Digital quantum simulation of bosonic systems and quantum complementarity

TL;DR

The paper develops a Gray-code–based digital quantum simulation framework for bosonic systems and applies it to interferometric variants of Afshar’s experiment to examine quantum complementarity. By encoding bosonic operators into Pauli strings and mapping multi-mode bosonic dynamics to qubit circuits, the authors simulate modified Unruh’s and Pessoa Júnior’s nested Mach–Zehnder interferometers on IBM quantum hardware, including blocker-induced absorption and the two-photon regime. They connect the results to an updated quantum complementarity principle (QCP), deriving wave- and particle-like quantifiers from QM postulates and illustrating that apparent conflicts with Bohr’s principle arise from ad hoc definitions rather than a breakdown of QM. The work also explores delayed-choice scenarios within the QCP framework, showing non-retrocausal interpretations and extending the analysis to two-photon states, thereby highlighting the broader applicability of DQS to foundational questions in quantum optics and bosonic many-body physics.

Abstract

Digital quantum simulation (DQS) has emerged as a powerful approach to investigate complex quantum systems using digital quantum computers. Such systems, like many-particle bosonic systems and intricate optical experimental setups, pose significant challenges for classical simulation methods. In this paper, we utilize a general formalism for the DQS of bosonic systems, which consists of mapping bosonic operators to Pauli operators using the Gray code, in order to simulate interferometric variants of Afshar's experiment -- an intricate optical experiment -- on IBM's quantum computers. We investigated experiments analogous to Afshar's double-slit experiment performed by Unruh and Pessoa Júnior, exploring discussions on the apparent violation of Bohr's complementarity principle when considering the entire experimental setup. Based on the aforementioned experiments, we construct a variation of a delayed-choice setup. We also explore another experiment starting with a two-photon initial state. Finally, we analyze these experiments within the framework of an updated quantum complementarity principle, which applies to specific quantum state preparations and remains consistent with the foundational principles of Quantum Mechanics.

Figures (16)

• Figure 1: Beam splitter with input modes A and B and output modes C and D, characterized by transmission and reflection coefficients $T$ and $R$, respectively. Each reflection adds a phase of $\pi/2$ rad to the reflected beam.
• Figure 2: Mach-Zehnder interferometer composed of two BBSs. By utilizing a BBS$_j$ (with $j=0,1$), the amplitudes of transmission ($T_j$) and reflection ($R_j$) --- which specify the probability amplitudes of the quantons in each MZI arm --- can be controlled. Mirrors (M) convert the horizontal (vertical) spatial mode into the vertical (horizontal) mode while introducing a global phase factor of $e^{i\pi/2}$; the phase shifter applies a phase factor of $e^{i\phi_{\text{E}}}$ to the vertical spatial mode, and $D_0$ and $D_1$ represent the detectors.
• Figure 3: The Afshar's experiment is a modified version of Young double-slit experiment. In this experiment, Afshar proposed different ways to quantify the wave- ($W$) and particle-like ($P$) behaviors in order to violate Bohr's complementarity principle (BCP). After measuring the positions of dark fringes (maximum destructive interference) at the screen $s_2$, Afshar replaced this screen by very thin wire grid at the dark fringes previously determined. According to Afshar, this procedure is proposed for maximally quantifying ($W=1$) the wave-like behavior in a non-destructive way by verifying that this process does not significantly affect the photon counts at $s_3$. The area $s_3$ encompasses two positions, specifically detector $D_A$ and detector $D_B$, located beyond the lens $L$, which direct the photons to these regions. In an auxiliary experiment, Afshar closes slit A (B) and verifies that only the corresponding detector $D_B$ ($D_A$) clicks. As only one detector register clicks, he concluded that, with both slits open, the slit-detector relation is still valid. This scenario is characterized by him as a quantification of particle behavior. With this in mind, since a maximum wave-like behavior is captured by the wire grid and a maximum particle-like behavior is obtained by the slit-detector relation after a click, the author concludes that wave-particle duality relation is violated as both behaviors are obtained maximally in only one experiment leading to a breach in the BCP.
• Figure 4: Modified Unruh's experiment proposed by Pessoa Júnior in Ref. Pessoajr2013. The version proposed by Unruh in Ref. Unruh2004 just doesn't take into account the phase element $\phi_{\text{H}}$. The main objective is to construct an analogy of Afshar's experiment using the Mach-Zehnder interferometer (MZI). The MZI$_1$ is related to the double-slit, the MZI$_2$ is related to the $s_2$ in Fig. \ref{['fig:afshar']} and the BS$_3$ makes the lens condition to redirect the photons to the detectors. The $B_0$ blocker in the paths is analogous to closing each of the slits in the experimental part responsible for creating the path-detector relationship. One of the issues raised by Unruh is regarding the dark fringes (related to the H-mode path) when blocked by the blocker $B_1$---this blocker can be related to the wire grid that quantify the wave behavior in Afshar's experiment. This procedure completely destroys the detector-path relationship. Modified Unruh's experiment proposed by Pessoa Júnior uses the analysis of phase shifts $\phi_{\text{E}}$ and $\phi_{\text{H}}$ to better investigate the problem and find a reconciliation with the results obtained in Afshar's experiment. In the experiment of Ref. Afshar2007, including the wire grid while closing each slit did not completely destroy the path-detector relationship.
• Figure 5: Quantum circuit for a BS, for which it is possible to control the transmission, $T$, and the reflection, $R$, coefficients with $\lambda=\arctan\left(\frac{R}{T}\right)/2$. Each qubit represents one of the input modes. Following the execution of the circuit, the qubits that represent the input modes $q_0,q_1$ are converted into the output mode $q_0^\prime,q_1^\prime$. For example, in Fig. \ref{['fig:BS']} the input modes A ($q_0$) and B ($q_1$) are converted into the output modes C ($q_0^\prime$) and D ($q_1^\prime$). The entire quantum circuit is represented by the box $U_{\text{BS}}$.