OAM tomography with Heisenberg–Weyl observables

Quantum state tomography (QST), or quantum state reconstruction, is a technique used to extract the maximum available information from a quantum state in order to reconstruct its density matrix (for discrete variables) or its phase-space representation (for continuous variables) [1].

Quantum tomography has been performed experimentally on various physical systems, e.g., sub-levels of H⁺ and He atoms [2], coherent states [3], squeezed vacuum states [4–6], vibrations of molecules [7] and large angular momentum states of Cs atoms [8].

The simplest QST, on polarisation qubits, consists of measuring the well-known Stokes parameters [9]. This technique was demonstrated on a non-maximally entangled bi-photon state [10]. It can be extended to n qubits [11] and was experimentally demonstrated for two qubits [11].

Scaling QST from qubits to qudits is challenging for several reasons. First, one has to find the equivalent of Pauli operators in d dimensions. One popular choice are the Gell–Mann matrices λ_i, the generators of SU(d). Second, in order to reconstruct ρ we need to measure experimentally the expectations values ⟨λ_i⟩, which is not trivial [12].

A different approach to qudit tomography has been introduced recently by Asadian et al [13]. The idea is to use the Heisenberg–Weyl observables (HW) instead of λ_i. In the same article [13] the authors also proposed a method to measure HW observables based on deterministic quantum computation with one qubit (DQC1) model [14]. This scheme is particularly attractive since it requires to measure only a qubit (ancilla) instead of measuring the qudit (see next section).

In this article we apply the Asadian et al scheme to an experimentally important case, namely photonic orbital angular momentum (OAM). Since the seminal paper of Allen et al [15], orbital angular momentum of light became a new field of exploration [16, 17]. The orbital angular momentum of a photon is due to the helical phase front along the propagation direction with quantized angular momentum ℓℏ, with $\ell \in \mathbb{Z}$. A review of OAM states can be found in references [18–20].

Due to their weak coupling to the environment, photons carrying OAM are a natural choice to implement protocols for qudits. Applications include QKD [21], object identification [22], enhanced phase sensitivity [23], imaging with super-diffraction-limit resolution [24] and metrology [25, 26].

All these applications rely on quantum tomography to characterise the OAM state. Current methods for OAM tomography [17, 27–29] require computer-generated holograms, usually displayed by spatial light modulators (SLMs), and single-mode fibers, acting as mode filters. SLMs have low efficiency due to the pixelated surface and the existence of dead zones (the opaque areas between pixels). Mode filters further reduce the efficiency.

Here we describe a different setup for OAM tomography. Our protocol simplifies the measurement step, as we need to measure only a path qubit. In our case the complexity is shifted to the ability to apply generalised Pauli gates on OAM states.

The article is structured as follows. In section 2 we briefly describe qudit tomography with Heisenberg–Weyl observables. In section 3 we apply this method to quantum tomography for OAM states. Our scheme uses a Mach–Zehnder interferometer with tuneable phases and generalised Pauli gates ${Z}_{d}^{l}{X}_{d}^{m}$. Finally, we sketch future perspectives of our setup in section 4.

Performing tomography of the density matrix ρ requires a suitable basis. In the following we assume the Hilbert space $\mathcal{H}$ is finite dimensional, $\mathrm{dim} \mathcal{H}=d$. For example, in d = 2 we use the Pauli matrices $\left\{{X}_{i},i=0, \dots ,3\right\}{:=}\left\{{\mathbb{I}}_{2},X,Y,Z\right\}$, such that:

$$\begin{equation}\rho =\frac{1}{2}\sum _{i=0}^{3}{r}_{i}{X}_{i}\end{equation} \tag{ 1 }$$

The Pauli matrices are Hermitian and form a basis in the space $\mathcal{M}\left(2\right)$ of linear operators in two-dimensions. The coefficients r_i = Tr (ρX_i) = ⟨X_i⟩ are the expectation values of the Pauli operators on the state ρ:

$$\begin{equation}\rho =\frac{1}{2}\left[\begin{bmatrix}{cc}1+\langle Z\rangle \hfill & \langle X\rangle -i\langle Y\rangle \hfill \\ \langle X\rangle +i\langle Y\rangle \hfill & 1-\langle Z\rangle \hfill \end{bmatrix}\right]\end{equation} \tag{ 2 }$$

In d dimensions a possible basis is the set of generalised Gell–Mann matrices λ_i [12], the standard generators of SU(d). However, it is not straightforward to measure the expectation values ⟨λ_i⟩ of these observables in d dimensions. In practice one measures a different set of projectors which are experimentally accessible. For example, in reference [29] these are projectors on the states $\vert \ell \rangle ,\frac{1}{\sqrt{2}}\left(\vert {\ell }_{1}\rangle {\pm}\vert {\ell }_{2}\rangle \right),\frac{1}{\sqrt{2}}\left(\vert {\ell }_{1}\rangle {\pm}i\vert {\ell }_{2}\rangle \right)$, ℓ₁ < ℓ₂. The expectations values ⟨λ_i⟩ of the Gell–Mann matrices can be calculated from the density matrix after tomographic reconstruction (see also reference [12]). In our method, which is the direct implementation of the theoretical protocol, the expectation values of the basis elements are obtained by direct measurement.

We adopt a different approach and use the Heisenberg–Weyl (HW) observables introduced in reference [13]. Consider the generalised Pauli operators X_d and Z_d in d dimensions:

$$\begin{equation}{X}_{d}: \vert j\rangle {\mapsto}\vert j\oplus 1\rangle \end{equation} \tag{ 3 }$$

$$\begin{equation}{Z}_{d}: \vert j\rangle {\mapsto}{\omega }^{j}\vert j\rangle \end{equation} \tag{ 4 }$$

where ⊕ is addition mod d and ω = e^2πi/d a root of unity of order d. These operators are unitary but not Hermitian, ${X}_{d}^{{\dagger}}={X}_{d}^{-1}$, ${Z}_{d}^{{\dagger}}={Z}_{d}^{-1}$, ${X}_{d}^{d}={\mathbb{I}}_{d}={Z}_{d}^{d}$; therefore they are not observables. Nevertheless, we can use them to construct the Heisenberg–Weyl (HW) observables [13]:

$$\begin{equation}{Q}_{lm}=\frac{1+i}{2}{Z}_{d}^{l}{X}_{d}^{m}{\omega }^{-lm/2}+\mathrm{h}.\mathrm{c}.\end{equation} \tag{ 5 }$$

The operators Q_lm are Hermitian (by construction) and orthogonal [13]:

$$\begin{equation}\mathrm{Tr}\left\{{Q}_{lm}{Q}_{{l}^{\prime }{m}^{\prime }}\right\}=d{\delta }_{l{l}^{\prime }}{\delta }_{m{m}^{\prime }}\end{equation} \tag{ 6 }$$

This gives us a set of d² linearly independent observables which form a basis in the space of d-dimensional linear operators $\mathcal{M}\left(d\right)$. Therefore any density operator can be written as:

$$\begin{equation}\rho =\frac{1}{d}\sum _{l,m=0}^{d-1}\langle {Q}_{lm}\rangle {Q}_{lm}\end{equation} \tag{ 7 }$$

with

$$\begin{equation}\langle {Q}_{lm}\rangle =\mathrm{Tr}\left(\rho {Q}_{lm}\right)\end{equation} \tag{ 8 }$$

Thus tomography of a d-dimensional density matrix ρ reduces to the measurement of the expectation values ⟨Q_lm⟩. Asadian et al [13] showed how to measure these values using the DQC1 technique. Deterministic quantum computation with one qubit is an efficient method to estimate the normalised trace of an operator [14]. We briefly discuss this method in the following.

Consider the circuit in figure 1. We use a two-dimensional ancilla (a qubit) to perform tomography of a d-dimensional density matrix ρ of a qudit. The initial state of the qubit–qudit system is separable:

$$\begin{equation}{\rho }_{\mathrm{i}\mathrm{n}}=\vert 0\rangle \langle 0\vert \otimes \rho \end{equation} \tag{ 9 }$$

On the qubit ancilla we perform a Hadamard H, a phase shift P_φ, then a controlled unitary C(U) that couples the qubit and the qudit. Finally, we perform another Hadamard H and we then measure the qubit in the Z-basis (figure 1). After a straightforward calculation we obtain the expectation value of the qubit, see appendix A:

$$\begin{equation}\langle Z\rangle =\frac{1}{2}\mathrm{Tr}\left\{\rho \left({\mathrm{e}}^{\mathrm{i}\varphi }U+{\mathrm{e}}^{-\mathrm{i}\varphi }{U}^{{\dagger}}\right)\right\}\end{equation} \tag{ 10 }$$

Importantly, no measurement is performed on the qudit, as we trace out this system. Since the C(U) gate provides an effective qubit–qudit coupling (i.e., it entangles the two subsystems), a measurement on the qubit gives us information about the qudit state.

Zoom In Zoom Out Reset image size

Choosing $U={Z}_{d}^{l}{X}_{d}^{m}$ and taking $\varphi ={\varphi }_{lm}{:=}\frac{\pi }{4}-\frac{\pi }{d}lm$, we have ${Q}_{lm}=\frac{1}{\sqrt{2}}{\mathrm{e}}^{\mathrm{i}{\varphi }_{lm}}U+\mathrm{h}.\mathrm{c}$., therefore

$$\begin{equation}\langle {Q}_{lm}\rangle =\mathrm{Tr}\left(\rho {Q}_{lm}\right)=\sqrt{2}\langle Z\rangle \end{equation} \tag{ 11 }$$

Thus we can perform tomography of a d-dimensional density matrix ρ only by measuring the expectation value ⟨Z⟩ of a qubit. All the coefficients ⟨Q_lm⟩ are obtained by changing the phases φ_lm and the controlled operator $C\left({Z}_{d}^{l}{X}_{d}^{m}\right)$, l, m = 0, ..., d − 1, between the ancilla and the qudit.

We now apply the previous method to photons carrying orbital angular momentum, a popular choice for qudits. This requires the following ingredients:

(i) A fully-controllable qubit ancilla which can be measured in the Z-basis; fully-controllable means we can apply H and ${P}_{{\varphi }_{lm}}$ gates;

(ii) The capability to apply a controlled gate C(U) between qubit and OAM qudit;

(iii) The capability to implement ${Z}_{d}^{l}{X}_{d}^{m}$ operators on the OAM qudit, l, m = 0, ..., d − 1.

In the following we discuss how to implement these requirements.

(i) A priori there are several ways to implement the ancilla. The qubit can be a different quantum system (photon, atom in a cavity etc) or it can be another degree of freedom of the photon. Because we need to apply a controlled gate C(U) and since it is experimentally difficult to implement a photon–photon or photon–atom interaction, the natural choice is to use the path (or spatial mode) of the photon as the qubit ancilla; in the following we use path qubit and mode qubit interchangeably. In this case the Hadamard gate H is a beamsplitter [30–33], and therefore the two H gates define a Mach–Zehnder interferometer (MZI), as in figure 2.

The phase-gate ${P}_{{\varphi }_{lm}}$ on the path qubit (figure 1) is equivalent to a phase shift φ_lm in one arm of the MZI interferometer, independent on the OAM value. This can be done by having a variable path difference between the two arms of the MZI interferometer, figure 2.

Zoom In Zoom Out Reset image size

(ii) Using a path qubit as the ancilla also solves the second problem. In this case the controlled gate C(U) on the OAM is particularly simple: we need to apply the U gate on the OAM only on path 1.

(iii) Finally, we need to implement ${Z}_{d}^{l}{X}_{d}^{m}$ gates. For OAM states, the standard implementation of the ${Z}_{d}^{l}$ gate consists of two Dove prisms rotated with a relative angle α = πl/d [17, 34, 35], figure 3(a)

$$\begin{equation}{Z}_{d}^{l}: \vert k\rangle \to {\mathrm{e}}^{\mathrm{i}2\alpha k}\vert k\rangle ={\omega }^{lk}\vert k\rangle \end{equation} \tag{ 12 }$$

Zoom In Zoom Out Reset image size

The last building block is the X_d gate which performs a cyclic permutation of the basis states (3). Cyclic permutations for OAM states are difficult to implement and until recently only specific examples for d = 3, 4 were known [36, 37]. Two methods to construct arbitrary X_d gates for OAM states have been proposed recently [38, 39].

The first optical element of the X_d gate is a spiral-phase plate (SPP), see reference [38]. A spiral-phase plate of order k, SPP(k), shifts all OAM values by k units, $k\in \mathbb{Z}$:

$$\begin{equation}\mathrm{S}\mathrm{P}\mathrm{P}\left(k\right): \vert {j\rangle }_{\mathrm{O}\mathrm{A}\mathrm{M}}{\mapsto}\vert j+{k\rangle }_{\mathrm{O}\mathrm{A}\mathrm{M}}\end{equation} \tag{ 13 }$$

When we apply SPP(1) to the basis states |j⟩, the last state will be out of range, |d − 1⟩ ↦ |d⟩; for simplicity, we omit the OAM subscript. For the cyclic permutation (3) we need to put this value back to |0⟩.

This is where the OAM sorter S_d [40] comes into play. The sorter S_d demultiplexes (i.e., sorts) different OAM values into distinct paths and acts as a generalised polarising beam-splitter (PBS) for OAM. Thus after applying SPP(1) and the sorter S_d, we apply SPP(−d) only to path 0. Finally, the inverse sorter ${S}_{d}^{-1}$ multiplexes back all OAM values on the same path (figure 3(b)).

In order to implement the ${X}_{d}^{m}$ gate there are two strategies. First, one could apply m times the X_d gate described above (the serial setup). However, this is rather inefficient. Here we propose a more efficient way (the parallel setup).

In the parallel version we first shift all states by m units with an SPP(m). Consequently, m OAM values will be out of range, |d⟩, ..., |d − 1 + m⟩. Next a sorter S_d demultiplexes all OAM values to different paths. Since the sorter works cyclically on paths [40], the first m paths have OAM values outside of range. As before, in this case we shift back only these paths by SPP(−d). The last element is the inverse sorter ${S}_{d}^{-1}$ which multiplexes back all OAM states to a single path (i.e., it disentangles the path and OAM degrees of freedom).

The setup for the parallel ${X}_{d}^{m}$ gate performs the following sequence (see figure 3):

$$\begin{align}\hfill \vert {i\rangle }_{\mathrm{O}\mathrm{A}\mathrm{M}}\vert {0\rangle }_{m}& +m{\to }\vert i+{m\rangle }_{\mathrm{O}\mathrm{A}\mathrm{M}}\vert {0\rangle }_{m}\hfill \\ \hfill & {S}_{d}{\to }\vert i+{m\rangle }_{\mathrm{O}\mathrm{A}\mathrm{M}}\vert i\oplus {m\rangle }_{m}\hfill \\ \hfill & -{d}^{\left[r\right]}{\to }\vert i\oplus {m\rangle }_{\mathrm{O}\mathrm{A}\mathrm{M}}\vert i\oplus {m\rangle }_{m}\hfill \\ \hfill & {S}_{d}^{-1}{\to }\vert i\oplus {m\rangle }_{\mathrm{O}\mathrm{A}\mathrm{M}}\vert {0\rangle }_{m}\hfill \end{align} \tag{ 14 }$$

with r = 0, ..., m − 1.

The resources required to implement ${X}_{d}^{m}$ in the serial vs. the parallel setup are summarised in table 1. Both cases use the same number of SPP(−d); the serial setup has m SPP(1) and the parallel one has a single SPP(m) (which are equivalent resources). In contrast, the parallel setup needs a constant number of sorters (two) irrespective of m, whereas the serial setup requires 2m sorters ${S}_{d},{S}_{d}^{-1}$. Thus the parallel setup saves 2m − 2 sorters.

Resource-wise, the sorter S_d requires two Fourier gates F_d and ${F}_{d}^{{\dagger}}$ acting on path (spatial modes) and path-dependent phase shifts between them. The Fourier gates can be implemented using beam-splitters and phase-shifters [41–43] or as multimode interference devices in integrated optics [44–46].

We can extend our tomography method to an arbitrary number of qudits. For two qudits we need to perform d⁴ coincidence measurements of the observables Q_lm and Q_l'm' for the two photons [11, 29], see figure 4. For N-qudits the density matrix is reconstructed from a set of d^2N coincidence measurements of single-qudit observables ${Q}_{{l}_{1}{m}_{1}}, \dots ,{Q}_{{l}_{N}{m}_{N}}$.

Table 1. Number of optical elements required to implement ${X}_{d}^{m}$ gate in the serial and parallel setup.

	SPP(1)	SPP(m)	SPP(−d)	${S}_{d},{S}_{d}^{-1}$
Serial ${X}_{d}^{m}$	m	0	m	2m
Parallel ${X}_{d}^{m}$	0	1	m	2

Zoom In Zoom Out Reset image size

All protocols using photonic OAM as qudits require an efficient tomography step. Current methods for OAM tomography use spatial light modulators to display computer-generated holograms and single-mode fibers as mode filters [27–29]. Although configurable, SLMs have a lower efficiency compared to SPPs. Moreover, they are bulky and difficult to miniaturize, given the current drive towards integrated photonics.

In this article we describe a new scheme for OAM tomography based on reference [13] which avoids these problems. In our scheme we measure only a path qubit, thus simplifying the measurement step. The complexity of the scheme resides in the controlled application of Heisenberg–Weyl operators.

Future implementations of our setup can benefit from reconfigurable SPPs [47, 48]. These liquid crystal devices could be used as switchable SPPs in the ${X}_{d}^{m}$ gate.

A possible extension of our scheme is quantum tomography for radial modes. Since sorters for radial modes r have been experimentally demonstrated [49, 50], the missing element is the optical equivalent of an SPP for radial modes, |r⟩ ↦ |r + 1⟩. Although adding or subtracting arbitrary units of radial quantum number is an open question, recent developments in this direction are promising [51, 52].

The authors acknowledge support from a grant of the Romanian Ministry of Research and Innovation, PCCDI-UEFISCDI, project number PN-III-P1-1.2-PCCDI-2017-0338/79PCCDI/2018, within PNCDI III. RI acknowledges support from PN 19060101/2019-2022. S A also acknowledges support by the Extreme Light Infrastructure Nuclear Physics (ELI-NP) Phase II, a project co-financed by the Romanian Government and the European Union through the European Regional Development Fund and the Competitiveness Operational Programme (1/07.07.2016, COP, ID 1334).

Here we calculate the outcome of the circuit from figure 1. After the H and P_φ gates, the density matrix of the system is:

$$\begin{equation}{\rho }^{\prime }=\frac{1}{2}\left(\vert 0\rangle \langle 0\vert +{\mathrm{e}}^{-\mathrm{i}\varphi }\vert 0\rangle \langle 1\vert +{\mathrm{e}}^{\mathrm{i}\varphi }\vert 1\rangle \langle 0\vert +\vert 1\rangle \langle 1\vert \right)\otimes \rho \end{equation} \tag{ A1 }$$

We now apply the control gate C(U) and obtain

$$\begin{equation}{\rho }^{{\prime\prime}}=\frac{1}{2}\left(\vert 0\rangle \langle 0\vert \otimes \rho +\vert 1\rangle \langle 1\vert \otimes U\rho {U}^{{\dagger}}+{\mathrm{e}}^{\mathrm{i}\varphi }\vert 1\rangle \langle 0\vert \otimes U\rho +{\mathrm{e}}^{-\mathrm{i}\varphi }\vert 0\rangle \langle 1\vert \otimes \rho {U}^{{\dagger}}\right)\end{equation} \tag{ A2 }$$

The two subsystems (the qubit ancilla and the qudit) are now entangled. After applying the second Hadamard gate H on the qubit, the density matrix becomes

$$\begin{align}\hfill {\rho }_{\mathrm{o}\mathrm{u}\mathrm{t}}& =\frac{1}{4}\left\{\vert 0\rangle \langle 0\vert \otimes \left(\rho +U\rho {U}^{{\dagger}}+{\mathrm{e}}^{\mathrm{i}\varphi }U\rho +{\mathrm{e}}^{-\mathrm{i}\varphi }\rho {U}^{{\dagger}}\right)+\vert 1\rangle \langle 0\vert \otimes \left(\rho -U\rho {U}^{{\dagger}}-{\mathrm{e}}^{\mathrm{i}\varphi }U\rho +{\mathrm{e}}^{\mathrm{i}\varphi }\rho {U}^{{\dagger}}\right)\right.\hfill \\ \hfill & \left.\quad + \vert 0\rangle \langle 1\vert \otimes \left(\rho -U\rho {U}^{{\dagger}}+{\mathrm{e}}^{\mathrm{i}\varphi }U\rho -{\mathrm{e}}^{-\mathrm{i}\varphi }\rho {U}^{{\dagger}}\right)+\vert 1\rangle \langle 1\vert \otimes \left(\rho +U\rho {U}^{{\dagger}}-{\mathrm{e}}^{\mathrm{i}\varphi }U\rho -{\mathrm{e}}^{-\mathrm{i}\varphi }\rho {U}^{{\dagger}}\right)\right\}\hfill \end{align} \tag{ A3 }$$

In DQC1 we discard the qudit and measure only the qubit ancilla. Hence we trace out the qudit degrees of freedom and we obtain the reduced density matrix ρ₂ of the qubit

$$\begin{align}\hfill {\rho }_{2}& =\frac{1}{4}\left(\vert 0\rangle \langle 0\vert \left(2+\mathrm{Tr}\left\{{\mathrm{e}}^{\mathrm{i}\varphi }U\rho +{\mathrm{e}}^{-\mathrm{i}\varphi }\rho {U}^{{\dagger}}\right\}\right)+\vert 1\rangle \langle 1\vert \left(2-\mathrm{Tr}\left\{{\mathrm{e}}^{\mathrm{i}\varphi }U\rho +{\mathrm{e}}^{-\mathrm{i}\varphi }\rho {U}^{{\dagger}}\right\}\right)\right.\hfill \\ \hfill & \quad \left.+ \vert 1\rangle \langle 0\vert \mathrm{Tr}\left\{-{\mathrm{e}}^{\mathrm{i}\varphi }U\rho +{\mathrm{e}}^{-\mathrm{i}\varphi }\rho {U}^{{\dagger}}\right\}+\vert 0\rangle \langle 1\vert \mathrm{Tr}\left\{{\mathrm{e}}^{\mathrm{i}\varphi }U\rho -{\mathrm{e}}^{-\mathrm{i}\varphi }\rho {U}^{{\dagger}}\right\}\right)\hfill \end{align} \tag{ A4 }$$

Thus we recover the expectation value of the qubit in the Z-basis, equation (10):

$$\begin{equation}\langle Z\rangle =\frac{1}{2}\mathrm{Tr}\left\{\rho \left({\mathrm{e}}^{\mathrm{i}\varphi }U+{\mathrm{e}}^{-\mathrm{i}\varphi }{U}^{{\dagger}}\right)\right\}\end{equation} \tag{ A5 }$$