Operational quasiprobabilities for continuous variables

Jeongwoo Jae, Junghee Ryu, and Jinhyoung Lee

Phys. Rev. A 96, 042121 – Published 31 October 2017

Abstract

We generalize the operational quasiprobability involving sequential measurements proposed by Ryu et al. [Phys. Rev. A 88, 052123 (2013)] to a continuous-variable system. The quasiprobabilities in quantum optics are incommensurate, i.e., they represent a given physical observation in different mathematical forms from their classical counterparts, making it difficult to operationally interpret their negative values. Our operational quasiprobability is commensurate, enabling one to compare quantum and classical statistics on the same footing. We show that the operational quasiprobability can be negative against the hypothesis of macrorealism for various states of light. Quadrature variables of light are our examples of continuous variables. We also compare our approach to the Glauber-Sudarshan $P$ function. In addition, we suggest an experimental scheme to sequentially measure the quadrature variables of light.

Received 21 June 2017
Corrected 12 March 2018

DOI:https://doi.org/10.1103/PhysRevA.96.042121

Physics Subject Headings (PhySH)

Quantum foundations Quantum information processing with continuous variables Quantum states of light

General PhysicsQuantum Information, Science & Technology

Corrections

12 March 2018

Erratum

Publisher's Note: Operational quasiprobabilities for continuous variables [Phys. Rev. A 96, 042121 (2017)]

Jeongwoo Jae, Junghee Ryu, and Jinhyoung Lee

Phys. Rev. A 97, 039905 (2018)

Authors & Affiliations

Jeongwoo Jae¹, Junghee Ryu^2,*, and Jinhyoung Lee^1,†

¹Department of Physics, Hanyang University, Seoul 133-791, Republic of Korea
²Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore

^*rjhui82@gmail.com
^†hyoung@hanyang.ac.kr

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Vol. 96, Iss. 4 — October 2017

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Images

Figure 1
Measurement setups when two observables are considered. On implementing the measurements, four cases are possible: (a) the reference of no measurement, (b), (c) the single measurements of the two alternative observables, and (d) the sequential measurement of them. Each setup is denoted by the tuple $(m, n)$ , which is associated with the degrees of Hermite polynomials $H_{m} (x)$ and $H_{n} (x)$ .
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Figure 2
Venn diagram for theoretical predictions of operational quasiprobability distributions $W (x_{1}, x_{2})$ for measurements $M_{1}$ and $M_{2}$ . In a classical model assuming the NSIT and AoT conditions, $W$ becomes a joint probability $P (x_{1}, x_{2} | M_{1}, M_{2})$ . However, the $W$ can still be non-negative even though the condition of no signaling in time is violated.
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Figure 3
Operational quasiprobabilities for some states, $W (\vec{α}, \vec{β}) = W (α_{r}, α_{i}, β_{r}, β_{i})$ , for fixed $\vec{α}$ . The gray plane implies $W = 0$ , and the blue regions (below the plane) are negative values of $W$ . We plot $W (\vec{0}, \vec{β})$ for (a) the vacuum, (b) the number state $|2〉$ , and (c) the squeezed state of average photon number ${\bar{n}}_{sq} = 5$ . The plots of (a) and (b) are symmetric in the azimuthal direction. (d) $W$ of number state $|2〉$ is plotted by changing the $M_{1}$ basis from $\vec{α} = (0, 0)$ to $(1, 1)$ . The positive (red) regions in turn appear around $\vec{β} = (1, 1)$ .
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Figure 4
Negativity $N$ for the vacuum, coherent ( $•$ ), number ( $▪$ ), squeezed vacuum ( $⧫$ ), plus and minus cat states ( $▾$ ), and thermal state ( $▴$ ) as a function of average photon number $\bar{n}$ . The $N$ of coherent, squeezed vacuum, and cat states is saturated around $\bar{n} = 20$ , and in particular the coherent state is saturated to $N = 0.250045$ , denoted by dashed line. For the vacuum state $\bar{n} = 0, N = 0.145420$ (solid line). In contrast, for the thermal state, $N$ is decreasing as $\bar{n}$ is increasing. (a) The negativities of plus and minus cat states are the same, and they are close to that of the coherent state around $\bar{n} = 10$ .
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Figure 5
Optical scheme to sequentially measure quadrature variables. (a) Quadrature variables of input state $\hat{ϱ}$ are obtained by jointly measuring homodyne detection (HD). Their local oscillators (LOs) 1 and 2 are locked to measure $x$ and $p$ quadratures. We employ a beam splitter to entangle the input state with ancillary zero states ${|0〉}_{x, p}$ . We consider the ancillary states to be eigenstates of zero quadrature, i.e., $\hat{x} {|0〉}_{x} = 0$ and $\hat{p} {|0〉}_{p} = 0$ . The obtained statistics is a scaled Husimi $Q$ function $Q (\sqrt{2} μ)$ of the measurement basis $μ = x + i p$ . The postmeasurement state $\hat{ϕ}$ is obtained as the coherent state $| μ 〉 〈 μ |$ when photodetector (PD) 1 is not clicked. (b) Sequential measurement of the quadrature variables can be realized by performing the $Q$ -function measurement consecutively at two different times $t_{1}, t_{2}$ .
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