Quantum Linear Time-Translation-Invariant Systems: Conjugate Symplectic Structure, Uncertainty Bounds, and Tomography

TL;DR

This paper develops a comprehensive framework for quantum linear time-invariant systems, revealing their fundamental noise properties, uncertainty bounds, and measurement techniques, applicable to complex non-Markovian scenarios without internal state knowledge.

Contribution

It introduces a general quantization scheme for quantum LTI systems, characterizes their conjugate symplectic structure, and derives new uncertainty bounds and measurement methods.

Findings

Quantum noise is fundamentally linked to the conjugate symplectic structure.

Tighter Heisenberg uncertainty bounds are established for quantum LTI systems.

Complex squeezing in lossy systems can be detected only with symplectodyne measurements.

Abstract

Linear time-translation-invariant (LTI) models offer simple, yet powerful, abstractions of complex classical dynamical systems. Quantum versions of such models have so far relied on assumptions of Markovianity or an internal state-space description. We develop a general quantization scheme for multimode classical LTI systems that reveals their fundamental quantum noise, is applicable to non-Markovian scenarios, and does not require knowledge of an internal description. The resulting model is that of an open quantum LTI system whose dilation to a closed system is characterized by elements of the conjugate symplectic group. Using Lie group techniques, we show that such systems can be synthesized using frequency-dependent interferometers and squeezers. We derive tighter Heisenberg uncertainty bounds, which constrain the ultimate performance of any LTI system, and obtain an invariant representation of their output noise covariance matrix that reveals the ubiquity of "complex squeezing" in lossy systems. This frequency-dependent quantum resource can be hidden to homodyne and heterodyne detection and can only be revealed with more general "symplectodyne" detection. These results establish a complete and systematic framework for the analysis, synthesis, and measurement of arbitrary quantum LTI systems.