Efficient representation of Gaussian states for multimode non-Gaussian quantum state engineering via subtraction of arbitrary number of photons

Abstract

We introduce a complete description of a multi-mode bosonic quantum state in the coherent-state basis (which in this work is denoted as "K" function ), which-up to a phase-is the square root of the well-known Husimi "Q" representation. We express the K function of any N -mode Gaussian state as a function of its covariance matrix and displacement vector, and also that of a general continuous-variable cluster state in terms of the modal squeezing and graph topology of the cluster. This formalism lets us characterize the non Gaussian state left over when one measures a subset of modes of a Gaussian state using photon number resolving detection, the fidelity of the obtained non-Gaussian state with any target state, and the associated heralding probability, all analytically. We show that this probability can be expressed as a Hafnian, re-interpreting the output state of a circuit claimed to demonstrate quantum supremacy termed Gaussian boson sampling. As an example-application of our formalism, we propose a method to prepare a two-mode coherent-cat-basis Bell state with fidelity close to unity and success probability that is fundamentally higher than that of a well-known scheme that splits an approximate single-mode cat state-obtained by photon number subtraction on a squeezed vacuum mode-on a balanced beam splitter. This formalism could enable exploration of efficient generation of cat-basis entangled states, which are known to be useful for quantum error correction against photon loss.

Limit of Gaussian operations and measurements for Gaussian state discrimination and its application to state comparison

“…The technology of Gaussian operations and measurements is nowadays relatively well established and easily implementable, but this limited set of transformations is insufficient for many quantum information protocols. For example, by exploiting pure Gaussian transformation, quantum computation cannot show a quantum advantage [21], entanglement cannot be distilled [22], quantum error correction against Gaussian noise cannot be realized [23,24], and the capacity of optical communication cannot be reached [25,26].…”

Two-mode light states before and after delocalized single-photon addition

“…For example, the NOON state is a useful entangled resource, which has been generated [6] and applied to improve phase sensitivity in quantum metrology [7]. A large number of theoretical and experimental studies have demonstrated that entanglement between Gaussian entangled states can be increased via Gaussian or non-Gaussian operations [8][9][10][11][12][13][14][15][16][17][18][19][20]; these operations can be divided into local [21][22][23][24] and nonlocal [25,26]. Many nonlocal operations have the effect of delocalization, which can entangle the input separable states or change the entanglement of the input entangled states.…”

Non-Gaussian photonic state engineering with the quantum frequency processor

“…But producing such states-and non-Gaussian CV states more generallyis an extremely challenging endeavor, with proof-ofprinciple GKP realizations so far limited to non-photonic platforms [19,20]. The discovery and analysis of Gaussian boson sampling (GBS) [21,22], however, has provided a valuable framework for preparing non-Gaussian optical states [23][24][25][26]. Also straddling the interface between CV and DV-in that it leverages both CV fields and single-photon detection-GBS circuits can in principle produce arbitrary non-Gaussian states through ancilla modes and postselection on particular detection patterns, analogous to the probabilistic gates of linear-optical quantum computation (LOQC) in the DV paradigm [1,2].…”

“…Also straddling the interface between CV and DV-in that it leverages both CV fields and single-photon detection-GBS circuits can in principle produce arbitrary non-Gaussian states through ancilla modes and postselection on particular detection patterns, analogous to the probabilistic gates of linear-optical quantum computation (LOQC) in the DV paradigm [1,2]. The design [18,[26][27][28] and implementation [29][30][31] of GBS-type circuits for non-Gaussian state preparation have so far focused on the path degree of freedom (DoF), a natural choice given its long history in optics and well-known unitary decomposition procedure [32,33]. But other DoFs offer promise as well.…”

“…Leveraging and expanding on the Kfunction formalism of Ref. [26], we describe a resourceefficient method for computing the output of a QFP excited by Gaussian inputs and measured with photonnumber-resolving (PNR) detectors applied to a subset of frequency modes. As examples of this general approach, we design QFP circuits intended to produce Schrödinger cat states in one undetected bin and explore the impact of the number of components and ancilla modes on circuit performance, according to a cost function which balances both state fidelity and success probability.…”