The stellar decomposition of Gaussian quantum states

Abstract

We introduce the stellar decomposition, a novel method for characterizing non-Gaussian states produced by photon-counting measurements on Gaussian states. Given an $(m+n)$-mode Gaussian state $G$, we express it as an $(m+n)$-mode "Gaussian core state" $G_{\mathrm{core}}$ followed by an $m$-mode Gaussian transformation $T$ that only acts on the first $m$ modes. The defining property of the Gaussian core state $G_{\mathrm{core}}$ is that measuring the last $n$ of its modes in the photon-number basis leaves the first $m$ modes on a finite Fock support, i.e. a core state. Since $T$ is measurement-independent and $G_{\mathrm{core}}$ has an exact and finite Fock representation, this decomposition exactly describes all non-Gaussian states obtainable by projecting $n$ modes of $G$ onto the Fock basis. For pure states we prove that a physical pair $(G_{\mathrm{core}},T)$ always exists with $G_{\mathrm{core}}$ pure and $T$ unitary. For mixed states, we establish necessary and sufficient conditions for $(G_{\mathrm{core}},T)$ to be a Gaussian mixed state and a Gaussian channel. We also develop a semidefinite program to extract the "largest" possible Gaussian channel when these conditions fail. Finally, we present a formal stellar decomposition for generic operators, which is useful in simulations where the only requirement is that the two parts contract back to the original operator. The stellar decomposition leads to practical bounds on achievable state quality in photonic circuits and for GKP state generation in particular. Our results are based on a new characterization of Gaussian completely positive maps in the Bargmann picture, which may be of independent interest.

Accepted for publication in Quantum. 54 pages, 8 figures

References in corpus (62)

• Gaussian Quantum Information
• Quantum Teleportation is a Universal Computational Primitive
• Quantum state reconstruction of the single-photon Fock state
• Generation of a superposition of odd photon number states for quantum information networks
• On the impossibility of distilling Gaussian states with Gaussian operations
• Ultra-Large-Scale Continuous-Variable Cluster States Multiplexed in the Time Domain
• The characterization of Gaussian operations and Distillation of Gaussian States
• Quantum circuits with many photons on a programmable nanophotonic chip
• Gaussian Boson Sampling
• Efficient Classical Simulation of Continuous Variable Quantum Information Processes
• Non-gaussian statistics from individual pulses of squeezed light
• Gaussian transformations and distillation of entangled Gaussian states
• Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States
• A No-Go Theorem for Gaussian Quantum Error Correction
• Proposal for a loophole-free Bell test using homodyne detection
• Controllable generation of highly nonclassical states from nearly pure squeezed vacua
• Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer
• Non-Gaussian Quantum States and Where to Find Them
• Generation of Optical Coherent State Superpositions by Number-Resolved Photon Subtraction from Squeezed Vacuum
• Generation of large-amplitude coherent-state superposition via ancilla-assisted photon-subtraction
• Experimental nonclassicality of single-photon-added thermal light states
• Resource theory of quantum non-Gaussianity and Wigner negativity
• Remote creation of hybrid entanglement between particle-like and wave-like optical qubits
• Perspective: Toward large-scale fault-tolerant universal photonic quantum computing
• Quantum homodyne tomography of a two-photon Fock state
• Breeding the optical Schroedinger's cat state
• Convex resource theory of non-Gaussianity
• A detailed study of Gaussian Boson Sampling
• Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states
• Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes
• Generating superposition of up-to three photons for continuous variable quantum information processing
• Quantum-optical state engineering up to the two-photon level
• Non-Gaussian quantum states of a multimode light field
• Maximal Violation of Bell Inequalities using Continuous Variables Measurements
• Propagating Gottesman-Kitaev-Preskill states encoded in an optical oscillator
• All-optical generation of states for "Encoding a qubit in an oscillator"
• Generating Grid States From Schrödinger Cat States without Post-Selection
• Optical synthesis of large-amplitude squeezed coherent-state superpositions with minimal resources
• Single-Mode Displacement Sensor
• Conversion of Gaussian states to non-Gaussian states using photon-number-resolving detectors
• Synthesis and tomographic characterization of the displaced Fock state of light
• Stellar representation of non-Gaussian quantum states
• Experimental generation of multi-photon Fock states
• Mode-Wise Entanglement of Gaussian States
• Non-Gaussian and Gottesman-Kitaev-Preskill state preparation by photon catalysis
• A General Theory of Phase-Space Quasiprobability Distributions
• Resource theory of non-Gaussian operations
• Faithful hierarchy of genuine $n$-photon quantum non-Gaussian light
• Generation of highly pure Schrödinger's cat states and real-time quadrature measurements via optical filtering
• Entangling macroscopic light states by delocalized photon addition
• The Complexity of Bipartite Gaussian Boson Sampling
• Efficient representation of Gaussian states for multi-mode non-Gaussian quantum state engineering via subtraction of arbitrary number of photons
• Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson
• Generation of Flying Logical Qubits using Generalized Photon Subtraction with Adaptive Gaussian Operations
• Efficient backcasting search for optical quantum state synthesis
• Schrödinger's cat in an optical sideband
• Towards quantum frequency combs: boosting the generation of highly nonclassical light states by cavity-enhanced parametric down-conversion at high repetition rates
• Holomorphic representation of quantum computations
• Fast optimization of parametrized quantum optical circuits
• Riemannian optimization of photonic quantum circuits in phase and Fock space
• A Quadratic Speedup in the Optimization of Noisy Quantum Optical Circuits
• What's my phase again? Computing the vacuum-to-vacuum amplitude of quadratic bosonic evolution

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