Schrödinger cat states in continuous variable non-Gaussian networks

doi:10.1016/j.physleta.2020.126762

Physics Letters A

Volume 384, Issue 29, 19 October 2020, 126762

https://doi.org/10.1016/j.physleta.2020.126762 Get rights and content

Highlights

•
Continuous variable network with non-Gaussian gate prepares Schrodinger cat state.
•
Lowest order non-Gaussian resource: a cubic phase state is sufficient.
•
Superposition of two copies of an arbitrary input state can be created.
•
Intuitively clear interpretation of cat state emergence is presented.
•
Heisenberg picture as it is applied to Gaussian networks fails for “cat gate”.

Abstract

We show how continuous variable network with embedded non-Gaussian element can effectively prepare Schrödinger cat state using cubic phase state as elementary non-Gaussian resource, an entangling Gaussian gate, and homodyne measurement. The gate prepares superposition of two “copies” of an arbitrary input state well separated on the phase plane. A key feature of the cat-breeding configuration is that the measurement outcome provides multivalued information about the target system variables, which makes irrelevant the Heisenberg picture as it is applied to Gaussian networks. We present an intuitively clear interpretation of the emerging cat state, extendable to the circuits with other non-Gaussian elements.

Introduction

Continuous variable (CV) quantum networks are a promising area of quantum information (quantum communication, quantum computing and simulation) [1], [2]. Since the pioneering demonstrations of the CV quantum teleportation [3] and quantum dense coding [4], there was achieved an impressive progress in the theory and implementation of multimode CV cluster schemes, including preparation of large-scale CV cluster states multiplexed in the time or wavelength domain [5], [6], [7], [8], [9].

The properties of the CV schemes with embedded non-Gaussian elements are actively explored today [10], [11], [12].

In this letter we show that a CV circuit seeded with an elementary non-Gaussian element may conditionally evolve to a Schrödinger cat state by the proper choice of the projective measurement. The Schrödinger cat state prepared in our scheme is eventually due to cubic nonlinearity, and may conditionally emerge as a superposition of two “copies” of an arbitrary input state well separated in the coordinate and momentum plane. The “cat gate” uses the same resources as the well discussed cubic phase gate first introduced in [13], [14], that is, the cubic phase state of an ancillary oscillator as a non-Gaussian resource, an entangling Gaussian gate, and homodyne measurement.

The cubic phase state preparation is a goal of extensive experimental and theoretical research [13], [14], [15], [16], [17], [18], and the two-mode entangling Gaussian operations and homodyne measurements are currently performed with high efficiency.

Unlike the previously introduced CV measurement-induced schemes with non-Gaussian elements, we consider the ancillary oscillator measurement whose outcome provides a multivalued information about the relevant physical variables of the target subsystem, which is a key point. This feature does not emerge in Gaussian quantum networks and manifests a superior complexity of the non-Gaussian schemes.

The approach used in our proposal is scalable, that is, it can be extended to more complex CV non-Gaussian elements. From an heuristic point of view, one can easily identify some configurations where the CV cat states may arise using illustrative schemes similar to the one presented below.

By sequentially applying the cat-breeding operations together with standard Gaussian gates that generate shift, rotation, squeezing, shearing deformation, etc., one can transform an initial CV network quantum state to a multicomponent Schrödinger cat state of an arbitrary complexity.

The Schrödinger cat states are an object of a relentless interest since their first introduction [19]. Besides their fundamental importance, some proposals for fault-tolerant quantum information processing directly rely on the cat-like states [13], [20].

In general, one can prepare CV Schrödinger cat states by making use of the unitary evolution assisted by a non-linear interaction [21]. The schemes based on a hybrid measurement-induced evolution also can create cat-like states. The optical Schrödinger cat states were generated in low-photon regime using homodyne detection and photon number states as resources [22], [23], [24]. Unlike our scheme, the latter proposals are specifically aimed at the creation of superpositions of two coherent states.

The CV Gaussian and non-Gaussian measurement-induced quantum networks are a rapidly growing field, and our proposal to prepare Schrödinger cat states using CV schemes may be an alternative to other approaches.

Section snippets

Cat state from ancilla cubic phase state, entanglement and measurement

The target and ancillary oscillators evolution is schematically shown in Fig. 1 which in graph form illustrates transformations of the relevant canonical variables. The ancilla is assumed to be initially prepared in the cubic phase state [13], [14] via the action of unitary evolution operator $\exp (i γ q_{2}^{3})$ upon the momentum eigenstate ${| 0 〉}_{p_{2}}$ . In the Heisenberg picture this may be written as $q_{2} = q_{2} (0), p_{2} = p_{2} (0) + 3 γ q_{2}^{2} (0),$ where ${p_{2} (0) | 0 〉}_{p_{2}} = 0$ .

In order to illustrate the system evolution before the

Measurement-induced evolution of non-Gaussian circuits and the Heisenberg picture

There is a variety of measurement-based CV Gaussian schemes of quantum information, such as quantum teleportation and dense coding, quantum repeaters, cluster model of quantum computing, etc. The Gaussian schemes with measurement and feedforward are effectively described both in the Schrödinger and the Heisenberg picture. The last one offers a possibility to include noise sources and imperfections in the scheme [5], [6], [7], [8], [9], [25] and provides in many cases an intuitively clear

Conclusions

We have presented a CV measurement-induced non-Gaussian gate (“cat gate”) which is able to produce conditionally a superposition of two “copies” of an arbitrary input state, well separated on the phase plane – a Schrödinger cat state.

In analogy to some other CV schemes, the gate exploits the same key elements, that is, ancillary cubic phase state and entangling $C_{Z}$ Gaussian gate, but applies homodyne measurement which provides a multivalued information about the target system physical

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

I thank Nikolay N. Bezuglov for fruitful discussions. This research was supported by the Russian Foundation for Basic Research (RFBR) under the projects 18-02-00648-a and 19-02-00204-a.

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Quantum measurement-induced gate based on entanglement with ideal cubic phase state used as a non-Gaussian resource is able to produce Shrödinger cat state in the form of two high fidelity “copies” of the target state on the phase plane (Masalaeva and Sokolov, 2022 [21]). In this work we examine the effect of finite initial squeezing of the resource state on the gate performance. We present an exact solution for the gate output state and demonstrate that there exists a degree of squeezing, available in experiment, such that the output cat state quality almost does not improve with the further increase of squeezing. On the other hand, the probability of the expected ancilla measurement outcome decreases with squeezing. Since an overall efficiency of the conditional scheme should account for the probability of success, we argue that such measures of non-Gaussianity of the resource state, as Wigner logarithmic negativity and non-Gaussianity, may not be directly applicable to assess the efficiency of non-Gaussian gates, which are based on quantum entanglement and subsequent projective measurement.
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Citation Excerpt :
In the following, we consider as a Schrodinger cat state a superposition of two mutually displaced in phase space “copies” of the initial state of the target oscillator, thus applying a wider definition of the cat state than a superposition of two coherent states. An important feature of the gate [21] is that the Schrödinger cat state emerges when the ancillary oscillator measurement is compatible not with one, but with two different values of the target oscillator physical variables. This feature does not appear in Gaussian quantum networks and, given that a non-Gaussian resource state is used, needs the measurement which is consistent with this criterion.
A measurement-induced continuous-variable logical gate is able to prepare Schrödinger cat states if the gate uses a non-Gaussian resource state, such as cubic phase state (I.V. Sokolov (2020) [21]). Our scheme provides an alternative to hybrid circuits which use photon subtraction and (or) Fock resource states and photon number detectors. We reveal the conditions under which the gate conditionally prepares quantum superposition of two undistorted “copies” of an arbitrary input state that occupies a finite area in phase space. A detailed analysis of the fidelity between the gate output state and high-quality Schrödinger cat state is performed. A clear interpretation of the output state quantum statistics in terms of Wigner function in dependence on the gate parameters and measurement outcome is presented for a representative set of input Fock states.
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Measurement-assisted non-Gaussian gate for Schrödinger cat states preparation: Fock resource state versus cubic phase state
2023, arXiv
Schrödinger cat states prepared by logical gate with non-Gaussian resource state: effect of finite squeezing and efficiency versus monotones
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View full text

Schrödinger cat states in continuous variable non-Gaussian networks

Highlights

Abstract

Introduction

Section snippets

Cat state from ancilla cubic phase state, entanglement and measurement

Measurement-induced evolution of non-Gaussian circuits and the Heisenberg picture

Conclusions

Declaration of Competing Interest

Acknowledgements

Quantum computation with continuous variables

Phys. Rev. Lett.

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Rev. Mod. Phys.

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Phys. Rev. A

Gaussian quantum information

Rev. Mod. Phys.

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Nat. Photonics

Wavelength-multiplexed quantum networks with ultrafast frequency combs

Nat. Photonics

Implementation of a quantum cubic gate by adaptive non-Gaussian measurement

Phys. Rev. A

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Phys. Rev. A

Quantum hypergraph states in continuous variables

Phys. Rev. A