Comparative analysis of various protocols for high-precision tomography of qudits
Yu. I. Bogdanov, B. I. Bantysh, N. A. Bogdanova, K. B. Koksharov, V. F. Lukichev
备注：
8 pages, 6 figures, The 14th International Conference 'Micro- and Nanoelectronics - 2021' (ICMNE-2021) with the Extended Session 'Quantum Informatics', Zvenigorod, October 4-8, 2021
Quantum tomography is an important tool for obtaining information about the quantum state from experimental data. In this study, we conduct a comparative analysis of various quantum tomography protocols, including protocols based on highly symmetric sets of state vectors, on mutually unbiased bases (MUB), and two-level transformations. Using the universal distribution for the fidelity of quantum state tomography, we estimate the fidelity and efficiency of quantum measurements for qudits of various dimensions as applied to a set of random states uniformly distributed over the Haar measure.
Quantum state tomography (QST) is an essential tool for characterizing an unknown quantum state. Recently, QST has been performed for entangled qudits based on orbital angular momentum, time-energy uncertainty, and frequency bins. Here, we propose a QST for time-bin qudits, with which the number of measurement settings scales linearly with dimension $d$. Using the proposed scheme, we performed QST for a four-dimensional time-bin maximally entangled state with 16 measurement settings. We successfully reconstructed the density matrix of the entangled qudits, with which the average fidelity of th…
We consider the task of performing quantum state tomography on a $d$-level spin qudit, using only measurements of spin projection onto different quantization axes. After introducing a basis of operators closely related to the spherical harmonics, which obey the rotational symmetries of spin qudits, we map our quantum tomography task onto the classical problem of signal recovery on the sphere. We then provide algorithms with $O(rd^3)$ serial runtime, parallelizable down to $O(rd^2)$, for (i) computing a priori upper bounds on the expected error with which spin projection measurements along $r$ …
Quantum tomography makes it possible to obtain comprehensive information about certain logical elements of a quantum computer. In this regard, it is a promising tool for debugging quantum computers. The practical application of tomography, however, is still limited by systematic measurement errors. Their main source are errors in the quantum state preparation and measurement procedures. In this work, we investigate the possibility of suppressing these errors in the case of ion-based qudits. First, we will show that one can construct a quantum measurement protocol that contains no more than a s…
Manipulation of qudits in optical tables is a difficult and nonscalable task. The use of integrated optical circuits opens new possibilities for the generation, manipulation, and characterization of high dimensional states besides the ease of transmission of these states through an optical fiber. In this work we propose photonic circuits to perform minimum quantum state tomography of path qudits and show how to determine all the constituents parameters of these circuits (beam splitters and phase shifters). Our strategies were based on the symmetries of the involved POVMs (positive operator-val…
Recently quantum tomography has been proposed as a fundamental tool for prototyping a few qubit quantum device. It allows the complete reconstruction of the state produced from a given input into the device. From this reconstructed density matrix, relevant quantum information quantities such as the degree of entanglement and entropy can be calculated. Generally orthogonal measurements have been discussed for this tomographic reconstruction. In this paper, we extend the tomographic reconstruction technique to two new regimes. First we show how non-orthogonal measurement allow the reconstruction…
In this paper, we discuss the minimal number of observables, where expectation values at some time instant determine the trajectory of a d-level quantum system ("qudit") governed by the Gaussian semigroup. We assume that the macroscopic information about the system in question is given by the mean values of n selfadjoint operators $Q_1,...,Q_n$ at some time instants $t_1<t_2<...<t_r$, where $n<d^2-1$ and $r\leq {\rm deg} μ(λ,\bBBL)$. Here $μ(λ,\bBBL)$ stands for the minimal polynomial of the generator of the Gaussian flow.
We present the experimental quantum tomography of 7- and 8-dimensional quantum systems based on projective measurements in the mutually unbiased basis (MUB-QT). One of the advantages of MUB-QT is that it requires projections from a minimal number of bases to be performed. In our scheme, the higher dimensional quantum systems are encoded using the propagation modes of single photons, and we take advantage of the capabilities of amplitude- and phase-modulation of programmable spatial light modulators to implement the MUB-QT.
We propose a method to perform the quantum state tomography (QST) of an $\mathit{n}$-partite qudit state embedded in single photons using the Hong-Ou-Mandel (HOM) interference between the target state and probe state. This method requires only passive beam splitters for the HOM interference and removes all active optical devices in the target modes to control the measurement bases needed in conventional QST. Hence, it is applicable to various degree of freedom of the target state without altering the measurement setup. Moreover, a faithful estimation is realized even with classical probe light…
In this work, we propose three-dimensional photonic circuit designs that guarantee a considerable reduction in the complexity of circuits for the purpose of performing quantum state tomography of N-dimensional path qudits. The POVM (Positive Operator-Valued Measure) chosen in this work ensures that, for odd dimensions, such process is minimal. Our proposal consists of organizing the waveguides that form the circuit as a square array formed by N vertical sectors composed of N waveguides each, arranged in the vertical direction. Based on the symmetry of the chosen POVM, the interferometer acting…
We show that a generic $N$-qudit pure quantum state is uniquely determined by only $2$ of its $\lceil\frac{N+1}{2}\rceil$-particle reduced density matrices. Therefore we give a method to uniquely determine a generic $N$-qudit pure state of dimension $D=d^N$ with $O(D)$ local measurements, which is an improvement comparing with the previous known approach using $O(D\log^2 D)$ or $O(D\log D)$ measurements.
Neutral atom systems are an appealing platform for the development and testing of quantum control and measurement techniques. This dissertation presents experimental investigations of control and measurement tools using as a testbed the 16-dimensional hyperfine manifold associated with the electronic ground state of $^{133}$Cs atoms. On the control side, we present an experimental realization of a protocol to implement robust unitary transformations in the presence of static and dynamic perturbations. We also present an experimental realization of inhomogeneous quantum control. Specifically, w…
Multi-qudit systems are studied in tomographic probability representations of quantum qudit states. Results of calculations for Bell-type numbers within the framework of classical probability theory and in quantum tomography are compared. Violation of Bell-type inequalities are shown explicitly using the method of averaging in tomographic picture of quantum states.
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…
The techniques of low-rank matrix recovery were adapted for Quantum State Tomography (QST) previously by D. Gross et al. [Phys. Rev. Lett. 105, 150401 (2010)], where they consider the tomography of $n$ spin-$1/2$ systems. For the density matrix of dimension $d = 2^n$ and rank $r$ with $r \ll 2^n$, it was shown that randomly chosen Pauli measurements of the order $O(dr \log(d)^2)$ are enough to fully reconstruct the density matrix by running a specific convex optimization algorithm. The result utilized the low operator-norm of the Pauli operator basis, which makes it `incoherent' to low-rank ma…
It has previously been shown that quantum state tomography can be used to increase the key rate of quantum key distribution (QKD) for the special case of qubits (i.e.,d=2). Here, we consider the case of higher dimensions, i.e., qudits, and show that, for a prime number d>2, quantum state tomography can also improve the key rate of d-dimensional qudit-based QKD schemes, compared to the standard (d+1)-basis protocol. We apply our results to previous QKD experiments based on OAM (orbital angular momentum) encoding and demonstrate the advantage of tomography-based QKD protocols. Moreover, we compa…
Photons carrying orbital angular momentum (OAM) are excellent qudits and are widely used in several applications, such as long distance quantum communication, $d$-dimensional teleportation and high-resolution imaging and metrology. All these protocols rely on quantum tomography to characterise the OAM state, which currently requires complex measurements involving spatial light modulators and mode filters. To simplify the measurement and characterisation of OAM states, here we apply a recent tomography protocol [Asadian et al., \pra {\bf 94}, 010301 (2016)]. Our scheme for OAM tomography in $d$…
Using the the convex semidefinite programming method and superoperator formalism we obtain the finite quantum tomography of some mixed quantum states such as: qudit tomography, N-qubit tomography, phase tomography and coherent spin state tomography, where that obtained results are in agreement with those of References \cite{schack,Pegg,Barnett,Buzek,Weigert}.
The exponential growth in Hilbert space with increasing size of a quantum system means that accurately characterising the system becomes significantly harder with system dimension d. We show that self-guided tomography is a practical, efficient, and robust technique of measuring higher-dimensional quantum states. The achieved fidelities are over 99.9% for qutrits (d=3) and ququints (d=5), and 99.1% for quvigints (d=20), the highest values ever realised for qudits. We demonstrate robustness against experimental sources of noise, both statistical and environmental. The technique is applicable to…
Spatial qudit states can be realized by using multi-slits to discretize the transverse momentum of a photon. The merit of this kind of spatial qudit states is that the implementation of higher dimensional qudits is relatively easy. As we have recently shown, the quantum states of these spatial qudits can be analyzed by scanning a single interference pattern. This method of single scan tomography can also be applied at higher dimensions, but the reconstruction becomes more sensitive to smaller details of the scanned patterns as the dimensions increase. In this paper, we investigate the effect o…
Efficient Experimental Qudit State Estimation via Point Tomography
Universidad de Concepción（康塞普西翁大学）；Millennium Institute for Research in Optics（光学研究千年研究所）；University of Vienna（维也纳大学）；Vienna Center for Quantum Science and Technology (VCQ)（维也纳量子科学与技术中心）；Christian Doppler Laboratory for Photonic Quantum Computer（光子量子计算克里斯蒂安·多普勒实验室）；ICFO - Institut de Ciencies Fotoniques（ICFO-光子科学研究所）；The Barcelona Institute of Science and Technology（巴塞罗那科学技术研究院）；Universidad Católica de la Santísima Concepción（神圣康塞普西翁天主教大学）；Fundação Getulio Vargas（热图利奥·瓦加斯基金会）
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