1. Introduction

Quantum metrology [1–3] employs fundamentally quantum systems as measurement probes to overcome the “standard quantum limit” [4–8], a statistical bound on the precision of any measurement with uncorrelated particles. Already, ideas from this field of research have been implemented in gravitational wave detectors [4,9] and could become important in a range of precision measurement device such as magnetometers [10–12] and atomic clocks [13,14].

One of the classic problems used to demonstrate the advantage of quantum metrology is the measurement of a phase difference $\theta$ between the two arms $a$ and $b$ of a balanced interferometer. For the best measurement scheme using $N$ separable photons, the error in the estimate of $\theta$, $\delta \theta$, scales as $\delta \theta \propto 1/\sqrt {N}$. This is the “standard quantum limit" [4,5] in this problem, which is reflective of the measurement process being independent for each photon. On the other hand, the best general measurement strategy for this estimation is to send all the photons in a superposition of being all in path $a$ and all in path $b$. This highly entangled configuration is called the “$N00N$ state" [15–20] and with it, one can in theory achieve the Heisenberg scaling [5,21] of $\delta \theta \propto 1/N$, which is the best scaling allowed by quantum mechanics, although its performance is quickly ruined by loss [22]. Low photon number $N00N$ states have been realized in the laboratory multiple times [23–30]. It is often thought that super-sensitivity of this type comes at a price and that a quantum state super-sensitive to a given transformation is necessarily less sensitive to transformations generated by non-commuting operators. For example, a $N00N$ state where $N$ photons are in a superposition of all being in $+z$ eigenstates and all being in $-z$ eigenstates has stellar sensitivity to rotation around the $z$ axis. However, its sensitivity to rotation around the $x$ and $y$ axes is bounded by the “shot-noise limit" of $1/\sqrt {N}$. The 2-photon $N00N$ state is an extreme case which is completely insensitive to rotations around the $y$ axis. The idea of the trade-off in super-sensitivity is a misconception which was first cleared with the introduction of the compass state [31,32], a state of the quantum harmonic oscillator equally super-sensitive to displacement in position $\hat {x}$ and momentum $\hat {p}$.

In this work, we experimentally generate a counterpart of the compass state in the polarization of a light beam, the “tetrahedron state.” Unlike in the $N00N$ state example, we no longer consider the characterization of a single rotation, described by the action of the group U(1) and generated by the angular momentum operator $\hat {J}_z$, but instead we must look at general rotations described by the actions of the group SU(2), which can be generated by a triad of angular momentum operators $\{\hat {J}_x, \hat {J}_y, \hat {J}_z\}$. The tetrahedron state exhibits super-sensitivity for rotations generated by any linear combination of these three operators, corresponding to rotations around any axis, despite the non-commutativity of these generators. Beyond polarization rotations, SU(2) operations include the action of a generalized interferometer (with arbitrary transmission and reflection) on two modes of the electromagnetic field, the action of arbitrary magnetic fields on spins, reference-frame alignment, and many other processes. We will consider scenarios where the axis of the rotation is known but not necessarily aligned with the $z$-axis, as well as scenarios where the axis itself is also unknown. The main difference present in the latter is that the characterization of such processes requires the estimation of a minimum of three parameters instead of a single one [33,34]. The multi-parameter quantum estimation has proven to be more intricate than its single-parameter counterpart [5,35–37], with nuances such as parameter compatibility [38–40], and is important for many real-world problems [41,42]. This makes for a richer problem and requires us to look in different corners of the Hilbert space for suitable entangled probe states.

The tetrahedron state on which we report has several intriguing properties. It was initially studied as being the opposite of SU(2)-coherent states, earning the epithet “anticoherent” [43–45], making it in some sense the most quantum state. Using the Hilbert–Schmidt norm as the distance measure between states, it is the furthest state from the closest SU(2)-coherent state of the same dimension. Therefore, it has been termed as the “queen of quantumness” [46]. In the context of polarization, it manifests “hidden polarization” [47], as its classical (first-order) polarization properties are ignorant of higher-order polarization features [48]. It also minimizes the cumulative multi-poles of its polarization distribution, for which it has earned the moniker “king of quantumness” [49,50], and is the optimal state for rotation sensing, explaining the sobriquet “quantum rotosensor” [51,52]. The tetrahedron state has numerous other extremal properties [53], including maximizing delocalization measures in SU(2) phase space such as the Wehrl entropy [54]. It has also been studied for its entanglement properties [55–59], which have been generalized to multi-dimensional systems with multiple qudits [60]. While its polarization-sensing properties require the tetrahedron state to be in light’s polarization degree of freedom [61], it can also exist in light’s spatial degrees of freedom [62–64], and can be envisioned in other symmetric multi-qubit systems. We report on creating this unique quantum state in the polarization degree of freedom of light.

2. Theory

2.1 Quantum Fisher Information

The quantum Fisher information (QFI) [65] is a powerful tool that, in the context of multi-parameter estimation, can provide a lower bound on the covariance matrix of the estimated parameters for any measurement scheme. Let ${\tilde {\boldsymbol{\theta}}}$ be the estimator for a set of parameters $\boldsymbol {\theta}$ describing a specific SU(2) rotation $\hat {R}(\boldsymbol {\theta})$; then, the covariance matrix

It is noteworthy that the bound in Eq. (6) is independent of both the parameters to estimate $\boldsymbol {\theta}$ and of the initial rotation of the state (i.e., $|{\psi}\rangle$ and $\hat {R}\left (\boldsymbol{\phi }\right )|{\psi}\rangle$ will share the same bound for all $\boldsymbol{\phi }$). Furthermore, if we choose a Cartesian parametrization for the rotation

2.2 Polarization as a Spin

A single photon’s transverse polarization can be represented as a spin-$1/2$ particle, with the polarization states $|{H}\rangle\; (m=1/2)$ and $|{V}\rangle \;(m=-1/2)$ forming an orthonormal basis for the Hilbert space. In general, a collection of $N$ photons can be decomposed into the tensor sum of particles with spin $\leq N/2$:

2.3 The Majorana Representation

A helpful way to visualize states in the full-symmetric $\text {spin-}N/2$ sector is with the Majorana representation [70,71]. The idea is that one can always write these states as a product of single-particle creation operators over the two polarization modes acting on the vacuum state:

2.4 Choosing the Optimal State for Multi-Parameter Estimation

2.4.1 The Spin Coherent States

Spin coherent states, depicted in Fig. 1(a), are notable separable states in the symmetric subspace of $N$ photons. They consist of all $N$ photons sharing the same polarization properties. One of those states is

 

Fig. 1. Majorana representations of (a) spin coherent state, (b) 4-photon $N00N$ state, and (c) the tetrahedron state. (d) Conceptual illustration of the tetrahedron’s state creation process. The H polarization of a 3-photon $N00N$ state (three transparent red points on the vertices of an equilateral triangle on the equator) is partially attenuated resulting in the triangle of points with lower latitude (red). Simultaneously, an H polarized heralded single-photon is probabilistically added (green). The result is a 4-photon state represented by four points on the vertices of a tetrahedron.

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To remedy this situation, we could imagine dividing our available photons into three different spin-coherent states aligned with the $x$, $y$, or $z$ axes and using them sequentially. Because the quantum Fisher information is additive for uncorrelated measurements [36], the bound in Eq. (6) can be found adding up the different spin covariance matrices for each of these states. Assuming $N$ is divisible by three, the resulting matrix and the corresponding s-QCRB are

2.4.2 The $N00N$ States

The $N00N$ state is an entangled state of $N$ photons. The particular $N00N$ state aligned with the $z$ axis can be written as

We can attain a quantum scaling, however, by splitting the $N00N$ state into three batches, as with the spin coherent state. If the photons are split equally between $N00N$ states aligned with the $x$, $y$, and $z$ axes, we get

2.4.3 The Second-Order Unpolarized States

It was proven in Refs. [53,66] that the optimal states for the task at hand are pure states that exhibit the following properties:

We can quickly see that states satisfying Eq. (19) outperform all the other schemes imagined here by looking at their spin covariance matrix and the s-QCRB:

2.5 The Tetrahedron State

The second-order unpolarized state with the smallest photon number ($N=4$) is the one we call the “tetrahedron state." It is named as such because it has the same symmetries as the tetrahedron under rotation, which is apparent in its Majorana representation [depicted in Fig. 1(c)], where the points correspond to the vertices of a regular tetrahedron inscribed in the Bloch sphere. The state can be written as

3. Methods and Results

3.1 Experimental Apparatus

Our overall approach to our tetrahedron state creation scheme is intuitive in the Majorana picture. When we rewrite the tetrahedron state as a product of creation operators acting on the vacuum and we re-arrange the terms we get the following expression:

We create these two components independently with SPDC. The $N00N$ state is created by an interference between a weak single-mode squeezed vacuum generated through SPDC, and a coherent state [26,81,82]. The relevant section experimental apparatus is depicted in Fig. 2(a). We use a pulsed Ti-sapphire laser with a pulse width of 140 fs, a center wavelength of 807 nm, a repetition rate of $80$ MHz, and an average intensity of $3.5$ W. With it, we perform SHG in a $1$ mm BBO crystal to get a beam of over $500$ mW at $404$ nm. This beam, blue in Fig. 2, serves as a pump for both our SPDC processes. In the $N00N$ state source, we perform SPDC with a $2$ mm BBO crystal. This SPDC source is optimized in Type 1 collinear configuration. We measure a pair creation rate of $53$ kHz with a symmetric coupling efficiency of $13.4{\% }$. These numbers include all forms of loss in the experiment, including detector inefficiencies, but are corrected to account for the necessity to probabilistically split the photons in a collinear geometry. These photon pairs interfere with a coherent state in a polarization interferometer with a Mach–Zehnder geometry. The coherent state is light from the attenuated original laser beam, left over from the SHG process.

 

Fig. 2. Experiment overview. (a) An 808 nm pulsed beam (red) is used to generate a 404 nm beam (blue) through second harmonic generation (SHG) in a 1 mm barium borate (BBO) crystal. Part of this blue beam is sent to generate co-propagating pairs of horizontally polarized 808 nm photons (red) through type-I spontaneous parametric downconversion (SPDC) in a 2 mm BBO crystal while the red beam is attenuated and vertically polarized. The two beams are recombined on a polarizing beamsplitter (PBS) and polarization rotated to form a 3-photon $N00N$ state in the H/V basis. (b) Another blue beam then a pumps a second SPDC (2 mm BBO) to generate another photon pair. Here, after a 3 nm narrowband filter, a detection at a single-photon counting module (SPCM) labeled “T" heralds the presence of a photon in the upper path. (c) Both the $N00N$ state and heralded single photon enter a displaced polarization Sagnac interferometer. The heralded photon strictly takes the clockwise path while the $N00N$ state is split equally amongst the paths. The half-waveplate (HWP) in the counter-clockwise path is set at $45 ^{\circ }$ to transmit all the light in the upwards output while the HWP in the clockwise path is set at $22.5 ^{\circ }$ to attenuate the V polarization of the $N00N$ state by half and transmit the heralded single-photon half the time (see Fig. 1 for context). After being recombined, all the photons pass through a 3 nm narrowband filter to make sure their spectra are the same. (d) The state is transmitted via single-mode (SM) fiber to a tomography apparatus for projection onto an arbitrary polarization axis. Each output path is coupled to a multi-mode fiber splitter to allow for partial photon number counting capabilities (a maximum of three detections per polarization). Five-fold detection events between T and any four SPCMs in the tomography setup are recorded.

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Importantly, the phase between these two beams should be stable throughout the experiment. We actively stabilize this phase using the locking port in Fig. 2(a). The squeezed vacuum and the coherent states are minutely off from their intended polarizations so that we can collect a bit of light from the locking port. The superposition of two horizontally polarized single photons from the squeezed vacuum and two vertically polarized single photons from the coherent state forms a 2-photon $N00N$ state since those two terms are mostly indistinguishable in any other degrees of freedom. We measure 2-photon coincidence events on a basis where the correct phase between two terms would result in a signal that is in the middle of a fringe. Because of the drifts in the rates, we re-calibrate the set point every 8–12 hr of data collection.

The horizontally polarized single photon, is also created through SPDC, and depicted in Fig. 2(c). This time we use a non-colinear Type I geometry. One of the photons of the pair is detected immediately and acts as a herald, while the other goes to be part of the tetrahedron state. Throughout this experiment, this SPDC source produced photon pairs at a rate of $32$ kHz and with a heralding efficiency of $11.5{\% }$. Using a heralded single photon here as opposed to a low-intensity coherent state was necessary to suppress the contamination from higher-order terms where more than one photon would be created in this mode. In order to ensure that the photons from these different sources are as indistinguishable as possible, all the photons in the tetrahedron state are fed into the same single-mode fiber right after transmission through a $3$ nm narrowband filter. The resulting loss is already included in the efficiency figures above. The herald photon is similarly filtered and collected.

As we will discuss in Section 4, the state we generated is notably far from ideal. The main source of imperfections is the contributions of the higher-order terms from the coherent state and the SPDC sources. The 4-photon tetrahedron state is created in a single spatial mode and a temporal mode coincidentally with a single “herald” photon in another spatial mode. It is only when we detect these five photons simultaneously that we consider that a tetrahedron state has been created and detected. Ideally, in the absence of photon loss of any kind, the number of detected photons is the same as the number of created photons. In a laboratory setting, however, photon loss is unavoidable. As a result, some events with successful post-selection have more than the five intended photons and some photons are lost. The polarization state of four of those remaining photons is not in a tetrahedron state. A detailed model for the loss in this experiment is described in Supplement 1 S1 and S2. To calibrate the model, we performed sets of measurements to find the values of the relevant parameters. These measurements are described in Supplement 1 S3. The parameters include $\eta$ and $\mu$, the squeezing parameters of collinear and non-collinear SPDC sources, respectively. For both sources, the coincidence rate would be proportional to the absolute value squared of these squeezing parameters. To these are added $t$ and $\tau$, the transmission rates for the collinear SPDC and non-collinear SPDC photons, respectively. These include the detection efficiency. The coherent state amplitude, $\alpha$ (the single photon rate is proportional to $|\alpha |^2$), is set to $\sqrt {2\eta t}$ to create the 3-photon $N00N$ state [81]. Loss present in the coherent state path is already captured by this value of $\alpha$. The measured values for $\eta$, $\mu$, $t$, and $\tau$ are 0.078, 0.14, 0.16, and 0.12, respectively. Transmission rates are quite low and this causes the other undesired events to become non-negligible.

We calculated analytical expressions for the probabilities of possible 5-photon events in Supplement 1 S4 and Supplement 1 S5. For example, the probability of the desired 5-photon event that generates a tetrahedron state is proportional to $\eta ^3 t^3 \mu ^2 \tau ^2$. One of the biggest sources of imperfection is measuring an extra photon from the coherent state used to produce the $N00N$ state when one of the photons from the SPDC sources is lost. In these cases, the probability of the case when one of the photons from the collinear SPDC is lost is proportional to $\eta ^4 t^3(1-t) \mu ^2 \tau ^2$, and the probability of the case when the horizontally polarized photon from the non-collinear SPDC is lost is proportional to $\eta ^4 t^4 \mu ^2 \tau (1-\tau )$. All the cases mentioned above result in a 5-photon coincidence and we do not have a method to distinguish them. Substituting the values for our experimental setup, the two undesired events mentioned above have probabilities roughly of the same order of magnitude compared with the probability of generating a tetrahedron state. Several undesired events like the two mentioned above reduce the fidelity and the metrological potential of our state. One possible solution to lower the contributions from the undesired terms was to lower the squeezing parameters and the coherent state amplitude by reducing the single-photon rates. However, the data collection process, explained in Section 3.2, already took eight days even with the current single-photon rates because of the rarity of the 5-photon events and the necessity of regular calibrations every 8–12 hr. Due to these impracticalities, we decided to collect data with the current rates.

3.2 Tomography with Hidden Differences

The Hilbert space for 4-photons’ polarization is $\left (\text {spin-}1/2\right )^{\bigotimes 4} = \left (\text {spin-}2\right ) \bigoplus 3\left (\text {spin-}1\right )\bigoplus 2\left (\text {spin-}0\right )$. Ideally, in this case, the photons are indistinguishable, which constrains their polarization to be full-symmetric and contained in the $\text {spin-}2$ sector. In practice, however, we can expect some small mismatch in the modes of the photons from different sources, which will allow other spin sectors to be populated. Importantly, the parasitic distinguishability between the photons cannot be accessed, i.e., we have no practical way of favorably selecting photons from one source over another, which means that the only measurements we can perform have projectors symmetric under particle exchange. As a result, only the parts of the density matrix depicted in Fig. 3(a) can be reconstructed. They specifically consist of the spin sectors in the tensor sum decomposition of the Hilbert space. The coherences between these sectors are inaccessible. Furthermore, we are only sensitive to the sum of the different spin sectors with the same spin value and choose to report their average in each of the different sectors. Importantly, although our knowledge of the full state of the system is fundamentally limited in this way, the accessible information is enough to predict the result of any symmetric measurement done on the same state.

 

Fig. 3. (a) Layout of the full density matrix. Here $\rho _\text {sym}$ is the fully symmetric spin-2 sector (blue). Spin-1 (red) and spin-0 (green) sectors are also shown. (b) The full tomographically reconstructed density matrix, $\rho _\text {exp}$. In hidden differences tomography, coherences between spin sectors are neglected and sectors of the same spins are chosen to be identical. (c) Real and imaginary parts of the reconstructed $\rho _\text {sym}$. The density matrix of the ideal tetrahedron state is shown for comparison (dashed). Elements of the density matrix are labeled with spin notation $| {j, m}\rangle$. (d) Real and imaginary parts of a spin-1 sector. The population in each spin-0 sector is less than $10^{-10}$.

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The technique we use to reconstruct what part we need of the density matrix is called “tomography with hidden differences” [83–85]. It consists of a sequence of measurements on the polarization of our state akin to Stern–Gerlach measurements for a spin. The relevant section of the experimental apparatus is depicted in Fig. 2(d). We send photons through a PBS and then through networks of beamsplitters (BS) with every output leading to a SPCM. When four photons are simultaneously detected in coincidence with the herald photon in Fig. 2(b), we consider it a successful event and count the number of photons transmitted versus reflected at the PBS. We perform this polarization measurement in 13 different bases, chosen to be roughly uniformly spread on the Bloch sphere by rotating the quarter-waveplate (QWP) and HWP before the PBS. With only three SPCMs on each output of the PBS, we cannot detect two of the five potential 4-photon outcomes, when the photons are all transmitted, or all reflected. As for the other projectors, they need to be weighted by the likelihood that each photon lands on a different detector. A detailed discussion of this reweighing is included in Supplement 1 S2. With 13 measurement bases, however, we can still perform 39 different projections, which is enough to estimate the 35 linearly independent parameters of the accessible section of the density matrix using a maximum likelihood method.

Overall, 2434 successful events were recorded in 85 hr of counting time. These counting hours were distributed over the course of eight days. Every 8–12 hr, minor realignments were made to readjust the couplings in single-mode fibers and to reset the set-point of the locked interferometer.

Figure 3 shows the tomographic reconstruction of the density matrix. For ease of comparison with the theoretical tetrahedron state, we rotate our reconstructed density matrix by an optimal angle $\phi$ around the H/V axis in post-processing. We set $\phi$ by maximizing the fidelity ($F = \langle{\psi _{tetra}}|e^{-i\hat {J}_z\phi }\hat{\rho} e^{i\hat {J}_z\phi }| {\psi _{tetra}}\rangle$) between the unrotated reconstructed state and the tetrahedron state and find a value of $\phi = 0.135$ for a fidelity with the tetrahedron state of $(0.46 \pm 0.02)$.

Error bars on the all the entries of the density matrix were determined by a Monte Carlo simulation of the tomographic reconstruction process, where the simulated number of detection events for each projection was drawn from a Poisson distribution centered on the actual measurement results.

4. Discussion

Despite the low fidelity between the theoretical tetrahedron state and our reconstructed density matrix, it retains many of the notable features of the former.

As expected, most of the population ($87 \pm 4{\% }$) is in the full-symmetric, spin-2 subspace [$\rho _{\mathrm {sym}}$ in Fig. 3(a)]. The excess population in the other sectors is small but still non-zero, indicating that there were small mode mismatches between the different photons. Indeed, the visibility of a 2-photon $N00N$ state created with a very low rate ($\sim 1kHz$) was measured to be around $88{\% }$, indicating some mode mismatch between the SPDC photons and coherent state photons. This measurement is described in more detail in Supplement 1 S6. As intended, the two basis elements with the highest population in $\rho _{\mathrm {sym}}$ are $|{2,2}\rangle$ and $|{2,-1}\rangle$. The ratio between these populations, however, is $(0.73 \pm 0.12)$, which is significantly larger than the theoretical value of $1/2$. We unfortunately measure significant population ($\approx 10{\% }$) in the states $|{2,0}\rangle$ and $|{2,-2}\rangle$ that are meant to be empty. Finally, the largest coherence measured is between $|{2,2}\rangle$ and $|{2,-1}\rangle$. Theoretically, this is the only non-zero coherence and it takes a value of $\sqrt {2}/3\approx 0.47$. Here, we measure the significantly lower value of $(0.13 \pm 0.02)$, also much lower than $(0.32 \pm 0.02)$, the maximum possible value given the corresponding population. The coherence quoted has no imaginary part specifically because we allowed an optimal rotation around the H/V axis in the reconstructed density matrix. The purity of the measured state is $\text {Tr}[\rho _{\text {exp}}^2] = (0.32 \pm 0.02)$ and the purity of the symmetric part of the density matrix is $(0.43 \pm 0.02)$.

Most of these discrepancies can be explained by considering the contributions of terms created with six or more photons but detected as a successful five-fold coincidence due to photon loss at the detectors or in the optical apparatus. Using the model described in Supplement 1 S1, we compute the theoretical contribution of higher-order photon creation events with the presence of loss in our apparatus, and compare them with our experimental results. Because loss acts symmetrically on every photon, the model does not predict the moderate portion of the state not contained in the full-symmetric sector. It is, however, a great predictor for our results within said sector. When calibrated for our apparatus, we predict a state which has a $91{\% }$ fidelity with the symmetric portion of our experimental state. The state in question can be seen in Supplement 1 S4. This result indicates that the imperfection of our schemes are very well understood and that we can reliably infer the technical requirements required for the creation of a higher quality state. In particular, while keeping the same tetrahedron state detection rate, but while imagining switching the cascade of BS terminated with SPCMs (with assumed detection efficiency of $\sim 65{\% }$) for state-of-the-art photon-number-resolving detectors with near-unity efficiency [86], our model predicts a state with a fidelity of 0.80 with the ideal tetrahedron state, a notable improvement over the fidelity of 0.54 for the state predicted with our current parameters or over the fidelity of 0.53 for the re-normalized fully symmetric sector of our experimental density matrix.

Despite these imperfections, the main eigenstate of the reconstructed density matrix is very similar to the theoretical tetrahedron state with $92{\% }$ fidelity. We can interpret this to mean that our method succeeds in preparing a near-optimal tetrahedron state around half the time, but fails for the remaining half, in good part due to events when more photons were created than intended. A comparison of their Majorana representations is depicted in Fig. 4(a).

 

Fig. 4. (a) Majorana representations for the ideal tetrahedron state (red) and the eigenstate of the reconstructed density matrix with the largest eigenvalue (blue). (b) Longitude–latitude map of the Wigner representation for the fully symmetric portion of the ideal tetrahedron state. Points on the tetrahedron representation correspond to those on the map. The $y$-axis is the polar angle and the $x$-axis is the azimuthal angle in spherical coordinates. Polar angle is defined to be $\pi /2$ at the top corner and azimuthal angle is defined to be 0 at the leftmost corner. (c)–(f) The reconstructed Wigner representation of the fully symmetric portion of the state (not of the main eigenstate) when a $2\pi /3$ rotation is applied about each of the four points of the tetrahedron.

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Another feature of note is that our state keeps some of the tetrahedral symmetry that gives the tetrahedron state its name. This can be readily seen in the Wigner function [87–89] of our reconstructed state plotted in Fig. 4. Because this Wigner function has the domain of a sphere, we display four different projections, each centered on a different vertex of a tetrahedron. These four different projections of the Wigner function of the theoretical tetrahedron state are the same [shown in Fig. 4(b)] by virtue of its symmetry. The different projections of the symmetric portion of the reconstructed state are plotted in Figs. 4(c)–4(f). Although the features are not as pronounced, we notice that, the troughs and the peaks of the Wigner function align with those of the theoretical state. Indeed, despite the contamination of the noise coming from higher-order events, the reconstructed Wigner function qualitatively keeps, with some imperfections, the theoretical symmetry of the tetrahedron state. We note that a deviation from the optimally chosen phase $\phi$ would simply rotate the Wigner function around the $z$-axis and would not change the symmetry of the results.

We can also directly assess how well our state satisfies the unpolarized properties of Eq. (19). One direct measure of this property is by expanding the state’s quasiprobability distribution on a sphere in terms of multi-pole moments. Unpolarized states have vanishing first and second moments. Our state’s first and second moments are $1.6\times 10^{-2}$ and $3.1\times 10^{-2}$, respectively. This is much smaller than for a comparable “classical” state, with first- and second-order multi-pole moments summing to $2/5$ and $2/7$, respectively, and is even much smaller than for a state whose Majorana constellation is chosen randomly [90], with first- and second-order moments summing to $2.8\times 10^{-1}$ and $2.3\times 10^{-1}$, respectively.

The qualitative features can be further observed in Fig. 5, which compares the performance of the experimental state and the main eigenstate with the ideal tetrahedron state for the measurement of single rotations angles, that is the measurement of $\theta _x$, $\theta _y$, or $\theta _z$ when knowing that the other two angles take a value of zero. We plot the numerically calculated result of a projection on the original tetrahedron state following the rotation. It can be seen that both for the experimental state and the main eigenstate the visibility is not as high as the ideal tetrahedron state since they do not have perfect overlap with the tetrahedron state, especially the experimental state has very low visibility due to low fidelity and low purity. However, all axes still have three maxima per rotation, a signature of tetrahedral symmetry. For example, the 4-photon spin coherent state would have a single maximum, while the 4-photon $N00N$ state would exhibit four maxima for the $z$-axis, and two for the $x$-axis and the $y$-axis. The main experimental eigenstate closely approximates the tetrahedron state with similar levels of visibility for all rotations shown in the figure.

 

Fig. 5. Numerically calculated projection on the original tetrahedron state after rotations around the $x,y$, and $z$ axes on Bloch sphere for the following states: ideal tetrahedron state (dashed orange); experimental density matrix obtained from the tomography (solid black); and the main eigenstate of the experimental density matrix (solid red). The $x$-axis of the figure is the rotation amplitude in radians and the $y$-axis is the overlap between the rotated state and the ideal tetrahedron state.

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Finally, returning to multi-parameter estimation in Fig. 6, we compare the s-QCRB calculated for our experimental state with the theoretical estimation strategies outlined in Section 2.4. As mentioned in that section, spin-coherent states have shot-noise scaling. Although $N00N$ states have Heisenberg scaling for one axis, other axes have shot-noise scaling and they dominate the resulting s-QCRB. To obtain Heisenberg scaling with $N00N$ states, one has to use three batches aligned in the $x$, $y$, and $z$ axes sequentially. Since one does not need a sequential strategy for Platonic states, they yield an s-QCRB three times better than sequential $N00N$ states. The figure has two data points, one for the experimental density matrix reconstructed from the tomography, and one for the main eigenstate of the density matrix. The s-QCRB for them are calculated theoretically from the reconstructed density matrix and the error bars are obtained using Monte Carlo simulations. Having an s-QCRB of $0.91 \pm 0.09$, the full density matrix narrowly beats the strategy with coherent states aligned on different axes (which has an s-QCRB of $1.125$), but fails to surpass the strategy of using sequential $N00N$ states (which has an s-QCRB of 0.675), indicating that a quantum advantage can be obtained using our state but that the experimental fidelity needs to be improved before a better-than-$N00N$-state advantage can be harnessed in a real-world metrological task. The dominant eigenstate on the other hand has an s-QCRB of $0.4 \pm 0.05$, very close to the ideal value of 0.375, beating all other strategies under discussion.

 

Fig. 6. Scalar quantum Cramér–Rao bound $\mathcal {C}$ as a function of total number of photons. Solid lines represent the different strategies enumerated in Section 2.4: spin-coherent states (purple triangles); $N00N$ states (red squares for simultaneous; orange pentagons for sequential); and Platonic states (yellow circles). Sequential strategies use three batches of states aligned in the $x, y$, and $z$ axes, whereas simultaneous strategies use only a single batch of identical states. The s-QCRB decreases with shot-noise scaling for sequential coherent states and simultaneous $N00N$ states, and decreases with Heisenberg scaling for sequential $N00N$ states and Platonic states. The s-QCRB of the experimentally reconstructed density matrix of the tetrahedron state is represented by a cross symbol. The black star represents the s-QCRB of its main eigenstate.

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Improvements in existing technology can render the creation of Platonic states more feasible. In particular, as we discussed, multi-photon emissions are one of the main reasons for our low fidelity. These could be reduced by reducing the loss in our experiment. The loss upon detection, in particular, has a great impact on our result as we have seen. Another avenue to reduce the contributions of these terms would be to reduce their probability of creation in the first place, prior to the loss. One approach for doing this is to change our scheme to rid ourselves of the coherent state. The coherent state always leaves the possibility of a single extra photon being created, a relatively likely event which an experiment relying only on SPDC would not allow. Recent work on automating the process of finding quantum optics experiments have yielded alternate schemes for the creation of the tetrahedron state which rely only on parametric downconversion [91]. A particular challenge in this experiment is that the photons from different sources need to be spectrally indistinguishable. To achieve this we use narrowband filters in our experiment. However, this comes at the cost of decreasing the coupling efficiency; thus, increasing the multi-photon contamination. These challenges could be simultaneously overcome by the use of modern quantum dot single photon sources [92,93]. Such sources have recently been used for multi-photon interference experiments with up to 20 photons [94]. With these on-demand photon sources one can imagine building the Platonic states using the methods of Refs. [23,85,95]. In this approach, independent photons are non-deterministically combined into the same spatial mode one at a time, each with a polarization state corresponding to a Majorana point. Although the state-creation methods of Ref. [95] are probabilistic, given the high degree of indistinguishability and almost-zero multi-photon emission rates of quantum dot sources, the post-selected multi-photon states created from this method should not suffer the low fidelity we face here. Thus, as quantum technology continues to improve it should be possible to harness the full potential of the Platonic states.

5. Conclusion

In conclusion, we demonstrated a scheme for the creation of the tetrahedron state, a 4-photon polarization state optimal in quantum mechanics for the estimation of polarization rotations. While a correctly aligned $N00N$ state can measure the rotation angle of a rotation with a known axis with Heisenberg scaling, one would need three $N00N$ state copies, each aligned with the different Cartesian axis to measure the three parameters of a spin rotation with the same scaling. This resource increase leads to a degradation of the asymptotic performance by a factor of three in the appropriately weighted sum of the measured parameters’ variances. The tetrahedron state, and other second-order unpolarized states, can be used to simultaneously measure all of the parameters of the rotation with Heisenberg scaling and are therefore not affected by this performance hit. Generalizations of these states tailored to transformations described with a minimum of $d$ parameters would yield $d$-fold improvements in the weighted sum of variances of said $d$ parameters. In this work, we created the tetrahedron state in the polarization of four photons by combining three photons themselves in a $N00N$ state attenuated along the horizontal axis with a horizontally polarized single photon. Implementing this required interfering the output of two SPDC sources and a coherent state. The main source of noise in our experiment comes from events where more photons were created than those needed for the tetrahedron state. These events could be made less significant by reducing the optical losses, which in our experiment could be achieved with better sources and detectors. Furthermore, in principle, the rate of these insidious events can always be made proportionally arbitrarily small by decreasing the overall tetrahedron state creation rate and increasing the counting time accordingly. In part because of the low 4-photon rate of our experiment, we were unable use the quantum state we created to measure parameters of a polarization rotation. Nevertheless, at the proof-of-concept level, we have created and characterized a state that shares many of the interesting qualitative features and symmetries of the optimal tetrahedron state, showing that our method is fundamentally sound. We also have shown that we have a good understanding of the technical limitations of our method and have discussed how to improve upon them, paving the way for future experiments. Continued progress in developing methods to robustly engineer states in high-dimensional Hilbert spaces, both in optics and in other physical media, ensure that it will soon be possible to generate with high fidelity states such as these which will enable significant advantages in many multi-parameter metrology applications, ranging from optical ellipsometry to 3D magnetometry.