Quantum interferences reconstruction with low homodyne detection efficiency
- Martina Esposito^{1},
- Francesco Randi^{1},
- Kelvin Titimbo^{1, 2},
- Georgios Kourousias^{3},
- Alessio Curri^{3},
- Roberto Floreanini^{2},
- Fulvio Parmigiani^{1, 3, 4},
- Daniele Fausti^{1, 3},
- Klaus Zimmermann^{1, 2} and
- Fabio Benatti^{1, 2}Email author
Received: 4 November 2015
Accepted: 20 April 2016
Published: 4 May 2016
Abstract
Optical homodyne tomography consists in reconstructing the quantum state of an optical field from repeated measurements of its amplitude at different field phases (homodyne data). The experimental noise, which unavoidably affects the homodyne data, leads to a detection efficiency \(\eta<1\). The problem of reconstructing quantum states from noisy homodyne data sets prompted an intense scientific debate about the presence or absence of a lower homodyne efficiency bound (\(\eta> 0.5\)) below which quantum features, like quantum interferences, cannot be retrieved. Here, by numerical experiments, we demonstrate that quantum interferences can be effectively reconstructed also for low homodyne detection efficiency. In particular, we address the challenging case of a Schrödinger cat state and test the minimax and adaptive Wigner function reconstruction technique by processing homodyne data distributed according to the chosen state but with an efficiency \(\eta< 0.5\). By numerically reproducing the Schrödinger’s cat interference pattern, we give evidence that quantum state reconstruction is actually possible in these conditions, and provide a guideline for handling optical tomography based on homodyne data collected by low efficiency detectors.
Keywords
1 Introduction
The mathematical approaches to the quantum state reconstruction problem are divided in two main categories, the inverse linear transform techniques and the statistical inference techniques [4, 7, 8]. The first category is based on accessing the state ρ̂ by directly inverting the linear relation in (2) through back-projection algorithms [9]. The second category is instead based on looking for the most likely density matrix that generates the observed homodyne data by means of non linear algorithms like maximum likelihood estimation [10].
Here we focus on the first approach with particular attention to the problem of processing homodyne data with low detection efficiency. In detail we adopt quantum statistical methods based on minimax and adaptive estimation methods of the Wigner function [11, 12] which have been developed to intrinsically counteract the effects of detection inefficiencies. Usually, a very high detection efficiency and ad hoc designed apparatuses with low electronic noise are required [13]. However, the scientific debate about how to process homodyne data with low efficiency is of crucial importance [10–12, 14–23] towards applying optical homodyne tomography to study physical systems where high noise conditions are unavoidable [24–31]. In this context an intense discussion developed about the presence or absence of a lower homodyne efficiency bound (\(\eta= 0.5\)), under which quantum state reconstruction is not achievable [8, 11, 12, 14–17, 19, 20, 27]. In this framework, it has been mathematically demonstrated that the algorithms of minimax and adaptive estimation of the Wigner function to be used in the following allow the tomographic reconstruction of quantum states of light for any homodyne detection efficiency η, excluding the existence of a lower bound to the efficiency beyond which faithful reconstruction is impossible [11, 12]. Nevertheless the absence of a test of such algorithms for \(\eta<0.5\) could suggest that they might not be of practical use. In a precedent work [27] we tested the algorithm for the reconstruction of Gaussian states starting from homodyne data obtained with a commercially available detection apparatus associated with an efficiency of about 0.3. In that paper the discrimination between different Gaussian states (like coherent and squeezed states) has been proved. However, Gaussian states are a very special class of states characterized by Gaussian and always positive Wigner functions. This leaves open the question about whether intrinsically quantum features, like quantum interferences (characterized by negative portions of the Wigner function), can be retrieved in low efficiency conditions or whether they would be made practically invisible by optical losses. In this respect, a physically relevant example is provided by the so-called Schrödinger cat states, that is by linear superpositions of two coherent states [4]. The main challenge is to make it clear whether or not the reconstruction algorithm can distinguish the linear superposition from the statistical mixture of the constituent coherent states. Indeed, the first state exhibits purely quantum interference patterns, while they are absent in the second state. Such a challenge has so far not been taken on. In order to fill this gap, in the following we test the reconstruction algorithm by means of numerical experiments where we generate homodyne data according to the probability distribution corresponding to a given Schrödinger cat, but distorted by an independent Gaussian noise that simulates an efficiency lower than 0.5. By suitably enlarging the size of the set of numerically generated data, we are able to reconstruct the Wigner function of the linear superposition within errors that are compatible with the theoretical bounds. Our results show that the Schrödinger cat interference pattern can actually be unambiguously reconstructed even in low homodyne detection efficiency conditions, demonstrating also the concrete feasibility of the adopted tomographic approach for \(\eta<0.5\).
2 Methods
2.1 Wigner function reconstruction
3 Results
3.1 Interfering coherent states
Homodyne reconstruction is particularly useful to expose quantum interference effects that typically spoil the positivity of the Wigner function: it is exactly these effects that are often claimed not to be accessible by homodyne reconstruction in presence of efficiency lower than 50%, i.e. when η in (7) is smaller than 0.5 [18]. In contrast, in [12] it is theoretically shown that \(\eta<0.5\) only requires increasingly larger data sets for achieving small reconstruction errors. However, this claim was not put to test in those studies as the values of η in the considered numerical experiments were close to 1.
This relation between the various parameters of the reconstruction algorithm is the one which minimizes the theoretical upper bounds. In particular, it points to the way the estimator in (9) depends on the parameter β. If η is very small, γ diverges and \(\beta+2\gamma\simeq 2\gamma\): however, for \(\eta<1/2\) but not too small, the integration interval \([-1/h,1/h]\) shrinks with increasing β, therefore a larger β, through a smaller \(1/h\), can reduce the impact of numerical noise coming from too large an integration interval. More details on the role of β are given in Appendix C.
The choice to average over the M reconstructed Wigner functions has the benefit of reducing the noise and to show more clearly the interference pattern proper to the Schrödinger cat state. This pattern is exhibited, though more noisily, by each of the M reconstructed Wigner functions: if it were a mere artifact of the reconstruction, its visibility would not increase by averaging over one hundred independent reconstructions.
Calculated \(\pmb{\Delta_{h,n}^{\eta,r}(\hat{\rho})}\) for \(\pmb{M=100}\) samples of noisy quadrature data ( \(\pmb{\eta=0.45}\) ) for two different values of β . Comparison with the mathematical prediction of the upper bound Δ
β | \(\boldsymbol{\Delta_{h,n}^{\eta,r}(\hat {\rho})}\) | Δ |
---|---|---|
0.05 | 0.21 | 2.39 |
0.1 | 0.097 | 26.07 |
Despite common beliefs, the interference features clearly appear in the reconstructed Wigner function also for an efficiency lower than 50% and the reconstruction errors are compatible with the theoretical predictions. In the following, we make a quantitative study of the visibility of the interference effects.
3.2 A Schrödinger cat interference witness
The dependence of the errors on β can be understood as follows: when β decreases the integration interval in (9) becomes larger and approaches the exact interval \([-\infty, +\infty]\). However, this occurs at the price of increasing the reconstruction error. This can be noted both in Figure 3 (larger error bars) and in Figure 2 (increasing noise effects in the reconstructed Wigner function). This problem can be overcome with a larger number of data samples M, that would reduce the reconstruction noise and compensate the effect of decreasing β.
4 Conclusions
We have numerically shown that quantum interference effects can be reconstructed by means of optical homodyne tomography also in low efficiency conditions. In particular, we simulated quadrature data affected by high electronic noise associated with a detection efficiency lower than 50% and, based on the tomographic techniques developed in [12], we reconstructed the Wigner function of a Schrödinger cat state. The ability of doing so in presence of noisy data is a challenging task that any novel reconstruction technique is asked to overcome. Then, taking into account the decay properties of the Wigner function and its Fourier transform, we have checked that the reconstruction errors conform with the theoretical error bounds computed via \(L^{2}\) norms. In order to clearly exhibit the quantum interference effects, both qualitatively in the graphic reconstruction of the Wigner function, and quantitatively in the variance of an interference witness, larger sets of homodyne data are necessary as the detection efficiency gets smaller. Our results not only support the theoretical indications that homodyne reconstruction of quantum features is actually possible for efficiencies lower than 0.5, but also demonstrate for the first time the concrete applicability of this method for low efficiencies in terms of computational effort and number of needed homodyne data.
The functional relation between the parameters h and r on n also depends on an auxiliary parameter \(\beta>0\). This was introduced in [11] in order to characterize the localization properties on \(\mathbb{R}^{2}\) of the Fourier transforms of the Wigner functions and to possibly exploit them in the reconstruction methods. These have been which are indeed applied to the following class of density matrices: \( \mathcal{A}_{\beta,s,L} = \{\hat{\rho} : \int_{\mathbb{R}^{2}}\mathrm {d}q\,\mathrm {d}p \vert F [W_{\rho}](q,p)\vert^{2} \mathrm {e}^{2\beta(q^{2}+p^{2})^{s/2}} \leq (2\pi)^{2}L \}\).
Declarations
Acknowledgements
The authors are grateful to Francesca Giusti for insightful discussions and critical reading of the paper and thank referee number 2 of the manuscript in reference [27] for stimulating the discussions that led to this work. This work has been supported by a grant from the University of Trieste (FRA 2013).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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