Convex resource theory of non-Gaussianity

Ryuji Takagi; Quntao Zhuang

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Convex resource theory of non-Gaussianity

Ryuji Takagi, Quntao Zhuang

arXiv:1804.04669v2Apr 12, 2018quant-ph

TL;DR

The paper asks how to formulate and quantify useful non-Gaussian resources for universal continuous-variable quantum computation. It defines Gaussian protocols as free operations and the convex hull of Gaussian states as free states, then establishes a monotone and a probabilistic distillation protocol. The protocol increases genuine non-Gaussianity and can distill cubic phase states, while the theory bounds distillation performance.

Problem
Gaussian operations and states are experimentally accessible but cannot provide universal continuous-variable quantum computation, motivating a resource theory for the required non-Gaussianity.
Method
The paper defines Gaussian operations with feed-forward as free operations, the convex hull of Gaussian states as free states, and logarithmic Wigner negativity as a monotone.
Results
The theory supports bounds on state conversion and distillation, while a single-copy postselected protocol increases logarithmic negativity and can increase cubic phase-state fidelity.
Takeaways & Limitations
Genuine non-Gaussian states outside the convex hull of Gaussian states are the resource states for universal computation with Gaussian protocols, including cubic phase states.

Abstract

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Continuous-variable systems realized in quantum optics play a major role in quantum information processing, and it is also one of the promising candidates for a scalable quantum computer. We introduce a resource theory for continuous-variable systems relevant to universal quantum computation. In our theory, easily implementable operations---Gaussian operations combined with feed-forward---are chosen to be the free operations, making the convex hull of the Gaussian states the natural free states. Since our free operations and free states cannot perform universal quantum computation, genuine non-Gaussian states---states not in the convex hull of Gaussian states---are the necessary resource states for universal quantum computation together with free operations. We introduce a monotone to quantify the genuine non-Gaussianity of resource states, in analogy to the stabilizer theory. A direct application of our resource theory is to bound the conversion rate between genuine non-Gaussian states. Finally, we give a protocol that probabilistically distills genuine non-Gaussianity---increases the genuine non-Gaussianity of resource states---only using free operations and postselection on Gaussian measurements, where our theory gives an upper bound for the distillation rate. In particular, the same protocol allows the distillation of cubic phase states, which enable universal quantum computation when combined with free operations.

I. INTRODUCTION

The paper motivates a continuous-variable resource theory by identifying the limits of Gaussian schemes and proposing genuine non-Gaussianity as the resource for universal quantum computation.

Non-Gaussian states or operations are required for entanglement distillation, error correction, loophole-free Bell violations, and universal quantum computation.
Gaussian operations with adaptive feed-forward are experimentally accessible and can produce probabilistic mixtures of Gaussian states.
The convex hull of Gaussian states is chosen as the free-state set, while states outside it are genuine non-Gaussian resource states.
The paper introduces logarithmic negativity of the Wigner function as a computable monotone for genuine non-Gaussianity.It is defined as the logarithm of the integral of the absolute value of the Wigner function and is monotonic under free operations.
A probabilistic distillation protocol increases genuine non-Gaussianity using free operations and Gaussian-measurement postselection.The protocol requires a single input copy and also increases fidelity for imperfect cubic phase states when conditioned on success.

B. Continuous-variable quantum computation

The paper places continuous-variable computation in a framework where Gaussian operations are insufficient for universality, while adding a higher-than-quadratic unitary such as the cubic phase gate enables it.

Continuous-variable universality requires approximating arbitrary unitary evolutions generated by polynomial Hamiltonians.
Gaussian unitaries alone are not universal because they correspond to generators that are second order in quadrature operators.
Adding any unitary generated by a polynomial of order higher than two makes the operation set universal.
The cubic phase gate is a simple non-Gaussian choice because it is generated by the lowest polynomial order above two.
Experimental proposals use genuine non-Gaussian resource states with Gaussian operations and feed-forward to realize cubic phase gates.

III. RESOURCE THEORY FRAMEWORK

The resource theory defines free operations as accessible Gaussian protocols and free states as their invariant convex hull, distinguishing genuine non-Gaussian resources from broader nonnegative-Wigner states.

Resource theories quantify and manipulate a designated resource through free states and free operations.
Gaussian protocols combine Gaussian operations, homodyne measurements, and feed-forward conditioned on measurement outcomes.They also include conditioning on classical randomness generated through homodyne measurements on ancillae.
The convex hull of Gaussian states is selected because Gaussian states alone are non-convex and mixtures produced by free operations can be non-Gaussian.
States outside the convex hull are called genuine non-Gaussian states, while the convex-hull states are convex-Gaussian states.
The convex hull of all Gaussian pure states equals the full convex hull of Gaussian states and is the largest set preparable from Gaussian states by Gaussian protocols.
Convex-Gaussian states have nonnegative Wigner functions, and the inclusions G ⊊ Ḡ ⊊ W+ distinguish Gaussian, convex-Gaussian, and all nonnegative-Wigner states.
States in W+ \ Ḡ are bound genuine non-Gaussian states that do not enable universal computation with the chosen free operations and states.

B. Monotone

The paper defines genuine non-Gaussian monotones for bosonic states and focuses on logarithmic negativity of the Wigner function. This quantity has several monotonicity properties but is not faithful because some non-Gaussian states have zero logarithmic negativity.

Definition: A genuine non-Gaussian monotone maps bosonic quantum states to real numbers and must satisfy specified resource-theoretic conditions.The definition applies to density operators of N-mode bosonic systems.
Chosen monotone: The paper focuses on logarithmic negativity of the Wigner function because it is computable and relevant to universal quantum computation.The measure is selected from several possible monotone candidates.
Monotonicity properties: Logarithmic negativity is invariant under Gaussian unitaries and non-increasing under partial trace and Gaussian channels.These properties constrain how the measure behaves under standard Gaussian processing.
Monotonicity properties: Logarithmic negativity is additive on tensor-product states and non-increasing under free operations.Selective monotonicity additionally prevents the expected value from increasing after selective measurements.
Limitation: Logarithmic negativity vanishes exactly on states with non-negative Wigner functions, but it is not faithful because some such states lie outside the convex hull of Gaussian states.Those states are characterized as bound genuine non-Gaussian states rather than universal-computation resources.

IV. RESOURCE STATES

The paper compares genuine non-Gaussianity for several continuous-variable resource states using logarithmic negativity, including number, photon-added or subtracted, ON, and imperfect cubic phase states. Cubic phase states are modeled with finite squeezing because the ideal state is not physical.

Resource-state comparison: The comparison evaluates cubic phase, number, ON, photon-number-added, and photon-number-subtracted states using logarithmic negativity.The states are also compared at equal mean photon number where applicable.
Number states: Number states are common non-Gaussian sources and maximize non-convex non-Gaussianity at fixed photon number.Their Wigner functions are given in the appendix.
Photon addition and subtraction: Photon-number subtraction and addition conditionally transform a pure state into states proportional to a|ψ⟩ and a†|ψ⟩, respectively.The operations can be implemented experimentally, and the relevant Wigner functions are analytically calculable for zero-mean Gaussian inputs.
Photon addition and subtraction: For all squeezing values considered, the single-photon-added and single-photon-subtracted states have logarithmic negativity equal to that of the single-photon state.The paper states that this equality holds exactly.
Cubic phase states: The ideal cubic phase state is unnormalizable, so the paper instead studies finite-squeezing states produced by applying a cubic phase gate.Increasing squeezing approaches the ideal cubic phase state, and P = −6γe2s minimizes mean photon number for the comparison.

A. State conversions

The paper applies monotonicity of logarithmic negativity to state transformations under Gaussian protocols. In a cubic-phase-gate implementation, postselection can probabilistically convert squeezed inputs into imperfect cubic phase states, with success constrained by the input-to-output resource ratio.

Conversion bound: Under a Gaussian protocol, the probability-weighted output monotone provides a necessary condition for a state transformation.The probability density is P(q) = Tr[Φq ⊗ Mq(ρ)].
Cubic phase gate protocol: The cubic phase gate protocol starts with an input state and an ancillary ON state, applies continuous controlled-NOT, then homodyne measures the ancilla and applies Gaussian feed-forward.This realizes the transformation within the Gaussian-protocol framework.
Conversion bound: For an infinitely squeezed input, the initial logarithmic negativity equals that of the ancillary ON state, while the conditional output is proportional to a position eigenstate.This follows from additivity and zero logarithmic negativity of the squeezed input.
Numerical example: For γ = 0.1, the input ON state has logarithmic negativity 0.11, while examined conditional outputs have approximately 0.09.The values confirm the monotonicity relation and indicate efficient resource use for this case.
Cubic phase conversion: Postselecting outcomes q̃ ≈ 0 converts a squeezed state nondeterministically into an imperfect cubic phase state, with success probability bounded by the initial-to-output logarithmic-negativity ratio.The imperfect cubic phase state may subsequently be purified by another protocol.
ON-state resource: The logarithmic negativity of ON states for different N bounds the quality and success probability of protocols implementing the cubic phase gate.Larger N may help tune higher-order terms and improve the gate.

B. Distillation of genuine non-Gaussianity

The protocol uses a beam splitter, homodyne measurement, and postselection to probabilistically increase genuine non-Gaussianity. It exhibits a trade-off between output negativity and success probability and can also improve cubic-phase-state fidelity.

Protocol setup: A beam splitter mixes the input resource with vacuum, while homodyne outcomes determine which output states are retained by postselection.The retained outcome lies within an experimenter-tunable region.
Mechanism: The measurement-induced shifts make the vacuum Gaussian act as a filter that selects regions contributing strongly to Wigner negativity.The beam splitter creates a slight correlation, and the measurement kickback shifts the resource and Gaussian factors in opposite directions.
Monotonicity and trade-off: The resource theory supplies an upper bound on average logarithmic negativity without postselection and a trade-off between postselected negativity and success probability.Increasing the success region raises success probability but may reduce output negativity.
Numerical results: For imperfect cubic phase states, positive logarithmic-negativity increases occur for most tested success probabilities, while the increase and success probability remain in tension.The protocol also satisfies the selective monotonicity relation for average output negativity.
Cubic phase-state distillation: The same protocol can increase fidelity to a better imperfect cubic phase state, achieving a 13% increase at 1% success probability for initial squeezing s_ini = 1.For s_ini = 1.6, fidelity decreases, and repeated applications eventually saturate near the first-application fidelity because Wigner-function shape mismatch accumulates.

VI. CONCLUSIONS

The paper concludes that its Gaussian-protocol resource theory quantifies genuine non-Gaussianity and supports conversion and distillation analyses, while identifying several open directions.

Conclusions: The logarithmic negativity of the Wigner function is a valid monotone under Gaussian protocols.The paper also examines its properties and evaluates several genuine non-Gaussian resource states by mean photon number.
Conclusions: Number states, ON states, and cubic phase states show similar behavior when compared at equal mean photon number.
Conclusions: Monotonicity supplies a necessary condition for state conversions under Gaussian protocols.
Open directions: The paper identifies golden-unit definitions, faithful monotones, resource-generating power, and non-negative-Wigner-function-preserving operations as future research directions.It also notes that the operational meaning of bound genuine non-Gaussian states remains unknown.
Open directions: A protocol that increases the fidelity of imperfect cubic phase states eventually ceases to improve that fidelity under repeated applications.

Appendix A: Proof of Lemma 1

The appendix proves properties of Wigner-function logarithmic negativity using trace identities, Gaussian transformations, conditioning, and standard inequalities.

Proof of Lemma 1: The proof derives the stated identities from displacement conventions, Fourier-transform relations, and trace manipulations.The cyclic property of trace is justified using the Fubini-Tonelli theorem.
Proof of Lemma 1: The proof characterizes zero logarithmic negativity by a non-negative Wigner function up to measure-zero exceptions included in W+.
Proof of Lemma 1: The logarithmic negativity is shown to be non-increasing under Gaussian channels and Gaussian protocols.The argument uses Gaussian dilation, homodyne conditioning, and Gaussian channels dependent on measurement outcomes.
Proof of Lemma 1: The inequalities rely on linearity, the triangle inequality, monotonicity and concavity of the logarithm, and Jensen’s inequality.

Appendix D: Wigner functions for resource states

The appendix gives analytic Wigner-function constructions for several resource states and illustrates the cubic phase state with specified parameters.

Other resource states: The appendix states that Wigner functions for photon-added and photon-subtracted squeezed states are analytically calculable.
Cubic phase state: The Wigner function of a cubic phase state can be obtained analytically using the Airy function.
Cubic phase state: For γ = 0.05, P = 0, and s = 1, the cubic phase state’s Wigner function is shown for q > 0 and is symmetric under q → −q.
Cubic phase state: The mean photon number of |γ, P, s⟩ is calculated from the number operator and the state construction using cubic, displacement, and squeezing operations.
Cubic phase state: The calculation uses quadrature moments of a squeezed vacuum, including ⟨q^4⟩ = 3e^4s and vanishing mixed moments ⟨q^2p⟩ = ⟨pq^2⟩ = 0.

4. ON-state

The appendix introduces an analytically calculable Wigner function for the ON state.

4. ON-state: The ON-state Wigner function is expressed as a weighted combination of the vacuum Wigner function and an additional term depending on |a|^2.