Tomographic entanglement indicators in frequency combs and Talbot carpets

The measurement of any observable in a quantum mechanical system yields a histogram of the state of the system in the eigenbasis of that observable. Measurements of a judiciously chosen quorum of appropriate observables of a system that are informationally complete yield a set of histograms called a tomogram. Quantum state reconstruction seeks to obtain the density matrix from the tomogram. However, the reconstruction of the state of a radiation field from the corresponding optical tomogram could be both tedious and complex [1]. In many contexts, the reconstruction program is also limited by the presence of inherent aberrations in the source [2, 3]. It is worth noting that, even in the simple case of a two-level atom interacting with a radiation field, the state of the field subsystem was experimentally reconstructed from the corresponding tomogram at various instants during temporal evolution only as recently as 2017 [4]. With an increase in the number of field modes interacting with an atomic system, the inevitable entanglement that arises during dynamical evolution makes state reconstruction an increasingly formidable task. It would therefore be very helpful to read off information about a state, wherever possible, directly from the experimentally accessible histograms. In particular, estimating the extent of entanglement through simple manipulations of the relevant tomograms alone becomes an interesting and important exercise.

Entanglement is an essential resource in quantum information processing, and quantifying the entanglement between two subsystems of a bipartite system is necessary for this purpose. One of the standard measures of entanglement between the two subsystems A and B of a bipartite system is the subsystem von Neumann entropy ξ_SVNE = −Tr (ρ_i  log₂  ρ_i ) where ρ_i (i = A, B) is the reduced density matrix of the subsystem concerned. The computation of ξ_SVNE, however, requires a knowledge of the full density matrix, in contrast to the tomographic approach. The advantage of the latter approach has been illustrated in certain continuous-variable (CV) systems—for instance, in identifying qualitative signatures of entanglement between the radiation fields in the output ports of a quantum beam-splitter [5]. Further, a detailed analysis of entanglement has been carried out theoretically using several indicators obtained directly from tomograms in different contexts, e.g., close to avoided energy-level crossings [6]. The efficacy of such indicators has been assessed by comparison with standard entanglement measures [7, 8]. Further, these indicators have also been examined using the IBM quantum computing platform, by translating certain multipartite hybrid quantum systems into suitable equivalent circuits [9]. It is to be noted that, in these examples, measurements of multiple observables corresponding to the rotated quadrature operators

$$\begin{equation}{X}_{\theta }=(a{\text{e}}^{-\text{i}\theta }+{a}^{{\dagger}}{\text{e}}^{\text{i}\theta })/\sqrt{2},\end{equation} \tag{ 1 }$$

(where (a, a^†) are the photon annihilation and creation operators and θ ∈ [0, π)) are required, in principle.

In marked contrast, a different entanglement indicator has been used in certain recent experiments reported in the literature. In an experiment [10] involving an entangled pair of Talbot carpets generated from photons produced from spontaneous parametric down conversion (SPDC), an entanglement indicator I_D based on Bell-type inequalities has been used. The Talbot effect itself has been observed in several experimental settings since its discovery in 1836 [11]. In the quantum context, this effect has been demonstrated both with single photons and with entangled photon pairs [12], and also with single photons prepared in a lattice of angular momentum states [13]. Using a four-level quantum dot molecule, the re-formation of a plane wave into a periodic wave-form has been demonstrated [14]. Single photon interference owing to passage through a grating has been invoked to propose the creation of D-dimensional quantum systems encoded in the spatial degrees of freedom of light [15]. A proposal to realise a class of entangled states to demonstrate nonlocality in phase space using grating techniques [16], and the observation of the optical spatiotemporal Talbot effect [17], are further instances of continuing work on the Talbot effect for testing fundamental concepts and implementing quantum information processing.

Another interesting example of interferometry [18] pertains to biphoton frequency combs. Qubits are encoded in this CV system using the time–frequency continuous degrees of freedom of photon pairs generated through the SPDC process. In an attempt to provide higher capacity and increased noise resilience in quantum communication and computations in CV systems with time–frequency entanglement, a biphoton frequency comb with the potential to create a large Hilbert space (648 dimensions) has been investigated [19]. This approach provides a good platform, in principle, for improved quantum communication. The appearance of polarization-entangled biphoton frequency combs with over 1400 frequency modes has been demonstrated experimentally [20]. Such massive mode-entangled biphoton frequency combs are expected to increase the capacity and rate of quantum communication. Protocols have been proposed, involving measurements of appropriate modular variables, that facilitate a read-out of the encoded discrete quantum information from the corresponding logical states. Further, their experimental feasibility has also been discussed [21]. In reference [22], the authors assess the metrological power of different classes of biphoton states with a non-Gaussian time–frequency spectral distribution. These states could possibly be produced with atomic photon sources, bulk nonlinear crystals and integrated photonic waveguide devices. Further, new methods to characterize the spectral-temporal distribution of single photons have been examined in [23]. In reference [24], the authors investigate the interferometric signatures of different spectral symmetries of biphoton states. A sizable literature exists on the generation and manipulation of quantum states of light in nonlinear AlGaAs chips and their use in quantum networks. In particular, the control and the temporal delay between the photons comprising a pair provides a means of switching from symmetric to anti-symmetric quantum frequency states [25]. Quantum state engineering with biphoton states produced by SPDC, and the characterization of arbitrary states with frequency and time degrees of freedom, have been carried out in [26]. A second quantized description treating time and frequency as operators that can be used to define a universal set of gates, and to implement continuous variable quantum information protocols, has been examined in detail [27]. The frequency entanglement of biphoton states has been shown to simulate anyonic particle statistics [28]. Such photon pairs have been studied extensively to investigate entanglement properties and potential logic gate operations [29, 30].

In some of these experiments (see, for instance, [10, 18]), the tomographic approach mentioned earlier provides an efficient, readily usable procedure for capturing signatures of bipartite entanglement and understanding its qualitative features. It has been demonstrated recently [31] that it is possible to implement single-qudit logic gate operations using the Talbot effect. The next step would be to extend this investigation to the case of a pair of entangled Talbot carpets generated using a suitable combination of two such experimental setups. It is here that the tomographic approach comes into its own, as it has the advantage of simplifying considerably the necessary experimental arrangement. Further, we demonstrate the convenience and usefulness of the tomographic indicators in both the experimental situations under discussion, involving, respectively, entangled Talbot carpets and biphoton frequency combs. The versatility of the tomographic approach is brought out by the fact that optical tomograms are useful in the experiment on entangled Talbot carpets, while chronocyclic tomograms display the features of entanglement in the experiment on biphoton frequency combs.

The plan of the rest of this paper is as follows. In section 2, we review the salient features of the relevant tomograms, and describe the entanglement indicators that can be obtained directly from these tomograms. In section 3, we apply our procedure to assess the degree of entanglement between a pair of Talbot carpets. Whereas it has been suggested in [10] that an experimental setup which uses Bell-type inequalities could determine the degree of entanglement, we demonstrate that tomograms provide a simpler and more direct procedure for this purpose. Further, in section 4, we establish that the respective chronocyclic tomograms corresponding to the two biphoton states provide manifest entanglement signatures that distinguish between the states. Relevant technical details about the two biphoton states are presented in the appendices A and B. We conclude with some brief remarks in the final section.

2.1. Quadrature histograms

We begin by describing the salient features of optical tomograms. In a two-mode CV system, the infinite set of rotated quadrature operators [32–34] given by

$$\begin{align}\hfill {\mathbb{X}}_{{\theta }_{A}}=& \,({a}^{{\dagger}}{\text{e}}^{\text{i}{\theta }_{A}}+a{\text{e}}^{-\text{i}{\theta }_{A}})/\sqrt{2},\quad \mathrm{a}\mathrm{n}\mathrm{d}\hfill \\ \hfill {\mathbb{X}}_{{\theta }_{B}}=& \,({b}^{{\dagger}}{\text{e}}^{\text{i}{\theta }_{B}}+b{\text{e}}^{-\text{i}{\theta }_{B}})/\sqrt{2},\hfill \end{align} \tag{ 2 }$$

(where θ_a , θ_b ∈ [0, π)) constitute the quorum of observables that carries complete information about the state. Here (a, a^†) and (b, b^†) are the annihilation and creation operators corresponding respectively to subsystems A and B of the bipartite system, so that [a, a^†] = 1 and [b, b^†] = 1. The bipartite tomogram is

$$\begin{equation}w({X}_{{\theta }_{A}},{\theta }_{A};{X}_{{\theta }_{B}},{\theta }_{B})=\langle {X}_{{\theta }_{A}},{\theta }_{A};{X}_{{\theta }_{B}},{\theta }_{B}\vert {\rho }_{\mathrm{A}\mathrm{B}}\vert {X}_{{\theta }_{A}},{\theta }_{A};{X}_{{\theta }_{B}},{\theta }_{B}\rangle ,\end{equation} \tag{ 3 }$$

where ρ_AB is the bipartite density matrix and ${\mathbb{X}}_{{\theta }_{i}}\vert {X}_{{\theta }_{i}},{\theta }_{i}\rangle ={X}_{{\theta }_{i}}\vert {X}_{{\theta }_{i}},{\theta }_{i}\rangle$ (i = A, B). Here, $\vert {X}_{{\theta }_{A}},{\theta }_{A};{X}_{{\theta }_{B}},{\theta }_{B}\rangle$ stands for $\vert {X}_{{\theta }_{A}},{\theta }_{A}\rangle \otimes \vert {X}_{{\theta }_{B}},{\theta }_{B}\rangle$. The normalisation condition is given by

$$\begin{equation}{\int }_{-\infty }^{\infty }\,\mathrm{d}{X}_{{\theta }_{A}}{\int }_{-\infty }^{\infty }\,\mathrm{d}{X}_{{\theta }_{B}}w({X}_{{\theta }_{A}},{\theta }_{A};{X}_{{\theta }_{B}},{\theta }_{B})=1\end{equation} \tag{ 4 }$$

for each θ_a and θ_b . We note that θ_a = θ_b = 0 corresponds to the position quadrature. Since the investigation of entanglement in reference [10] only pertains to the intensity pattern on the screen, we have treated the screen as the tomographic slice. This sets the specific values of θ_a and θ_b equal to zero. As we are only interested in the corresponding slice of the tomogram, we adopt the following simplified notation. A slice of the tomogram (equivalently, the histogram in the position basis) for a bipartite state |ψ_AB⟩ is given by

$$\begin{equation}w({x}_{\mathrm{A}};{x}_{\mathrm{B}})=\vert \langle {x}_{\mathrm{A}};{x}_{\mathrm{B}}\vert {\psi }_{\mathrm{A}\mathrm{B}}\rangle {\vert }^{2}.\end{equation} \tag{ 5 }$$

Here, {|x_i⟩} (i = a, b) are the position eigenstates in the two subsystems, and |x_A; x_B⟩ denotes the corresponding product state. The marginal distributions obtained from the joint distribution above yield the corresponding slices w_A(x_A) and w_B(x_B) of the subsystem (or reduced) tomograms.

2.2. Chronocyclic tomogram

The analogy between an ultrashort light pulse and a quantum mechanical wave function leads [35] to a chronocyclic representation for the study of ultrashort pulses, where the time t and frequency ω are the conjugate observables. The state of a single photon of frequency ω is denoted in a spectral representation of infinitely narrow-band pulses by |ω⟩. The superposed state of a photon that has a frequency ω with a probability amplitude $\mathcal{S}(\omega )$ is given by $\int \mathrm{d}\omega \,\mathcal{S}(\omega )\vert \omega \rangle$. In an equivalent temporal representation of infinitely short-duration pulses {|t⟩}, this one-photon state is $\int \mathrm{d}t\,\tilde{\mathcal{S}}(t)\vert t\rangle$, where $\tilde{\mathcal{S}}(t)$ is the Fourier transform of $\mathcal{S}(\omega )$. A family of rotated observables (ω cos θ + t sin θ) can then be defined [36], where ω and t have been scaled by a natural time scale of the system to make them dimensionless. Measurements of these rotated observables form the basis of chronocyclic tomography, in which the set of histograms corresponding to these observables gives the tomogram of the state of a one-photon system. In this chronocyclic representation, a one-photon state can also be described in the time–frequency 'phase space' by a corresponding Wigner function [35]. Extension to multipartite states corresponding to two or more photons is straightforward. For instance, two photons of frequencies ω and ω' are given by the biphoton CV bipartite state |ω⟩ ⊗ |ω'⟩.

In reference [18], the experiment on biphoton frequency combs is carried out in the time–frequency domain. It follows naturally that, in the tomographic approach, chronocyclic tomograms need to be examined. This allows for four choices of tomographic slices, namely, frequency–frequency, frequency–time, time–frequency, and time–time. We have examined the various possibilities and found that the time–time tomographic slice corresponding to each biphoton entangled state is optimal for distinguishing between them. We note that a priori there is no set procedure in general for choosing the optimal slice (i.e., the analogues of θ_A and θ_B in quadrature tomograms). However, the physical quantities in the present instance are only frequency and time, and hence only four possibilities exist. The time–time slice of the bipartite chronocylic tomogram corresponding to state |ψ⟩, is given by,

$$\begin{equation}w(t;{t}^{\prime })=\vert \langle t;{t}^{\prime }\vert \psi \rangle {\vert }^{2},\end{equation} \tag{ 6 }$$

where |t; t'⟩ stands for |t⟩ ⊗ |t'⟩ in the temporal representation. This is the chronocyclic analogue of (5). The marginal distributions are obtained from (6) as described before.

2.3. Tomographic entanglement indicators

The extent of correlation between the subsystems can be deduced from the tomographic entropies. For instance, for the slice defined in (5), the bipartite tomographic entropy is given by

$$\begin{equation}{S}_{\mathrm{A}\mathrm{B}}=-{\int }_{-\infty }^{\infty }\,\mathrm{d}{x}_{\mathrm{a}}{\int }_{-\infty }^{\infty }\,\mathrm{d}{x}_{\mathrm{B}}w({x}_{\mathrm{A}};{x}_{\mathrm{B}})\,{\mathrm{log}}_{2}\,w({x}_{\mathrm{A}};{x}_{\mathrm{B}}).\end{equation} \tag{ 7 }$$

The subsystem tomographic entropy is given by

$$\begin{equation}{S}_{i}=-{\int }_{-\infty }^{\infty }\,\mathrm{d}{x}_{i}\,{w}_{i}({x}_{i}){\mathrm{log}}_{2}\,{w}_{i}({x}_{i})\,\,(i=\text{A,B}).\end{equation} \tag{ 8 }$$

Signatures of the extent of entanglement can be gleaned from the mutual information [7],

$$\begin{equation}{\varepsilon }_{\text{TEI}}={S}_{\mathrm{A}}+{S}_{\mathrm{B}}-{S}_{\mathrm{A}\mathrm{B}}.\end{equation} \tag{ 9 }$$

In the chronocyclic case, analogous definitions of entropies and mutual information hold for any tomographic slice given by (6).

For our purposes, we will largely be concerned with the experimental setup proposed in [10] except for the phase shifter used to calculate I_D. We note that while [10] and the present paper are both concerned with the extent of entanglement of two Talbot carpets, the procedures used are very different. In marked contrast to the procedure for entanglement assessment in [10], we use the tomographic approach to compute the entanglement indicator, namely, ɛ_tei. This can be obtained directly from a single slice of the tomogram provided by the screen in the experimental setup. This approach neither requires a phase shifter in the experimental setup, nor is it concerned with calculating Bell-type inequalities. The purpose of the present work, of course, is to compare the trends in I_D reported in [10] with trends in ɛ_TEI, the latter being a very different entanglement indicator.

In the proposed experiment (figure 1), light from a laser source passes through a nonlinear crystal (NLC). Entangled SPDC photon pairs are produced, with a spatial correlation given by

$$\begin{equation}R=\frac{{\kappa }_{+}^{2}-{\kappa }_{-}^{2}}{{\kappa }_{+}^{2}+{\kappa }_{-}^{2}}=-\frac{{{\Delta}}_{+}^{2}}{{{\Delta}}_{-}^{2}}.\end{equation} \tag{ 10 }$$

Here κ₊ is the width of the pump field frequency profile, κ₋ is the standard deviation in the phase matching of the two output photons, and

$$\begin{equation}\frac{1}{{{\Delta}}_{\pm }^{2}}=\frac{1}{{\kappa }_{+}^{2}}\pm \frac{1}{{\kappa }_{-}^{2}}.\end{equation} \tag{ 11 }$$

The light is then guided along two different paths A and B using appropriate mirrors. Along path i (i = A, B), a D-slit aperture D_i , a lens L_1i , a grating G_i , a lens L_2i and a screen Sc_i are placed as shown on the figure. Talbot carpets are seen on each screen, and the extent of entanglement between the two is to be assessed. The slit width in each aperture is δ, and the inter-slit spacing is s. Each grating has slit width σ and period ℓ. The screens are in the x–y plane.

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The entangled Talbot state is of the form

$$\begin{equation}\vert {\Psi}\rangle =\sum\limits _{{d}_{1},{d}_{2}=0}^{D-1}\,{C}_{{d}_{1},{d}_{2}}{\vert {d}_{1}\rangle }_{\mathrm{A}}\otimes {\vert {d}_{2}\rangle }_{\mathrm{B}},\end{equation} \tag{ 12 }$$

where

$$\begin{equation}{C}_{{d}_{1},{d}_{2}}=\mathcal{N}\,\mathrm{exp}\left\{-\frac{{s}^{2}}{4{{\Delta}}_{+}^{2}}\left({d}_{1}^{2}-2R{d}_{1}{d}_{2}+{d}_{2}^{2}\right)\right\},\end{equation} \tag{ 13 }$$

where $\mathcal{N}$ is the normalization constant such that $\sum _{{d}_{1},{d}_{2}}\vert {C}_{{d}_{1},{d}_{2}}{\vert }^{2}=1$. The basis states are now given by

$$\begin{align}\hfill {\langle {x}_{i}\vert d\rangle }_{i}& ={T}_{d}({x}_{i})={\mathcal{A}}_{d}\sum\limits _{m=-\infty }^{\infty }\hfill \\ \hfill & \quad \times \mathrm{exp}\left\{-\frac{{(2\pi m\sigma )}^{2}}{2{\ell }^{2}}-\frac{{({x}_{i}-ds-m\ell )}^{2}}{4{\delta }^{2}}\right\},\,\hfill \end{align} \tag{ 14 }$$

where i = A, B and ${\mathcal{A}}_{d}$ is the normalisation constant. It is straightforward to measure the intensity distribution at different points (x_A, x_B) along the x-axis on the two screens. The expression for this distribution can be obtained from Ψ(x_A, x_B) by using (14).

The extent of entanglement has been assessed from I_D for different values of the spatial correlation R and the number of slits D [10]. For this purpose, two pairs of local measurements A₁, A₂ (respectively, B₁, B₂) are performed on both A and B. Each of these four measurements has D outcomes with projective operators $\left\{{\vert {f}_{{\alpha }_{1}}\rangle }_{\mathrm{A}\,\mathrm{A}}\hspace{-2.5pt}\langle {f}_{{\alpha }_{1}}\vert \right\}$, $\left\{{\vert {f}_{{\alpha }_{2}}\rangle }_{\mathrm{A}\,\mathrm{A}}\hspace{-2.5pt}\langle {f}_{{\alpha }_{2}}\vert \right\}$, $\left\{{\vert {g}_{{\beta }_{1}}\rangle }_{\mathrm{B}\,\mathrm{B}}\hspace{-0.5pt}\langle {g}_{{\beta }_{1}}\vert \right\}$, and $\left\{{\vert {g}_{{\beta }_{2}}\rangle }_{\mathrm{B}\,\mathrm{B}}\hspace{-0.5pt}\langle {g}_{{\beta }_{2}}\vert \right\}$ corresponding to A₁, A₂, B₁, and B₂ respectively. Here

$$\begin{equation}{\vert {f}_{{\alpha }_{j}}\rangle }_{\mathrm{A}}=\frac{1}{\sqrt{D}}\;\sum\limits _{d=0}^{D-1}\,{\mathrm{e}}^{2\pi id(f+{\alpha }_{j})/D}{\vert \mathrm{d}\rangle }_{\mathrm{A}},\end{equation} \tag{ 15 }$$

$$\begin{equation}{\vert {g}_{{\beta }_{j}}\rangle }_{\mathrm{B}}=\frac{1}{\sqrt{D}}\;\sum\limits _{d=0}^{D-1}\,{\mathrm{e}}^{2\pi id(-g+{\beta }_{j})/D}{\vert \mathrm{d}\rangle }_{\mathrm{B}}\,(j=1,2)\end{equation} \tag{ 16 }$$

with α₁ = 0, α₂ = 0.5, β₁ = 0.25, β₂ = −0.25 and f, g = 0, 1, ..., D − 1. In what follows, we shall denote the factored product basis $\left\{{\vert {f}_{{\alpha }_{i}}\rangle }_{\mathrm{A}}\otimes {\vert {g}_{{\beta }_{j}}\rangle }_{\mathrm{B}}\right\}$ by $\left\{\vert {f}_{{\alpha }_{i}\mathrm{A}};{g}_{{\beta }_{j}\mathrm{B}}\rangle \right\}$. (In the tomographic approach that will be outlined later, these phase shifts α_j , β_j (j = 1, 2) need not be implemented, and the entanglement indicator can be deduced solely from the original intensity patterns on the screens). It can be seen [37] that for I_D ⩽ 2, the states are unentangled. Here,

$$\begin{equation}{I}_{\mathrm{D}}=\sum\limits _{k=0}^{[D/2]-1}\left(1-\frac{2k}{D-1}\right){J}_{k},\end{equation} \tag{ 17 }$$

where

$$\begin{align*}\hfill {J}_{k}=& \,P({A}_{1}={B}_{1}+k)\,-\,P({A}_{1}={B}_{1}-k-1)\hfill \\ \hfill & +P({B}_{2}={A}_{1}+k)\,-\,P({B}_{2}={A}_{1}-k-1)\hfill \\ \hfill & +P({B}_{1}={A}_{2}+k+1)\,-\,P({B}_{1}={A}_{2}-k)\hfill \\ \hfill & +P({A}_{2}={B}_{2}+k)\,-\,P({A}_{2}={B}_{2}-k-1)\hfill \end{align*}$$

and

$$\begin{equation}P({A}_{i}={B}_{j}+k)=\sum\limits _{p=0}^{D-1}\,P({A}_{i}=(p+k)\,\mathrm{mod}\,D,{B}_{j}=p).\end{equation} \tag{ 18 }$$

The joint probability distribution of the outcomes being A_i = p and B_j = q is denoted by P(A_i = p, B_j = q).

In the tomographic approach (as in the case of I_D), we set σ = 0.05ℓ, δ = 0.025s, ℓ = 1, s = 1/D, and κ₊ = 9ℓ to facilitate comparison. As outlined in section 2, we compute ɛ_TEI (see (9)) from the optical tomogram (5). In figure 2, ɛ_TEI, ξ_SVNE, and I_D are plotted versus D for various values of R. These values are chosen so as to facilitate comparison with [10]. It is clear that ɛ_TEI agrees well with ξ_SVNE. Further, ɛ_TEI captures the gross features of I_D. From figure 2, we see that when ɛ_TEI is computed from tomograms corresponding to other basis sets, the results do not change, and that the extent of entanglement significantly increases even with very small changes in R. (For D = 10, ɛ_TEI ∼ 0.3 for R = 0.998, and $\sim \,3.32$ for R = 1 correct to two decimal places).

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This feature can be understood by examining the manner in which the subsystem density matrix (in terms of which standard entanglement measures are computed) responds to small changes in R. This is presented in figure 3. We first compare the top left and the bottom left panels in figure 3. These show the subsystem density matrix for R = 0.998 and 1 respectively, in the basis set comprising the slits (i.e., where each slit is treated as a basis state). It is evident that the subsystem density matrix corresponding to the top left panel has non-zero off-diagonal contributions. In contrast, corresponding to the bottom left panel, the subsystem density matrix is diagonal. For R = 1, it is clear from the figure that only the diagonal elements survive, leading to a maximally mixed state. This therefore indicates that the bipartite pure state is maximally entangled for R = 1. This is clearly not the case in the top left panel, where the off-diagonal contribution is substantial. We now point out another feature that follows by comparing the top right and the bottom right panels. In both these panels, the subsystem density matrix is given in a different basis which is phase shifted relative to that corresponding to the left panels. (We choose this basis also because it connects with that used in [10] where a phase shifter creates the necessary shift of basis). It is clear from these two panels that a small change in R from 0.998 (top right) to 1 (bottom right) produces a maximally mixed subsystem state. This follows as in the earlier discussion from the diagonal form of the density matrix. A comparison between the bottom right and left panels establishes the basis-independence of the maximally mixed subsystem state for R = 1.

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We first recapitulate the salient features of the experiment [18] on two pairs of entangled biphoton states, denoted by |Ψ_α ⟩ and |Ψ_β ⟩, which are frequency combs comprising finite-width peaks. The entangled photons were generated using SPDC. In the setup, the resonant frequency of the cavity is denoted by $\bar{\omega }$, and the pump photons have frequency ω_p. If ω_S and ω_I denote the signal and idler frequencies, respectively, and Ω is their difference, it can be seen (appendix A) that |Ψ_α ⟩ and |Ψ_β ⟩ can be expressed as given below.

$$\begin{align}\hfill \vert {{\Psi}}_{\alpha }\rangle & ={\mathcal{N}}_{\alpha }^{-1/2}\int \mathrm{d}{\omega }_{\mathrm{S}}\int \mathrm{d}{\omega }_{\mathrm{I}}{f}_{+}({\omega }_{\mathrm{S}}+{\omega }_{\mathrm{I}}){f}_{-}({\Omega})\hfill \\ \hfill & \quad \times {f}_{\text{cav}}({\omega }_{\text{s}}){f}_{\text{cav}}({\omega }_{\text{i}})\vert {\omega }_{\text{s}}\rangle \otimes \vert {\omega }_{\text{i}}\rangle .\hfill \end{align} \tag{ 19 }$$

Here,

$$\begin{equation}{f}_{-}({\Omega})={\mathrm{e}}^{-{({\Omega}-{{\Omega}}_{0})}^{2}/4{({\Delta}{\Omega})}^{2}},\end{equation} \tag{ 20 }$$

where Ω₀ and ΔΩ are the mean and standard deviation of Ω. f_cav is the Gaussian comb

$$\begin{equation}{f}_{\text{cav}}(\omega )=\sum\limits _{n}{\mathrm{e}}^{-{(\omega -n\bar{\omega })}^{2}/2{({\Delta}\omega )}^{2}},\end{equation} \tag{ 21 }$$

where Δω is the standard deviation of each Gaussian, ${\mathcal{N}}_{\alpha }$ is the normalisation constant, and f₊(ω_S + ω_I) = δ(ω_p − ω_S − ω_I). We note that (19) features the product f_cav(ω_S)f_cav(ω_I), where f_cav(ω) is a superposition of Gaussians corresponding to odd and even values of n such that the two are in phase with each other.

The second biphoton state is given by

$$\begin{align}\hfill \vert {{\Psi}}_{\beta }\rangle & ={\mathcal{N}}_{\beta }^{-1/2}\int \mathrm{d}{\omega }_{\mathrm{S}}\int \mathrm{d}{\omega }_{\mathrm{I}}{f}_{+}({\omega }_{\mathrm{S}}+{\omega }_{\mathrm{I}}){f}_{-}({\Omega})\hfill \\ \hfill & \quad \times {g}_{\text{cav}}({\omega }_{\mathrm{S}}){f}_{\text{cav}}({\omega }_{\mathrm{I}})\vert {\omega }_{\text{s}}\rangle \otimes \vert {\omega }_{\text{i}}\rangle ,\hfill \end{align} \tag{ 22 }$$

where

$$\begin{equation}{g}_{\text{cav}}(\omega )=\sum\limits _{n}{(-1)}^{n}{\mathrm{e}}^{-{(\omega -n\bar{\omega })}^{2}/2{({\Delta}\omega )}^{2}}.\end{equation} \tag{ 23 }$$

Here, ${\mathcal{N}}_{\beta }$ is the normalisation constant. In contrast to |Ψ_α ⟩, (22) features the product g_cav(ω_s)f_cav(ω_i), where g_cav(ω) is a superposition of Gaussians corresponding to odd and even n such that the two are out of phase with each other. In [18], the authors address several related issues pertaining to these biphoton frequency combs. The focus is primarily on distinguishing between these entangled states, and further, manipulating them with quantum logic gate operations. Our interest in the present case is only in the former aspect. Whereas in [18] an interferometric setup is used for this purpose, in what follows, we will use the tomographic approach to distinguish between the two biphoton states. Since photon coincidence counts were used to experimentally distinguish between the two states, it is reasonable to expect that the time–time slices of the tomograms corresponding to the two biphoton states will capture the difference. The time–time slices are denoted by

$$\begin{equation}{w}^{x}({t}_{\text{S}};{t}_{\text{I}})=\langle {t}_{\text{S}};{t}_{\text{I}}\vert {{\Psi}}_{x}\rangle \langle {{\Psi}}_{x}\vert {t}_{\text{S}};{t}_{\text{I}}\rangle ,\end{equation} \tag{ 24 }$$

where (x = α, β), and |t_s;t_I⟩ stands for |t_s⟩ ⊗ |t_I⟩ in the time–time basis.

We work with the parameter values used in reference [18], namely, ω_p/(2π) = 391.8856 THz, $\bar{\omega }/(2\pi )=19.2$ GHz, Δω/(2π) = 1.92 GHz, Ω₀/(2π) = 10.9 THz, and ΔΩ/(2π) = 6 THz. The time–time slices of the tomograms of |Ψ_α ⟩ and |Ψ_β ⟩ have been obtained by substituting (19) and (22) in turn in (24) and simplifying the resulting expressions (see appendix B).

As expected, w^α (t_S; t_I) and w^β (t_S; t_I) are distinctly different from each other, as seen in figures 4(a)–(c). This difference arises because |Ψ_α ⟩ and |Ψ_β ⟩ correspond to combs that are clearly displaced with respect to each other, when expressed in the time–time basis.

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Next, we calculate the reduced tomograms ${w}_{i}^{x}({t}_{i})$ corresponding to subsystem i (where i = S, I and x = α, β) by integrating out the other subsystem. (For instance, ${w}_{\text{s}}^{x}({t}_{\text{s}})=\int \mathrm{d}{t}_{\text{i}}{w}^{x}({t}_{\text{s}};{t}_{\text{i}})$.) Using these full-system and subsystem chronocyclic tomograms in (7)–(9), we obtain the entanglement indicator ɛ_TEI corresponding to any chosen slice of the chronocyclic tomogram. (For ease of notation, we have dropped the explicit dependence of ɛ_TEI on the choice of both the tomogram slice and the specific state.) In the case of the time–time slice of the tomograms we get, finally, the values ɛ_TEI = 6.50 for the state |Ψ_α ⟩, and ɛ_TEI = 5.44 for the state |Ψ_β ⟩. Thus, ɛ_TEI clearly distinguishes between these two biphoton states. We emphasize that the methods used by us could, in principle, provide an alternative approach to the procedure adopted in the experiment.

The tomographic entanglement indicator ɛ_TEI proves to be a very useful tool which is also easily computed from the histogram of a relevant measured observable. In the case of the entangled Talbot states, ɛ_TEI computed from the histogram in the position basis, closely mimics the standard entanglement measure ξ_SVNE. We emphasize that a single slice suffices to estimate the extent of entanglement, and we do not require the rotated quadratures in this case. In fact, we have shown that ɛ_TEI is better than I_D, the Bell-like-inequality-based indicator. Further, we have unambiguously distinguished between a pair of biphoton states using the entanglement indicator ɛ_TEI. This paper demonstrates alternative procedures using the tomographic approach that are useful and efficient in a variety of experimentally relevant CV systems.

We acknowledge useful discussions with P Milman, Laboratoire Mat$\acute{\mathrm{e}}$riaux et Ph$\acute{\mathrm{e}}$nom$\grave{\mathrm{e}}$nes Quantiques, Universit$\acute{\mathrm{e}}$ de Paris. SL was supported, in part, by a grant from Mphasis tothe Centre for Quantum Information, Communication, and Computing (CQuICC). B S acknowledges the UK STFC 'Quantum Technologies for Fundamental Physics' program (Grant No. ST/T006404/1) for partial support. We thank the referees for useful suggestions and comments.

All data that support the findings of this study are included within the article (and any supplementary files).

In section 4, the expressions for the two biphoton states |Ψ_α ⟩ and |Ψ_β ⟩ which are distinguished from each other using tomograms are given in (19) and (22) respectively. For convenience, we give the expressions below.

$$\begin{align}\vert {{\Psi}}_{\alpha }\rangle & ={\mathcal{N}}_{\alpha }^{-1/2}\int \mathrm{d}{\omega }_{\text{S}}\int \mathrm{d}{\omega }_{\text{I}}{f}_{+}({\omega }_{\text{S}}+{\omega }_{\text{I}})\\ & \quad \times {f}_{-}({\Omega}){f}_{\text{cav}}({\omega }_{\text{S}}){f}_{\text{cav}}({\omega }_{\text{I}})\vert {\omega }_{\text{S}}\rangle \otimes \vert {\omega }_{\text{I}}\rangle ,\end{align}$$

where ${\mathcal{N}}_{\alpha }$ is the normalisation constant. Here, f₋(Ω) and f_cav(ω) are defined in (20) and (21) respectively. The other biphoton state

$$\begin{align}\vert {{\Psi}}_{\beta }\rangle & ={\mathcal{N}}_{\beta }^{-1/2}\int \mathrm{d}{\omega }_{\text{S}}\int \mathrm{d}{\omega }_{\text{I}}{f}_{+}({\omega }_{\text{S}}+{\omega }_{\text{I}})\\ & \quad \times {f}_{-}({\Omega}){g}_{\text{cav}}({\omega }_{\text{S}}){f}_{\text{cav}}({\omega }_{\text{I}})\vert {\omega }_{\text{S}}\rangle \otimes \vert {\omega }_{\text{I}}\rangle ,\end{align}$$

where g_cav(ω) is defined in (23), and ${\mathcal{N}}_{\beta }$ is the normalisation constant.

In what follows, we outline the procedure to show that these two states are indeed the two states which were shown to be distinguishable using photon coincidence counts, in the experiment reported in reference [18].

In the experimental setup, the sum of the frequencies of the signal and the idler photons matches the pump frequency, i.e., (ω_S + ω_I = ω_P). Hence, as stated in the supplementary material (https://stacks.iop.org/JPB/55/185501/mmedia) of [18], f₊(ω_S + ω_I) can be replaced by δ(ω_S + ω_I − ω_P). Integrating over the variable Ω₊(=ω_s + ω_i), appropriately changing the integration variables, noting that Ω = ω_s − ω_i, and dropping the normalisation constant, we get

$$\begin{equation}\vert {{\Psi}}_{\alpha }\rangle =\int \mathrm{d}{\Omega}{f}_{-}({\Omega}){f}_{\text{cav}}({\omega }_{\text{S}}){f}_{\text{cav}}({\omega }_{\text{I}})\vert {\omega }_{\text{S}}\rangle \otimes \vert {\omega }_{\text{I}}\rangle .\end{equation} \tag{ A.1 }$$

This can be identified as one of the states considered in the experiment, namely, the expression (B19) in [18], on changing the notation from Ω, ω_S, ω_I in (A.1) to ω₋, ω_S, ω_I respectively.

We now proceed to establish that the other biphoton state |Ψ_β ⟩ considered by us, is the same as the state $\vert {\psi }^{{\prime}}\rangle \,(={C}^{{\prime}}{Z}_{{t}_{\text{S}}}\vert {\tilde{+}\rangle }_{{\omega }_{\text{S}}}\otimes \vert {\tilde{+}\rangle }_{{\omega }_{\text{I}}})$ defined in [18]. Here, C'|t_S; t_I⟩ = |t_S + t_I; t_S − t_I⟩ where, for instance, |t_S⟩ ⊗ |t_I⟩ is denoted by |t_S; t_I⟩ with t_S and t_I being the time variables associated with the signal and the idler photons respectively, and ${Z}_{{t}_{\text{s}}}\vert {\tilde{+}\rangle }_{{\omega }_{\text{s}}}=\vert {\tilde{-}\rangle }_{{\omega }_{\text{s}}}$. It is convenient to express $\vert {\tilde{+}\rangle }_{{\omega }_{x}}$ and $\vert {\tilde{-}\rangle }_{{\omega }_{x}}\ (x=\text{S,}\text{I})$ as

$$\begin{align}\hfill \vert {\tilde{+}\rangle }_{{\omega }_{x}}=\int \mathrm{d}{\omega }_{x}\,& \int \mathrm{d}{t}_{x}\,\,\mathrm{exp}\left(-\frac{{t}_{x}^{2}}{2{\kappa }_{x}^{2}}-\frac{{\omega }_{x}^{2}}{2{({\Delta}\omega )}^{2}}\right)\hfill \\ \hfill & \sum\limits _{n}\,{\text{e}}^{\text{i}({\omega }_{x}+n\bar{\omega }){t}_{x}}\,\vert {\omega }_{x}+n\bar{\omega }\rangle ,\hfill \end{align} \tag{ A.2 }$$

and

$$\begin{align}\hfill \vert {\tilde{-}\rangle }_{{\omega }_{x}}=\int \mathrm{d}{\omega }_{x}\,& \int \mathrm{d}{t}_{x}\,\,\mathrm{exp}\left(-\frac{{t}_{x}^{2}}{2{\kappa }_{x}^{2}}-\frac{{\omega }_{x}^{2}}{2{({\Delta}\omega )}^{2}}\right)\hfill \\ \hfill & \sum\limits _{n}{(-1)}^{n}\,{\text{e}}^{\text{i}({\omega }_{x}+n\bar{\omega }){t}_{x}}\,\vert {\omega }_{x}+n\bar{\omega }\rangle .\hfill \end{align} \tag{ A.3 }$$

These expressions follow from the properties of the displacement operator, and the expressions (B1), (B2) and (B7) given in [18]. Here, κ_x (x = S, I) is the standard deviation in t_x . It follows from (A.2) and (A.3) that

$$\begin{align}\hfill \vert {\psi }^{\prime }\rangle & =\int \mathrm{d}t\,\int \mathrm{d}{t}^{\prime }\,\int \mathrm{d}\omega \,\int \mathrm{d}{\omega }^{\prime }\hfill \\ \hfill & \quad \times \mathrm{exp}\left(-\frac{{t}^{2}{({\Delta}{\omega }_{p})}^{2}+{t}^{\prime 2}{({\Delta}{\Omega})}^{2}}{2}-\frac{{\omega }^{2}+{\omega }^{\prime 2}}{2{({\Delta}\omega )}^{2}}\right)\hfill \\ \hfill & \sum\limits _{n,m}{(-1)}^{n}{\text{e}}^{\text{i}(n\bar{\omega }+\omega )(t+{t}^{\prime })}{\text{e}}^{\text{i}(m\bar{\omega }+{\omega }^{\prime })(t-{t}^{\prime })}\vert n\bar{\omega }+\omega \rangle \hfill \\ \hfill & \times \otimes \,\vert m\bar{\omega }+{\omega }^{\prime }\rangle ,\hfill \end{align} \tag{ A.4 }$$

where Δω_P is the standard deviation in ω_P. Integrating over the time variables t and t', writing $({\omega }_{\text{S}}=n\bar{\omega }+\omega )$, $({\omega }_{\text{I}}=m\bar{\omega }+{\omega }^{\prime })$ where n, m are integers, and using the fact that f₊ is a Gaussian function with a standard deviation Δω_P (Δω_P ≪ ΔΩ), it is straightforward to see that (A.4) can be expressed as |Ψ_β ⟩ in (22), unnormalised.

We are interested in the time–time slice of the tomograms corresponding to |Ψ_α ⟩ (19) and |Ψ_β ⟩ (22). As a first step, we calculate the explicit expressions for the states |Ψ_α ⟩ and |Ψ_β ⟩ in the Fourier transform basis (i.e., time–time basis) using f₊(ω_S + ω_I) = δ(ω_S + ω_I − ω_P) in (19) and (22). The biphoton state |Ψ_α ⟩ in the time–time basis is given by

$$\begin{align}\vert {{\Psi}}_{\alpha }\rangle & =\frac{1}{\sqrt{{\mathcal{M}}_{\alpha }{\tau }_{\text{P}}}}\,\int \mathrm{d}{t}_{\text{S}}\,\int \mathrm{d}{t}_{\text{I}}\\\ & \quad \times \mathrm{exp}\left.-\frac{{({t}_{\text{I}}-{t}_{\text{S}})}^{2}\,{({\Delta}\omega )}^{2}\,{({\Delta}{\Omega})}^{2}}{4\,({({\Delta}\omega )}^{2}+{({\Delta}{\Omega})}^{2})}\right.\\ & \quad \times {\left[\mathcal{F}({t}_{\text{I}}-{t}_{\text{S}})\right]}^{2}\vert {t}_{\text{s}};{t}_{\text{I}}\rangle ,\end{align} \tag{ B.5 }$$

and the time–time slice w^α (t_S; t_I) corresponding to |Ψ_α ⟩ is

$$\begin{align}{w}^{\alpha }({t}_{\text{S}};{t}_{\text{I}})& =\frac{1}{{\mathcal{M}}_{\alpha }{\tau }_{\text{P}}}\\ & \quad \times \mathrm{exp}\left.-\frac{{({t}_{\text{I}}-{t}_{\text{S}})}^{2}\,{({\Delta}\omega )}^{2}\,{({\Delta}{\Omega})}^{2}}{2\,({({\Delta}\omega )}^{2}+{({\Delta}{\Omega})}^{2})}\right.{\left\vert \mathcal{F}({t}_{\text{I}}-{t}_{\text{S}})\right\vert }^{4},\end{align}$$

where τ_p = 1 s (introduced for dimensional purposes),

$$\begin{equation*}\mathcal{F}({t}_{\text{I}}-{t}_{\text{S}})=\sum\limits _{n}\mathrm{exp}\left.\frac{i({t}_{\text{I}}-{t}_{\text{S}})\,n\,\bar{\omega }{({\Delta}{\Omega})}^{2}}{2({({\Delta}\omega )}^{2}+{({\Delta}{\Omega})}^{2})}\right.,\end{equation*}$$

and the normalisation constant is

$$\begin{equation}{\mathcal{M}}_{\alpha }=\frac{\pi }{{\mu }_{0}}\sum\limits _{m,n,{m}^{\prime },{n}^{\prime }}\,\mathrm{exp}\left(-\frac{{(n-{n}^{\prime }+{m}^{\prime }-m)}^{2}\,{\bar{\omega }}^{2}\,{({\Delta}{\Omega})}^{2}}{2\,{({\Delta}\omega )}^{2}\,({({\Delta}\omega )}^{2}+{({\Delta}{\Omega})}^{2})}\right),\end{equation} \tag{ B.6 }$$

where ${\mu }_{0}={\left(\frac{\pi {({\Delta}{\Omega})}^{2}{({\Delta}\omega )}^{2}}{2({({\Delta}{\Omega})}^{2}+{({\Delta}\omega )}^{2})}\right)}^{1/2}$.

Similarly, the other biphoton state |Ψ_β ⟩ in the time–time basis is given by

$$\begin{align}\vert {{\Psi}}_{\beta }\rangle & =\frac{1}{\sqrt{{\mathcal{M}}_{\beta }{\tau }_{\text{P}}}}\,\int \mathrm{d}{t}_{\text{S}}\,\int \mathrm{d}{t}_{\text{I}}\\\ & \quad \times \mathrm{exp}\left.-\frac{{({t}_{\text{I}}-{t}_{\text{S}})}^{2}\,{({\Delta}\omega )}^{2}\,{({\Delta}{\Omega})}^{2}}{4\,({({\Delta}\omega )}^{2}+{({\Delta}{\Omega})}^{2})}\right.\\ & \quad \times \mathcal{G}({t}_{\text{I}}-{t}_{\text{S}})\,\mathcal{F}({t}_{\text{I}}-{t}_{\text{S}})\vert {t}_{\text{S}};{t}_{\text{I}}\rangle ,\end{align} \tag{ B.7 }$$

and the time–time slice corresponding to |Ψ_β ⟩ is

$$\begin{align}& {w}^{\beta }({t}_{\text{S}};{t}_{\text{I}})=\frac{1}{{\mathcal{M}}_{\beta }{\tau }_{\text{P}}}\mathrm{exp}\left.-\frac{{({t}_{\text{I}}-{t}_{\text{S}})}^{2}\,{({\Delta}\omega )}^{2}\,{({\Delta}{\Omega})}^{2}}{2\,({({\Delta}\omega )}^{2}+{({\Delta}{\Omega})}^{2})}\right.\\ & \times {\left\vert \mathcal{G}({t}_{\text{I}}-{t}_{\text{S}})\,\mathcal{F}({t}_{\text{I}}-{t}_{\text{S}})\right\vert }^{2},\end{align}$$

where,

$$\begin{equation*}\mathcal{G}({t}_{\text{I}}-{t}_{\text{S}})=\sum\limits _{n}{(-1)}^{n}\mathrm{exp}\left.\frac{i({t}_{\text{I}}-{t}_{\text{S}})\,n\,\bar{\omega }{({\Delta}{\Omega})}^{2}}{2({({\Delta}\omega )}^{2}+{({\Delta}{\Omega})}^{2})}\right.,\end{equation*}$$

and the normalisation constant ${\mathcal{M}}_{\beta }$ is essentially the same as ${\mathcal{M}}_{\alpha }$ with an extra factor of (−1)ⁿ⁺ⁿ ' within the summation in (B.6).