Direct Tomography of High-Dimensional Density Matrices for General Quantum States of Photons

Abstract

Quantum state tomography is the conventional method used to characterize density matrices for general quantum states. However, the data acquisition time generally scales linearly with the dimension of the Hilbert space, hindering the possibility of dynamic monitoring of a high-dimensional quantum system. Here, we demonstrate a direct tomography protocol to measure density matrices in the spatial domain through the use of a polarizationresolving camera, where the dimension of density matrices can be as large as 580×580 in our experiment. The use of the polarization-resolving camera enables parallel measurements in the position and polarization basis and as a result, the data acquisition time of our protocol does not increase with the dimension of the Hilbert space and is solely determined by the camera exposure time (on the order of 10 milliseconds). Our method is potentially useful for the real-time monitoring of the dynamics of quantum states and paves the way for the development of high-dimensional, time-efficient quantum metrology techniques.

Digital holographic imaging via direct quantum wavefunction reconstruction

“…In addition, wavefunction reconstruction may find important applications in testing foundational issues [15] or quantum information science. [26] As the number of qubits grows quickly in various physical platforms, e.g., superconducting, ionic and atomic quantum processors, the need for efficient quantum state reconstruction seems urgent and the new method introduced may provide a promising way to tackle this issue.…”

Nonseparable States of Light: From Quantum to Classical

“…[163,164] In a similar way, the concept of nonseparability has been also used for spatiotemporal quantum-analogue pulses, for example, the Schmidt number or Schmidt rank was used to characterize the invariant propagation effect of a classical pulse. [101] Since the quantum state tomography method was modified to highdimensional state, [162] we can apply analogous method to more complex structured light, for example, the space-time nonseparable pulses, and many quantum concepts (fidelity, concurrence, linear entropy, etc.) were open to quantitatively characterize the spatiotemporal propagation dynamics.…”

“…[159][160][161] Moreover, the quantum Bell measurement method and Bell's inequality are also available to use for revealing the spin-to-orbital coupling in a classical beam. [158,99] Besides the pure state systems, the quantum mechanics of mixed state was also transferred to complex classical beams for exploring high-dimensional quantumanalogue system, [162] which can also realize the counter-intuitive quantum teleportation effect in classical beams system. [163,164] In a similar way, the concept of nonseparability has been also used for spatiotemporal quantum-analogue pulses, for example, the Schmidt number or Schmidt rank was used to characterize the invariant propagation effect of a classical pulse.…”

Nonseparable states of light: From quantum to classical

“…Moreover, the quantum Bell measurement method and Bell's inequality are also available to use for revealing the spin-to-orbital coupling in a classical beam [98,158]. Besides the pure state systems, the quantum mechanics of mixed state was also transferred to complex classical beams for exploring high-dimensional quantum-analogue system [162], which can also realize the counter-intuitive quantum teleportation effect in classical beams system [163,164]. In a similar way, the concept of nonseparability has been also used for spatiotemporal quantum-analogue pulses, e.g.…”

“…the Schmidt number or Schmidt rank was used to characterize the invariant propagation effect of a classical pulse [100]. Since the quantum state tomography method was modified to highdimensional state [162], we can apply analogous method to more complex structured light, e.g. the space-time nonseparable pulses, and many quantum concepts (fidelity, concurrence, linear entropy, etc.)…”