Convex resource theory of non-Gaussianity

Ryuji Takagi and Quntao Zhuang

Phys. Rev. A 97, 062337 – Published 25 June 2018

Abstract

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.

Received 19 April 2018

DOI:https://doi.org/10.1103/PhysRevA.97.062337

Physics Subject Headings (PhySH)

Quantum computation Resource theories

Quantum Information, Science & Technology

Authors & Affiliations

Ryuji Takagi^1,2,* and Quntao Zhuang^2,3,†

¹Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
³Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

^*rtakagi@mit.edu
^†quntao@mit.edu

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Vol. 97, Iss. 6 — June 2018

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Images

Figure 1
Genuine non-Gaussianity measured by logarithmic negativity of some resource states. PNS: photon-number subtraction. PNA: photon-number addition. Note that for single-PNS/PNA, we are plotting the maximum genuine non-Gassianity.
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Figure 2
(a) Schematic of the distillation protocol. The yellow box is a beam splitter with transmittance $t$ . “Homo” means homodyne measurement. The dashed line denotes the postsection by the homodyne measurement result ${\tilde{p}}_{v}$ . “Vac” denotes the vacuum ancilla. (b) Mechanism of the distillation protocol. The cross section of the Wigner function at $q = 0$ is shown as functions of $p$ . Dotted blue curve represents the input vacuum state with standard deviation $σ = 1$ and purple curve represents the input resource state. In this figure, the Wigner function of an imperfect cubic phase state is plotted. The partial homodyne shifts the Gaussian distribution by $- \sqrt{t / (1 - t)} {\tilde{p}}_{v}$ where $t$ is the transmittance of the beam splitter and ${\tilde{p}}_{v}$ is the outcome of the homodyne measurement. The case of a negative ${\tilde{p}}_{v}$ is shown. The Wigner function of the resource state is also shifted in the opposite direction by $- \sqrt{(1 - t) / t} {\tilde{p}}_{v}$ , but it is not shown in the figure because the shift is small when $1 - t ≪ 1$ . The Wigner function of the output state is the product of the shifted Gaussian distribution and the slightly shifted distribution of the resource state. The shifted Gaussian works as a filter function that focuses on the region which has a large contribution to the negativity. (c) Logarithmic negativity $N_{L}$ of imperfect cubic phase states $| γ, P, s 〉$ with $γ = 0.05, P = 0$ in terms of squeezing parameter $s$ . (d) Logarithmic negativity $N_{L} ({\tilde{p}}_{v})$ (blue dots) and the probability density $P ({\tilde{p}}_{v})$ (purple squares) for the case when the input state is $| γ, P, s_{in} 〉$ with $γ = 0.05, P = 0, s_{in} = 1$ in terms of the outcome of the homodyne measurement ${\tilde{p}}_{v}$ . The horizontal dotted line is the logarithmic negativity of the input state $N_{L}^{ini}$ .
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Figure 3
(a) Logarithmic negativity after postselection $N_{L}^{post}$ in terms of success probability $P_{suc}$ for different input squeezing parameters. Input states are $| γ, P, s_{ini} 〉$ with $γ = 0.05, P = 0$ . The initial logarithmic negativity for each $s_{ini}$ are $N_{L}^{ini} = 0.11, 0.38, 0.81$ for $s_{ini} = 0.2, 0.6, 1.0$ , respectively. (b) Average logarithmic negativity after the protocol $〈N_{L}^{fin}〉$ normalized by the initial logarithmic negativity. (c) Fidelity from the target state $| γ, P^{'}, s_{targ} 〉$ after postselection $F^{post}$ in terms of success probability $P_{suc}$ for different input squeezing parameters. The target squeezing parameter is set as $s_{targ} = 4.0$ . Input states are $| γ, P, s_{ini} 〉$ with $γ = 0.05, P = 0$ . The initial fidelity for each $s_{ini}$ are $F^{ini} = 0.04, 0.10, 0.18$ for $s_{ini} = 0.2, 1.0, 1.6$ , respectively.
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Figure 4
Wigner function of $| γ, P, s 〉$ with $γ = 0.05, P = 0, s = 1$ .
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