Higher dimensional quantum systems are an important avenue for new explorations in quantum computing as well as quantum communications. One of the ubiquitous resources in quantum technologies is entanglement. However, so far, entanglement has been certified in higher dimensional systems through suitable bounds on known entanglement measures. In this work, we have, for the first time, quantified the amount of entanglement in bi-partite pure qutrit states by analytically relating statistical correlation measures and known measures of entanglement, and have determined the amount of entanglement in our experimentally generated spatially correlated bi-partite qutrit system. We obtain the value of Negativity in our bi-partite qutrit to be 0.85 +/- 0.03 and the Entanglement of Formation(EOF) to be 1.23 +/- 0.01. In terms of quantifying the deviation from the maximally entangled state, the Negativity value demonstrates ~15 % deviation while the EOF value demonstrates ~24 % deviation. This serves as the first experimental evidence of such non-equivalence of entanglement measures for higher dimensional systems.
While it has recently been demonstrated how to certify the maximal amount of randomness from any pure two-qubit entangled state in a device-independent way [E. Woodhead et al., Phys. Rev. Research 2, 042028(R)(2020)], the problem of optimal randomness certification from entangled states of higher local dimension remains open. Here we introduce a method for device-independent certification of the maximal possible amount of $2\log_23$ random bits using pure bipartite entangled two-qutrit states and extremal nine-outcome general non-projective measurements. To this aim, we exploit the extended Be…
Multipartite entanglement is one of the core concepts in quantum information science with broad applications that span from condensed matter physics to quantum physics foundations tests. Although its most studied and tested forms encompass two-dimensional systems, current quantum platforms technically allow the manipulation of additional quantum levels. We report the experimental demonstration and certification of a high-dimensional multipartite entangled state in a superconducting quantum processor. We generate the three-qutrit Greenberger-Horne-Zeilinger state by designing the necessary puls…
Self-testing refers to the fact that, in some quantum devices, both states and measurements can be assessed in a black-box scenario, on the sole basis of the observed statistics, i.e. without reference to any prior device calibration. Only a few examples of self-testing are known, and they just provide non-trivial assessment for devices performing unrealistically close to the ideal case. We overcome these difficulties by approaching self-testing with the semi-definite programming hierarchy for the characterization of quantum correlations. This allows us to improve dramatically the robustness o…
We present the first experimental confirmation of the quantum-mechanical prediction of stronger-than-binary correlations. These are correlations that cannot be explained under the assumption that the occurrence of a particular outcome of an $n \ge 3$-outcome measurement is due to a two-step process in which, in the first step, some classical mechanism precludes $n-2$ of the outcomes and, in the second step, a binary measurement generates the outcome. Our experiment uses pairs of photonic qutrits distributed between two laboratories, where randomly chosen three-outcome measurements are performe…
A scheme for characterizing entanglement using the statistical measure of correlation given by the Pearson correlation coefficient (PCC) was recently suggested that has remained unexplored beyond the qubit case. Towards the application of this scheme for the high dimensional states, a key step has been taken in a very recent work by experimentally determining PCC and analytically relating it to Negativity for quantifying entanglement of the empirically produced bipartite pure state of spatially correlated photonic qutrits. Motivated by this work, we present here a comprehensive study of the ef…
We produce two identical keys using, for the first time, entangled trinary quantum systems (qutrits) for quantum key distribution. The advantage of qutrits over the normally used binary quantum systems is an increased coding density and a higher security margin. The qutrits are encoded into the orbital angular momentum of photons, namely Laguerre-Gaussian modes with azimuthal index l +1, 0 and -1, respectively. The orbital angular momentum is controlled with phase holograms. In an Ekert-type protocol the violation of a three-dimensional Bell inequality verifies the security of the generated ke…
Gate-based quantum computation has been extensively investigated using quantum circuits based on qubits. In many cases, such qubits are actually made out of multilevel systems but with only two states being used for computational purpose. While such a strategy has the advantage of being in line with the common binary logic, it in some sense wastes the ready-for-use resources in the large Hilbert space of these intrinsic multi-dimensional systems. Quantum computation beyond qubits (e.g., using qutrits or qudits) has thus been discussed and argued to be more efficient than its qubit counterpart …
We have recently shown that for the special case of a bipartite system with binary inputs and outputs there exist equalities in local theories which are violated by quantum theory. The amount of white noise tolerated by these equalities are twice that of inequalities. In this paper we will first introduce an inequality in bipartite qutrits systems which, if non-maximally entangled state is used instead of maximally entangled state, is violated more strongly by quantum theory. Hence reproducing the results obtained in the literature. We will then prove that our equalities in this case are viola…
We present a study of quantum contextuality of three-dimensional mixed states for the Klyachko-Can-Binicioğlu-Shumovsky (KCBS) and the Kurzyński-Kaszlikowski (KK) noncontextuality inequalities. For any class of states whose eigenvalues are arranged in decreasing order, a universal set of measurements always exists for the KK inequality, whereas none does for the KCBS inequality. This difference can be reflected from the spectral distribution of the overall measurement matrix. Our results would facilitate the error analysis for experimental setups, and our spectral method in the paper combined …
Reliable encoding of information in quantum systems is crucial to all approaches to quantum information processing or communication. This applies in particular to photons used in linear optics quantum computing (LOQC), which is scalable provided a deterministic single-photon emission and preparation is available. Here, we show that narrowband photons deterministically emitted from an atom-cavity system fulfill these requirements. Within their 500 ns coherence time, we demonstrate a subdivision into d time bins of various amplitudes and phases, which we use for encoding arbitrary qu-d-its. The …
We experimentally study the violation of the CGLMP inequality for entangled 2-qubit and 2-qutrit states with different degrees of entanglement using numerically optimized measurement settings. The qudits are encoded and manipulated in the frequency spectrum of broadband energy-time entangled photons by taking into account a spatial light modulator. The latter allows to discretize the spectrum into bins. By controlling each frequency bin individually, the generation of maximally and non-maximally entangled qutrits is verified through quantum state tomography.
In this report we study certification of quantum measurements, which can be viewed as the extension of quantum hypotheses testing. This extension involves also the study of the input state and the measurement procedure. Here, we will be interested in two-point (binary) certification scheme in which the null and alternative hypotheses are single element sets. Our goal is to minimize the probability of the type II error given some fixed statistical significance. In this report, we begin with studying the two-point certification of pure quantum states and unitary channels to later use them to pro…
In some quantum algorithms, arithmetic operations are of utmost importance for resource estimation. In binary quantum systems, some efficient implementation of arithmetic operations like, addition/subtraction, multiplication/division, square root, exponential and arcsine etc. have been realized, where resources are reported as a number of Toffoli gates or T gates with ancilla. Recently it has been demonstrated that intermediate qutrits can be used in place of ancilla, allowing us to operate efficiently in the ancilla-free frontier zone. In this article, we have incorporated intermediate qutrit…
In this article the elementary gates for ternary quantum logic circuit are studied. We propose the ternary controlled X (TCX) gate or ternary controlled Z (TCZ) gate as two-qutrit elementary gate, which is universal when assisted by arbitrary one-qutrit gates. It is primitive, efficient and easy to implement. Based on Cartan decomposition, we also give the one-qutrit elementary gates. Then the synthesis of some important ternary gates is investigated and the scheme of physical implementation for these ternary gates is discussed. Finally we extend these elementary gates to more general qudit ca…
Qutrit-Based Neural Quantum Kernels for Classification Tasks
BCAM - Basque Center for Applied Mathematics（巴斯克应用数学中心）；University of the Basque Country UPV/EHU（巴斯克大学）；TECNALIA, Basque Research and Technology Alliance (BRTA)（巴斯克研究与技术联盟）
Quantum computation is conventionally performed using quantum operations acting on two-level quantum bits, or qubits. Qubits in modern quantum computers suffer from inevitable detrimental interactions with the environment that cause errors during computation, with multi-qubit operations often being a primary limitation. Most quantum devices naturally have multiple accessible energy levels beyond the lowest two traditionally used to define a qubit. Qudits offer a larger state space to store and process quantum information, reducing complexity of quantum circuits and improving efficiency of quan…
Quantum computation is traditionally expressed in terms of quantum bits, or qubits. In this work, we instead consider three-level qu$trits$. Past work with qutrits has demonstrated only constant factor improvements, owing to the $\log_2(3)$ binary-to-ternary compression factor. We present a novel technique using qutrits to achieve a logarithmic depth (runtime) decomposition of the Generalized Toffoli gate using no ancilla--a significant improvement over linear depth for the best qubit-only equivalent. Our circuit construction also features a 70x improvement in two-qudit gate count over the qub…
Knot your average qutrit: Measurement-induced entanglement splitting and the cabling dictionary for GHZ and W States
Department of Astronomy, Astrophysics and Space Engineering, Indian Institute of Technology, Indore（印度印多尔理工学院天文、天体物理与空间工程系）；Department of Mathematics, Techno Main Salt Lake, Techno India Group（印度泰科印度集团加尔各答盐湖技术主校数学系）
Classical realism demands that system properties exist independently of whether they are measured, while noncontextuality demands that the results of measurements do not depend on what other measurements are performed in conjunction with them. The Bell-Kochen-Specker theorem states that noncontextual realism cannot reproduce the measurement statistics of a single three-level quantum system (qutrit). Noncontextual realistic models may thus be tested using a single qutrit without relying on the notion of quantum entanglement in contrast to Bell inequality tests. It is challenging to refute such …
Multi-valued quantum systems can store more information than binary ones for a given number of quantum states. For reliable operation of multi-valued quantum systems, error correction is mandated. In this paper, we propose a 5-qutrit quantum error-correcting code and provide its stabilizer formulation. Since 5 qutrits are necessary to correct a single error, our proposed code is optimal in the number of qutrits. We prove that the error model considered in this paper spans the entire $(3 \times 3)$ operator space. Therefore, our proposed code can correct any single error on the codeword. This c…
We present a method for measuring quantum states encoded in the temporal modes of photons. The basis for the multilevel quantum states is defined by the use of modes propagating in a dispersive medium, which is a fiber in this case. The propagation and time-resolved single photon detection allow us to define a positive-operator valued measure (POVM). The POVM depends on the amount of dispersion and the characteristics of a detector. This framework is numerically tested by performing quantum state tomography on a large number of states for a set of realistic experimental settings. Finally, the …
Device-independent certification of tripartite quantum networks with bilocal Bell inequalities
Center for Quantum-Enabled Computing, Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论与量子计算中心）；Institute of Theoretical Physics, University of Warsaw（华沙大学理论物理研究所）
We present a comprehensive framework for quantum state tomography (QST) of time-bin qudits sent through a fiber. Starting from basic assumptions, we define a positive-operator valued measure (POVM) which is then applied to the quantum state reconstruction problem. A realistic scenario is considered where the time uncertainty of the detector is treated as a source of experimental noise. The performance of the quantum tomography framework is examined through a series of numerical simulations conducted for different parameters describing the apparatus. The quality of state recovery, quantified by…
We determine the cost of performing Shor's algorithm for integer factorization on a ternary quantum computer, using two natural models of universal fault-tolerant computing: (i) a model based on magic state distillation that assumes the availability of the ternary Clifford gates, projective measurements, classical control as its natural instrumentation set; (ii) a model based on a metaplectic topological quantum computer (MTQC). A natural choice to implement Shor's algorithm on a ternary quantum computer is to translate the entire arithmetic into a ternary form. However, it is also possible to…
In recent years, quantum computers and algorithms have made significant progress indicating the prospective importance of quantum computing (QC). Especially combinatorial optimization has gained a lot of attention as an application field for near-term quantum computers, both by using gate-based QC via the Quantum Approximate Optimization Algorithm and by quantum annealing using the Ising model. However, demonstrating an advantage over classical methods in real-world applications remains an active area of research. In this work, we investigate the robustness verification of ReLU networks, which…
Sparse and weak-measurement certification of graph-edge entanglement in PXP scar wavepackets
School of Physics and Electronic Engineering, Shanxi University（山西大学物理与电子工程学院）；Collaborative Innovation Center of Extreme Optics, Shanxi University（山西大学极端光学协同创新中心）；Department of Physics, University of Science and Technology of China（中国科学技术大学物理系）；School of Instrument Science and Opto-Electronics Engineering, Beijing Information Science and Technology University（北京信息科技大学仪器科学与光电工程学院）；School of Physics, Beihang University（北京航空航天大学物理学院）；School of Applied Science, Beijing Information Science and Technology University（北京信息科技大学应用科学学院）；School of Automation (School of Artificial Intelligence), Beijing Information Science and Technology University（北京信息科技大学自动化学院（人工智能学院））；School of Space and Earth Sciences, Beihang University（北京航空航天大学空间与地球科学学院）
Quantum error-correcting code for higher dimensional systems can, in general, be directly constructed from the codes for qubit systems. What remains unknown is whether there exist efficient code design techniques for higher dimensional systems. In this paper, we propose a 7-qutrit error-correcting code for the ternary quantum system and show that this design formulation has no equivalence in qubit systems. This code is optimum in the number of qutrits required to correct a single error while maintaining the CSS structure. This degenerate CSS code can (i) correct up to seven simultaneous phase …
On-chip semi-device-independent quantum random number generator exploiting contextuality
Technical University of Denmark（丹麦技术大学）；Queen Mary University of London（伦敦大学玛丽女王学院）；IHP – Leibniz-Institut für innovative Mikroelektronik（IHP – 莱布尼茨创新微电子研究所）；Technische Universität Berlin（柏林工业大学）；University of Florence（佛罗伦萨大学）；QTI s.r.l.（QTI有限公司）
Quantum optical microcombs in integrated ring resonators generate entangled photon pairs over many spectral modes, and allow the preparation of high dimensional qudit states. Ideally, those sources should be programmable and have a high generation rate, with comb lines tightly spaced for the implementation of efficient qudit gates based on electro-optic frequency mixing. While these requirements cannot all be satisfied by a single resonator device, for which there is a trade-off between high generation rate and tight bin spacing, a promising strategy is the use of multiple resonators, each gen…
Quantum frequency combs from chip-scale integrated sources are promising candidates for scalable and robust quantum information processing (QIP). However, to use these quantum combs for frequency domain QIP, demonstration of entanglement in the frequency basis, showing that the entangled photons are in a coherent superposition of multiple frequency bins, is required. We present a verification of qubit and qutrit frequency-bin entanglement using an on-chip quantum frequency comb with 40 mode pairs, through a two-photon interference measurement that is based on electro-optic phase modulation. Ou…
High-dimensional entanglement offers promising perspectives in quantum information science. In practice, however, the main challenge is to devise efficient methods to characterize high-dimensional entanglement, based on the available experimental data which is usually rather limited. Here we report the characterization and certification of high-dimensional entanglement in photon pairs, encoded in temporal modes. Building upon recently developed theoretical methods, we certify an entanglement of formation of 2.09(7) ebits in a time-bin implementation, and 4.1(1) ebits in an energy-time implemen…
School of Computation, Information & Technology, Technical Univ. of Munich（慕尼黑工业大学计算、信息与技术学院）；Fraunhofer Institute for Cognitive Systems IKS（弗劳恩霍夫认知系统研究所IKS）
