Theory of Deterministic Photon-Loss Subspaces for Quantum Interferences

Abstract

Quantum coherence, quantum decoherence, and photon number reduction are coexistent in linear lossy optical systems. However, how these three elements combine together to determine the evolution of the quantum light remains unclear. Here, based on singular value decomposition (SVD), we propose the theory of deterministic photon-loss subspace (DPLS) for quantum interferences in lossy systems. By performing an SVD of scattering matrices with singular values either 0 or 1, a series of completely lossy and lossless input modes are first defined. According to n_1,...,n_i photons in the first,..., i-th lossy modes, the Hilbert space of the input states can be decomposed into a set of orthogonal subspaces H_((n_1,...,n_i ))^in, i.e., deterministic photon-loss subspaces (DPLSs). When the concept of DPLS is established, the input state can be projected onto these DPLSs. In each DPLS, the photons in lossy modes will be completely dissipated, while those in lossless modes experience a unitary evolution. The output state is a statistical mixture of the evolved outcomes of all projections, since decoherence is a concurrent process. Then, based on the DPLS theory, we not only revisited Anti-HOM interference and the distillation of quantum states, but also demonstrate a robust W-state generation for various input states in a three-port lossy system with one-dimensional DPLSs. Through investigating the loss-induced subspace structure of the system, our general theory for analyzing quantum state evolution in lossy systems explicitly reveals the interplay among quantum coherence, quantum decoherence, and photon number reduction. By engineering the loss, the constructed DPLSs can be used to precisely control quantum interferences in dissipative systems, with potential applications in quantum state preparation, quantum logic operations, and other quantum information processes.