancilla-assisted protocol for detecting the POVM using both an ancilla and a reference register. The key step involves a controlled-SWAP gate $V_{\text{POVM}}$ between the system and the register. The outcome is obtained by measuring the ancilla in the $Z$ basis, yielding $\langle Z_A \rangle = \operatorname{Tr}[\rho E(\Omega)] / N$, which matches the result from (a) with $100\%$ success probability at the cost of increased resource requirements.