A dimensionality and purity measure for high-dimensional entangled states

The working principle of our technique is visualised in Fig. 1(a), where a set of analysers probe distinct parts of a discrete Hilbert space. We can think of each analyser as a probe that scans a sparse set of modes, reminiscent of a conditional measurement that indicates whether there is entanglement within the subspace or not. By combining the information gathered from a number of such analysers, we infer how many dimensions the state occupies. We will demonstrate this procedure both theoretically and experimentally.

To gain access to various parts of the Hilbert space, we make use of high dimensional mode projectors that map onto the states

The set-up used to demonstrate our scheme is shown conceptually in Fig. 1 (d) with the corresponding detailed description in the Methods section. We measure the coincidences between the signal and idler photons for analyser projections on both arms as a function of the relative rotation angle of the holograms. To achieve this, we encoded the fractional OAM mode analyser on the SLM in the signal arm fixed at an angle $0$, while the conjugate mode was encoded in the idler arm and rotated at angles $\theta\in[0,2\pi]$.

To illustrate the operation of our technique we measured the coincidence-rates for six ($N=6$) analysers with $n=1,3,5,7,9$ and 11, and $M=n/2$, with example outcomes for $n=5$ and 9 shown as filled circles in Fig. 2(a) and (b), respectively. Here, no accidental count subtraction was performed on the measurements. Importantly, the periodicity in the detected probabilities confirms the azimuthal $n$-fold symmetry predicted by our theory (solid curve). Because the visibility is a monotonically increasing function of dimension and purity, a measured visibility returns a range of possible $(p,K)$ values, a “trajectory” or curve in the $(p,K)$ space. This is illustrated in Fig. 2(c) and (d), where the measured visibility (red horizontal plane) intercepts the visibility function along a curve (red curve) that restricts the possible solutions, $K$ and $p$, to those consistent with the measurement outcome. The set of such curves from measuring many visibilities (each with its own analyser/projection) then restricts the final solution to a narrow region in $(p,K)$, whose uncertainty (width) is determined primarily from the uncertainly in the visibility measurement. An example is shown in Fig. 2(e), where each solution trajectory is projected onto the $(p,K)$ plane. Final values and uncertainty of $(p,K)$ can be determined by an appropriate routine to find the interception of all such trajectories by a minimisation procedure, as shown in Fig. 2(f). Using this approach we infer the purity and dimensionality of the system to be $(p,K)=(0.45\pm 0.03,22.84\pm 0.62)$.

In Fig. 3 (a) we show the six measured visibilities as square data points together with the calculated visibility (solid red line) based on the inferred $(p,K)$, which clearly match very well. This is confirmation of the minimisation procedure for finding the intercept. In order to assess the procedure under high noise levels, we introduced background noise using a white light source and repeated the measurements, shown as the circle data points and associated the blue dashed line in Fig. 3. The average quantum contrast (see Supplementary Information), measured from the spiral spectrum in Fig. 3 (c) and (d), dropped from $Q=19.19$ to $Q=3.76$, resulting in a reduced purity and dimensionality of $(p,K)=(0.13\pm 0.01,17.73\pm 0.71)$.

As a form of validation of these results, we estimate values from other techniques, with the comparison given in Table 2. If the dimension and noise of a system is known, then it is possible to calculate the purity following $\hat{p}=(Q-1)/(Q-1+d)$ where $Q$ is the quantum contrast and $d$ the dimension zhu2019high . Likewise, if the state is assumed to be pure and not mixed and background subtraction is done to remove noise, then the spiral spectrum can be used to get an upper bound on the dimension. For the two noise cases in Table 2, low and high, we find purity estimates of $\hat{p}\approx 0.44$ (assuming $d=23$) and $\hat{p}\approx 0.13$ (assuming $d=19$), respectively, while estimates of the dimensionality return $\hat{K}\approx 22$ and $\hat{K}\approx 18$, respectively. These values are in excellent agreement with our results, which did not require any such assumptions, nor any noise adjustments. The (perhaps) surprisingly low purity in our results is due to the fact that no background subtraction due to accidental counts was performed.

A measure of dimensionality and purity, particularly in the presence of (inevitable) deleterious noise that degrades the purity, is crucial for many quantum protocols and studies. For example, there is a minimum purity needed to witness entanglement in a given dimension peres1996separability ; horodecki1997separability ; horodecki2009quantum , setting the transition from separable to entangled states. Likewise, knowing the purity is important in entanglement distillation processes since it informs whether the noise can be removed for a given dimension vollbrecht2003efficient ; horodecki1999reduction , while in entanglement based quantum communication there is a minimum purity Collins2002 associated with security ekert1991quantum . In turn, the dimensionality sets the information capacity of the state for quantum information processing and the error tolerance in quantum communication protocols, while high-dimensional states are important for fundamental tests of quantum mechanics where qubits will not suffice klyachko2008simple ; lapkiewicz2011experimental . While a full quantum state tomography would also provide the necessary information, such a measurement would take several days for the states studied here, as compared to less than an hour with our technique.

Unlike a conventional Schmidt decomposition, we do not assume the state is pure, and the dimension extracted from our technique is conditioned on the presence of entanglement: a maximally mixed and maximally entangled system cannot yield the same result. While our approach would benefit from knowledge of the modal spectrum, which can be measured very quickly for OAM Pires2010 ; kulkarni2017single , the outcome on purity and dimensionality are only modestly affected by typical spectrum shapes (see Supplementary Information), e.g., in our examples the uncertainty in dimensionality is $\approx 5\%$ with knowledge of the spectrum, increasing to $\approx 10\%$ without. In this work we used OAM as our demonstration example, but reiterate that the projections are applicable to general states in the spatial mode basis.

In summary, we have developed a simple yet powerful technique to measure the dimensionality and purity of high dimensional entangled photonic quantum systems. Our approach is robust, fast, and provides quantitative values rather than bounds or witnesses, and works on both pure and mixed states. Our scheme exploits visibility in fringes after joint projections, making it fast and easy to implement, returning the key parameters of the system in a fraction of the time that a full quantum state tomography would take. We believe that this tool will be useful to the active research in high-dimensional spatial mode entanglement and foster its wide-spread deployment in quantum based protocols.

The authors express their gratitude to Bienvenue Ndagano for his inputs. I.N. would like to acknowledge the Department of Science and Technology (South Africa) for funding.

The experiment was performed by I.N. and V.R.F., the theory was developed by I.N., F.Z,. H.H., and J.L., the data analysis was performed by I.N., V.R.F. and A.F. and the experiment was conceived by I.N., V.R.F., and A.F. All authors contributed to the writing of the manuscript. A.F. supervised the project.

The authors declare not competing interests.

Both $K$ and $\Delta\ell$ depend on the shape of the distribution $|\lambda_{\ell}|^{2}$. Examples of various types of distributions for $|\lambda_{\ell}|^{2}$ are shown in Fig. 4 for $K=21$. The distributions are: a square distribution, corresponding to a uniform (maximally entangled state) within a given $\ell$-range,

a Gaussian (normal) distribution

where $\gamma_{G}$ scales with the width of the distribution; a SPDC source miatto2011full ; torres2003quantum

where $\gamma_{S}$ is determined by the experimental conditions; and a Lorentz distribution

where $\gamma_{L}$ is a scaling parameter.

Note that all distributions in Fig. 4 have the same $K$ value, but differ in $\Delta\ell$, with $\Delta\ell=12,11.8,14.6\text{ and }41.45$ for the square, Gaussian, SPDC and Lorentz distributions, respectively.

In this work, we make use the SPDC and normal distributions. For convenience, we relate the the Schmidt number to the scaling parameters as, $\gamma_{S}\approx\sqrt{(K-1)/4}$ and $\gamma_{G}\approx 2.5066K$ for the SPDC and normal distributions, respectively.

Our fractional OAM analysers can be decomposed into the OAM basis using entangled photons through digital spiral imaging. In this scheme, one photon from an entangled pair interacts with the analyser while its twin is decomposed in the OAM basis. The entangled photon pair has a biphoton state

as defined in Eq.  (15). The probability amplitude for detecting the $m$th OAM mode, given a M charged fractional mode of $n$ superpositions, is

where $\mathcal{M}$ is a normalisation constant such that $\sum_{m}|\tilde{c}^{n}_{m}(\alpha)|^{2}=1$. Due to the orthonormality of the OAM basis, the overlap $\braket{m}{\ell}$ is simply the Kronecker delta function $\delta_{m,\ell}$, which evaluates as 0 if $\ell\neq m$ or 1 if $\ell=m$. Since from Eq.  (29) we know the expansion coefficients for the analyser in terms of the OAM basis, $\braket{M,\alpha}{{}_{n}}{\ell}$ evaluates as

These new weightings are simply the original coefficients of the analysers modulated by the spectrum of the entangled system. For a maximally entangled state, we obtain the expression $|\tilde{c}^{n}_{m}(\alpha)|^{2}=|c^{n}_{-m}(\alpha)|^{2}$, being the original weightings of the analyser, as desired.

In Fig. 5 we show the measured weightings for our SPDC system which has a normally distribution of OAM modes with $\Delta\ell=11$ centered at $\ell=0$. We show results for $\ket{M,\alpha}_{n}=\ket{0.5,0}_{1},\ket{1.5,0}_{3},\ket{2.5,0}_{5}$ for analysers $n=1,3$ and $5$ in Fig. 5(a),(b) and (c), respectively. It can be seen that the theory (points) and experiment (bars) are in good agreement. To obtain these results, two photons where generated from an SPDC source and modulated with SLMs (see experimental setup in the Methods section in the main text). One SLM was encoded with a fractional OAM mode projecting onto the state, $\ket{M,0}_{n}$, while the second SLM was encoded with OAM basis modes, $\ket{\ell}$.

Given a bipartite system of the form of Eq. (33), we want to know what the detection probability is, due to the relative rotations of our fractional OAM analysers acting the entangled photons. Suppose the first analyser projects onto the state $\ket{M,\theta_{1}}_{n}$, and the second analyser projects on the state $\ket{-M,\theta_{2}}_{n}$. A joint measurement on a two photon system using the two analysers is characterized by the product state $\ket{M,\theta_{1}}_{n}\ket{-M,\theta_{2}}_{n}$. The probability amplitude resulting from such a measurement is

Therefore we only need to know how to decompose each of the analysers in the OAM basis to obtain the detection probability for the joint measurements. Using Eq. (29), it follows that

We use this approach to numerically calculate the detection probabilities $|C_{n}(\theta_{1},\theta_{2})|^{2}$ by simply calculating the probability amplitudes for each analyser in the OAM basis with a desired rotation $\theta_{1,2}$ and multiplying them with the coefficients $\lambda_{\ell}$ that determine the quantum system being probed.

An alternative approach, can be to compute the overlap integral by considering the modal overlaps in the azimuthal degree of freedom, $\phi$, following

with $\Phi_{M}(\phi;\theta)/\sqrt{2\pi}$ being the transmission function of the fractional OAM analyser projects onto the state $\ket{M,\theta}_{n}$. We can rewrite probability amplitude $C_{n}(\theta_{1},\theta_{2})$ as an overlap integral given

where $\Phi_{\pm M}(\phi_{1,2},\theta_{1,2})$ are the phases of the fractional OAM analysers. Since $e^{-i\Phi_{M}(\phi_{1};\theta_{1})}$ has no $\ell$ dependence, we can introduce the summation into the second integral resulting in

It is convenient to define the periodic function

with angular harmonics $e^{i\ell\left(\phi_{1}-\phi_{2}\right)}$ determined by the coefficients $\lambda_{\ell}$, and use it to rewrite $C_{n}(\theta_{1},\theta_{2})$ as

Notice that the second integral is a convolution between $\Lambda(\phi_{1}-\phi_{2})$ and the second analyser. As a example, we consider a maximally entangled state ($\lambda_{\ell}:=\text{constant}$). In this case, $\Lambda(\phi_{1}-\phi_{2})=\delta(\phi_{1}-\phi_{2})$ and therefore

The integral now only depends in the transmission functions of the analysers with an analytical solution found in huang2018various .

We now calculate the probability $P_{n}(\theta_{1},\theta_{2})=\left|C_{n}(\theta_{1},\theta_{2})\right|^{2}$ as a function of relative orientation $\theta=(\theta_{1}-\theta_{2})$ between the two analysers and the dimensions, $K$, of an entangled system with some given OAM spectrum $|\lambda_{\ell}|^{2}$. The latter is embedded in the function $\Lambda(\phi_{1}-\phi_{2})$. Figures 6(a)-(d) show examples of the probability surfaces assuming a normal (Gaussian) spectrum $|\lambda_{\ell}|^{2}$ for superposition states $n=1,3,5$ and $7$. In the second row of, Fig. 6 (e)-(h), we show examples of the probability curves normalised to unity for several values of dimensionality $K$. Here, it can be seen that the frequency of the probabilities as a function of $\theta$ increases with $n$, owing to the n-fold symmetry in the phase profiles of the analysers.

Crucially, the exact shape and visibility of the curves depends on both the dimensions ($K$) of the state being probed and the number of superpositions ($n$). For all $n$’s, the visibility for a specific $K$ shows a decreasing trend as the number of superpositions $n$ are increased. Therefore the analysers are sensitive to the dimensions of the system.

We also found that the shape of the spectrum affects the measured probabilities, as illustrated in Fig. 7(a)-(d) for square (maximally entangled), normal (Gaussian) and SPDC, respectively.

Now that we have shown how the detected probabilities depended on the dimensions and superposition states measured, in the following section we study the relation between the visibility and dimensions quantitatively.

The visibilities are calculated from detection probabilities resulting from the projections of an entangled state with an initial OAM distribution $|\lambda_{\ell}|^{2}$ onto the states $\ket{M,0}_{n}\ket{-M,\theta}_{n}$, where $\theta\in[0,2\pi]$ is their relative rotation.

Contour plots of the visibilities with changing dimensions ($K$) and fractional OAM superpositions ($n$) for various OAM spectral shapes (Normal, SPDC, Uniform) are shown in Fig. 7 (a)-(c) for pure states. As shown, the assumed spectrum can affect the visibility that is measured for various superpositions ($n$). The visibilities for each analyser ($n$) and spectrum shape are monotonic with increasing $K$. In particular, for the uniform spectrum (maximal entanglement in K dimensions) the visibility is 1 above some $K=d_{n}$ and zero below this. We further exploit this property to determine the dimensionality of an entanglement system.

The visibilities that can be measured with our analysers are not only dependent on the effective dimensions of the system but also the purity. In particular, we consider the isotropic state,

which can decomposed into the high-dimensional entangled state, $\ket{\Psi_{d}}$, and the separable and mixed state, $1/d^{2}\mathbb{I}_{d^{2}}=1/d^{2}\sum_{\ell,\ell^{\prime}=-L}^{\ell,\ell^{\prime}=L}\ket{\ell}\ket{\ell^{\prime}}\bra{\ell^{\prime}}\bra{\ell}$, where $\mathbb{I}_{d^{2}}$ is the identity operator. Such states model quantum systems that have noise contributions from the environment. Here $p$ can be associated with the purity of the state ranging from a maximally mixed ($p=0$) to a pure state ($p=1$). Interestingly, the isotropic state is separable for $p\leq 1/(d+1)$ and entangled otherwise. Importantly the generalised Bell inequality can also be violated when $p>2/S_{d}$ where $S_{d}$ is the Bell parameter Collins2002 . We show that both $p$ and $d$ can be measured using our analysers. For convenience, we assume $d\approx K$, where K is the effective dimensionality of the entanglement. We will demonstrate that we can measure both $p$ and $K$ using our analysers.

Firstly, we calculate the detection probabilities from the overlap, $P_{n}(\theta;K,p)=\text{Trace}(\hat{M}\rho_{p})$ where $\hat{M}$ projects onto the states $\ket{M,0}_{n}\ket{-M,\theta}_{n}$. As a result, the detection probability can be written as

where $P_{n}(\theta,K)=\left|\sum_{\ell=-L}^{L}\lambda_{\ell}c^{n}_{\ell}(0)c^{n}_{-\ell}(\theta)\right|^{2}$ and $I_{n}(0;K)=\left|\sum_{\ell=-L}^{L}\left|c^{n}_{\ell}(0)\right|^{2}\ \right|^{2}$ is the overlap of the analysers with the maximally mixed state. Since the functions are periodic and obtain maximum and minimum values for $\theta=0$ and $\pi/n$, respectively, we obtain the expression

where $\Delta P_{n}(K)=P_{n}(0,p=1,K)-P_{n}(\pi/n,p=1,K)$. The visibilities can be calculated from

We show the dependence of the visibilities on the dimensions ($K$) and purity ($p$) in Fig. 8 (a-c) for the Normal, SPDC and uniform distribution, respectively. Each panel shows the visibilties from various analysers depending on the number of superposition ($n$). As shown the visibilities increase monotonically with increasing dimensions ($K$) as well as purity $p$ for each analyser where $p=1$ obtains a maximal visibility. However, as $n$ increases the visibilities decrease for all $p$ and $K$. Since the visibilities are monotonic in both $p$ and $K$ as well as $n$, we can exploit this property to map the dimensions of a quantum system. We favour this approach since the visibilities can be easily measured and require few measurements (peak and trough).

Using our procedure we measured the dimensions and purity of SPDC photons with varying noise levels (low and high). The results are summarised in Table 2. In the second and third column, we know what the input spectrum shape (SPDC) is and can therefore accurately optimise for the dimensions ($K$) and purity ($p$) of the state (see Results section). Further, if we guess the spectrum based on its shape (symmetry) we also obtain values that are similar to the expected results, with a relative error of up to $\approx 13\%$. This was done using the normal distribution as the function modelling the mode spectrum. Next, we verify our result using the values extracted from the spiral bandwidth.

To calculate the the expected dimensions, $\hat{K}$, we used the coincidences from the spiral spectrum in the OAM basis, i.e $C_{\ell_{A},m_{B}}$, where $\ell_{A}$ denotes the mode index of photon A and $m_{B}$ for photon B. Since we want the Schmidt number of the pure part of the state, we subtracted the accidentals and then used Eq. (16), yielding results with a low relative error of $3\%$, validating our results. Subsequently, we estimated the purity $\hat{p}$ (see Results section). Note that no accidental subtraction was performed in this case. Accordingly, to estimate the purity we measured the quantum contrast using

taken from the ratio between the average coincidences in the anti-diagonal entries, $\bar{C}=\sum_{\ell}C_{\ell,-\ell}$ and the average noise contribution from coincidences excluding the anti-diagonal entries, i.e $\bar{C}^{\prime}=\frac{1}{d^{2}-d}\left(C_{T}-d\ \bar{C}\right)$. Here $C_{T}$ corresponds to the total coincidences $C_{T}=\sum_{\ell,m}C_{\ell,m}$. Indeed, using the quantum contrast we obtained a purity that is comparable to that obtained from our method showing a relative error of only up to $2\%$.