Higher-dimensional symmetric informationally complete measurement via programmable photonic integrated optics

Standard measurements in quantum theory are projections onto a basis in $d$-dimensional Hilbert space. However, in the modern form of quantum theory, there exist more complex measurements, that can on their own be tomographically complete, with up to $d^{2}$ outcomes, while remaining irreducible D’Ariano et al. (2005). That is, they cannot be simulated by stochastically combining any other quantum measurements. The most prominent example of such generalised quantum measurements is a symmetric informationally complete (SIC) measurement Renes et al. (2004). Its defining property, in addition to tomographic completeness and irreducibility, is outcomes being fully symmetric in Hilbert space.

Much research has been devoted to theoretically constructing these measurements Fuchs et al. (2017). While they find many applications in quantum information, they are also connected to some interpretations of quantum theory Fuchs and Schack (2013) and well-known mathematics problems Appleby et al. (2017). In quantum information, SIC measurements are natural for many important tasks. Examples are state tomography Scott (2006); Stricker et al. (2022), entanglement detection Shang et al. (2018); Morelli et al. (2023) and random number generation Acín et al. (2016); Tavakoli et al. (2021). They are also relevant for quantum certification protocols Tavakoli et al. (2019), quantum key distribution Renes (2005); Tavakoli et al. (2020) and tests of nonlocality and contextuality Bengtsson et al. (2012); Tavakoli et al. (2021). Naturally, there has been considerable interest in experimentally realising SIC measurements. However, the implementation of generalised measurements comes with additional challenges. It requires control of ancillary quantum systems and entangling gate operations. In the case of SIC measurements, the challenge is particularly pronounced, as one must simultaneously resolve the maximal number of $d^{2}$ outcomes while also achieving uniform Hilbert space angles by control of populations and phases. Early photonics realisations of SIC measurements are based on simulating it by separately performing several standard measurements and post-selecting on successful outcomes Medendorp et al. (2011); Pimenta et al. (2013); Bent et al. (2015). Faithful implementations, using multiple photonic degrees of freedom for system and ancilla, have been achieved for the simplest, qubit, systems, see e.g. Bian et al. (2015); Zhao et al. (2015); Hou et al. (2018). However, it is well-known that already the interferometry required for realising SIC measurements beyond qubit dimensions is much more demanding Tabia (2012a).

Going beyond binary quantum systems is an important frontier in quantum information science. Higher-dimensional systems propel advantages in many communication protocols. In quantum key distribution, higher-dimensional systems can give better key rates and offer important improvements to the tolerance of noise and loss Hu et al. (2021); Bulla et al. (2023). Similar high-dimensional advantages are encountered in basic quantum information primitives such as entanglement distribution Ecker et al. (2019); Hu et al. (2020). In nonlocality experiments, high-dimensional systems can lead to larger Bell inequality and steering inequality violations Dada et al. (2011); Srivastav et al. (2022). Also in quantum computation, processors have been designed for higher dimensions Paesani et al. (2021); Chi et al. (2022).

It is a natural endeavor to develop measurement devices for beyond-binary quantum systems, that can faithfully implement not only standard but also generalised quantum measurements. Notably, a higher-dimensional generalised measurement has been reported Martínez et al. (2023), but it is conceptually different from a SIC measurement because it is neither tomographically complete nor motivated by established quantum information tasks. The biggest challenge in high dimensional SIC measurements is that the complexity of constructing measurements increases rapidly with increasing dimensions. A promising avenue to overcome this challenge is photonic integrated circuits (PICs), which provide high stability and scalability and are developing into a promising platform for quantum photonic experiments Wang et al. (2020); Feng et al. (2022a). Many quantum applications including logic gates Crespi et al. (2011); Feng et al. (2022b), higher-dimensional entanglement with standard measurements Wang et al. (2018); Lu et al. (2020); Chi et al. (2022), tests of Boson Sampling Crespi et al. (2013); Arrazola et al. (2021), quantum teleportation Llewellyn et al. (2020) and frustrated quantum interference Feng et al. (2023) have been demonstrated with PICs. Photonic quantum states can be directly generated, manipulated, and detected on integrated chips. Moreover, because of the high interferometric visibility and phase stability, quantum PIC with many passive and active components is particularly attractive to construct large-scale programmable quantum photonic processors. To date, quantum PICs have been expanded to contain thousands of components Bao et al. (2023) and dozens of photons Arrazola et al. (2021).

Here, we develop a PIC device that can faithfully implement generalised quantum measurements, in the form of SIC measurements on three-level quantum systems. Our device operates at high quality, clearly achieving the standards necessary both for demonstrating physical properties well beyond those of standard, projective, measurements, and for harvesting the advantages of generalised measurements in quantum information protocols. Specifically, this is demonstrated through detector tomography, through a range of verification and simulation tasks and through quantum random number generation. In addition, the device is versatile as it can be readily programmed to implement many other measurements. We demonstrate this feature by realising highly symmetric projections as well as the seminal mutually unbiased bases, and apply them in relevant information tasks.

A quantum measurement in dimension $d$ with $n$ outcomes is described by a collection of positive semi-definite $d\times d$ matrices $\{E_{a}\}_{i=1}^{n}$ such that $\sum_{a}E_{a}=\openone$. For irreducible standard measurements, $E_{i}$ are orthogonal projectors and the number of outcomes is at most $n=d$. For irreducible generalised measurements, one can reach $n=d^{2}$ outcomes. The SIC measurement corresponds to choosing $E_{a}=\frac{1}{d}\outerproduct{\psi_{a}}{\psi_{a}}$ where all distinct pairs of modulus overlaps are identical, namely $|\innerproduct{\psi_{j}}{\psi_{k}}|^{2}=\frac{1}{d+1}$ for every $j\neq k$. A standard instance of a qutrit SIC measurement corresponds to the following nine (unnormalized) vectors $\ket{\psi_{a}}$:

where $\omega=e^{\frac{2{\pi}i}{3}}$.

Generalized measurements can always be achieved through standard projection measurements combined with auxiliary systems which can be achieved through a so-called Naimark dilation Peres (1990). When $n>d$ and the measurement operators, $E_{a}$, are proportional to rank-one projectors onto a vector $e_{a}$, the $d\times n$ matrix $U=\left(e_{1}\ldots e_{n}\right)$ must be extended to a larger Hilbert space, in which its columns become orthogonal. Hence, we require an $(n-d)\times n$ matrix $W$ such that $U=\left(\begin{array}[]{c}V\\ W\end{array}\right)$ is unitary.

We consider the Naimark extension for the SIC measurement corresponding to the following unitary matrix,

Here, the rows 1, 4, and 7 correspond to the matrix $U$ containing the qutrit SIC states (1). This matrix is to be applied to an incoming qutrit and initialised ancilla qutrit ($\ket{\psi}\otimes\ket{0}$), with the former ordered on ports $(1,4,7)$. In order to reduce the complexity of implementing the unitary transformation $U$, we decompose the $9\times 9$ matrix into a sequence of three-dimensional subspace operations. These are shown in Fig. 1a, in terms of unitaries $P$ and $Q_{1-3}$ (see Supplementary Material), which are unit operations in suitable three-dimensional subspaces Tabia (2012b).

We use a reconfigurable silicon photonic integrated circuit to implement the high-dimensional SIC measurement scheme. Silicon photonics show sizable advantages in scaling and functionality with low-cost, CMOS-compatible fabrication and ultra-high integration density Thomson et al. (2016). As shown in Fig. 1b, the device we developed consists of a cascade of Mach-Zehnder interferometers (MZIs) and multiple phase shifters, which are programmed to realize the expected operations given below. The total chip size is just 3.3$\times$1.0$\,\rm{mm}^{2}$, and all phase shifters are characterized with average interference visibility over 0.99.

Depending on the function, the device is divided into two parts, the state preparation part (region I) and the measurement part (regions II, III, IV). In the state preparation part, any three-dimensional pure state can be constructed using two MZIs and phase shifters. The subsequent measurement apparatus, with three inputs and nine outputs, can be switched between several different settings. For example, by adjusting the phase shifter, one can switch from a nine-outcome SIC measurement to different sets of mutually unbiased basis measurements. In the case of SIC measurement, in our scheme, we only need to implement the unitary operation on three-dimensional subspaces. As shown in Fig 1, we sequentially perform the unitaries $P^{{\dagger}}$, $Q^{{\dagger}}$, and $P$ operations on the three-dimensional subspaces in the II, III, and IV regions, ultimately resulting in 9 outputs. In contrast, mutually unbiased bases, being standard projective measurements, require only a unitary operation of 3 inputs and 3 outputs. They can be implemented by resetting the phase of the phase shifter. The specific phase shifter settings and further details on the PIC device can be found in the supplementary materials.

In the experiment, a CW laser at 775 nm is used to pump a type II PPKTP crystal to generate correlated photon pairs Feng et al. (2022b). They are coupled into different singe-mode fibers, and one photon is detected directly to herald the existence of another. The photons show a coherence time of about 1.5 ps. The heralded single photons are coupled into the silicon quantum photonic processor through the single-mode fiber and on-chip grating coupler, and the qutrit states are generated and analyzed on the single chip.

A central question is to gauge how accurately the measurement implemented on our device corresponds to the SIC measurement. For this purpose, we discuss several different verification methods. Details are given in Supplementary Material.

We consider the use of the measurement device for protocols beyond the purpose of benchmarking the device. In our three case studies, we showcase also the programmability of the measurement device. Details on the protocols and our experiments are found in Supplementary Material.

We have reported on a photonic integrated circuit for implementing a three-dimensional symmetric informationally complete measurement. This demonstrates a faithful realisation, beyond qubit systems, of the perhaps most celebrated type of generalised quantum measurement. Importantly, the quality of the implemented measurement is showcased in a variety of tasks, where it significantly exceeds the limitations of standard, projective, measurements. In principle, our work can be extended to generalized measurements of any dimension combined with mature integrated optics. The results showcase the potential of using integrated photonics towards implementing sophisticated quantum measurements while maintaining both the high quality necessary for harvesting their advantages and the scalability necessary to access such operations in truly higher-dimensional systems. The scalability and programmability of integrated optics for high-dimensional generalized measurement can also be combined with existing light source preparation, transmission, and operation technologies to achieve quantum information tasks at high performance standards.

Funding.

This work was supported by the Innovation Program for Quantum Science and Technology (Nos. 2021ZD0301500, 2021ZD0303200, 2021ZD0301200), the National Natural Science Foundation of China (NSFC) (Nos. T2325022, 62275240, 62061160487, 12004373, and 12374338), Key Research and Development Program of Anhui Province (2022b1302007), CAS Project for Young Scientists in Basic Research (Grant No. YSBR-049), USTC Tang Scholarship, Science and Technological Fund of Anhui Province for Outstanding Youth (2008085J02), the Postdoctoral Science Foundation of China (No. 2021T140647) and the Fundamental Research Funds for the Central Universities. A.T. is supported by the Wenner-Gren Foundation and by the Knut and Alice Wallenberg Foundation through the Wallenberg Center for Quantum Technology (WACQT).

Acknowledgment.

This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.

Disclosures.

The authors declare no conflicts of interest.

Data availability.

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document.

See Supplement 1 for supporting content.