A. V. Burlakov, L. A. Krivitskiy, S. P. Kulik, G. A. Maslennikov, M. V. Chekhova
We proposed the procedure of measuring the unknown state of the three-level system - the qutrit, which was realized as the arbitrary polarization state of the single-mode biphoton field. This procedure is accomplished for the set of the pure states of qutrits; this set is defined by the properties of SU(2) transformations, that are done by the polarization transformers.
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Three-dimensional entanglement of orbital angular momentum states of an atomic qutrit and a single photon qutrit has been observed. Their full state was reconstructed using quantum state tomography. The fidelity to the maximally entangled state of Schmidt rank 3 exceeds the threshold 2/3. This result confirms that the density matrix cannot be decomposed into ensemble of pure states of Schmidt rank 1 or 2. That is, the Schmidt number of the density matrix must be equal to or greater than 3.
We produce and holographically measure entangled qudits encoded in transverse spatial modes of single photons. With the novel use of a quantum state tomography method that only requires two-state superpositions, we achieve the most complete characterisation of entangled qutrits to date. Ideally, entangled qutrits provide better security than qubits in quantum bit-commitment: we model the sensitivity of this to mixture and show experimentally and theoretically that qutrits with even a small amount of decoherence cannot offer increased security over qubits.
We present numerical evidence showing that any three-dimensional subspace of C^3 \otimes C^n has an orthonormal basis which can be reliably distinguished using one-way LOCC, where a measurement is made first on the 3-dimensional part and the result used to select an optimal measurement on the n-dimensional part. This conjecture has implications for the LOCC-assisted capacity of certain quantum channels, where coordinated measurements are made on the system and environment. By measuring first in the environment, the conjecture would imply that the environment-assisted classical capacity of any …
Entangled states in high dimensional systems are of great interest due to the extended possibilities they provide in quantum information processing. Recently, Sun et al. [Phys. Rev. A 82, 052323 (2010)] and Kim et al. [Nat. Phys. 8, 117 (2012)] pointed out that weak measurement and quantum weak measurement reversal can actively combat decoherence. We generalize their studies from qubits to qutrits under amplitude damping decoherence. We find that the qutrit-qutrit entanglement can be partially retrieved for certain initial states when only weak measurement reversals are performed. However, we …
An experimental demonstration of two-qutrit state tomography via one-qutrit symmetric informationally complete positive operator-valued measure (SIC-POVM) is presented. A two-qutrit state is encoded in the transversal paths of a spontaneous parametric down-converted photon pair. A spatial light modulator allows to implement the necessary 81 POVM elements to reconstruct the state. The quality of the reconstruction is evaluated by comparing independent measurements with the predicted results calculated with the reconstructed state. Entanglement in the system is calculated via negativity and gene…
A protocol of measurements providing direct, complete and single-valued recover of all a-priori unknown parameters of biphoton polarization qutrits
该研究提出一种测量协议，利用测量结果概率构造显式表达式，以完整恢复双光子偏振 qutrit 的所有先验未知参数。
Measurement Device Independent Quantum Private Query (MDI QPQ) with qutrits is presented. We compare the database security and client's privacy in MDI QPQ for qubits with qutrits. For some instances, we observe that qutrit will provide better security for database than qubit. However, when it comes to the question of client's privacy we have to take additional measures in case of qutrit. Hence we conclude that though in case of Quantum Key Distribution (QKD) higher dimension provides better security but in case of QPQ this is not obvious.
We investigate the quantum correlation in a qutrit-qubit mixed spin chain based on measurement-induced disturbance (MID) [S. Luo, Phys. Rev. A, 77, (2008) 022301]. We also compare MID and thermal entanglement measured by negativity and illustrate their different characteristics.
In this paper, by exploiting the weak measurement and quantum measurement re- versal (WMQMR) procedure, we propose a scheme to show how one can protect the geometric quantum discord (GQD) of a two-qutrit V-type atomic system each of which interacts with a dissipative reservoir independently. We examine the scheme for the GQD of the initial two-qutrit Werner and Horodecki states for different classes of weak mea- surement strengthes. It is found out that the presented protocol enables us to suppress decoherence due to the amplitude damping (AD) channel and preserve the quantum dis- cord of the …
For a general-form polarization biphoton qutrit, physically corresponding to a pair of arbitrarily polarized photons in a single frequency and wavevector mode, we explicitly find polarization Schmidt modes. A simple method is suggested for factorizing the state vector and the explicit expressions for the factorizing photon creation operators are found. The degrees of entanglement and polarization of a qutrit are shown to depend directly on the commutation features of the factorizing operators. Clear graphic representations for the Stokes vectors of the qutrit state as a whole, its Schmidt mode…
The possibility of recovering and protecting the entanglement of accelerated 2-qubit and 2-qutrit systems is discussed using weak-reverse measurements. The accelerated partial entangled states are more responsive to be protected than the accelerated maximum entangled states. The accelerated coded local information in qutrit system is more robust than that encoded in a 2-qubit system, and it can be conserved even for larger values of accelerations. Meanwhile, the non-accelerated information in qubit system is not affected by the local operation compared with that depicted on qutrit system. For …
Mermin inequalities are derived for systems of three-state particles (qutrits) employing three local measurement settings. These establish perfect correlations which violate local realistic bounds more strongly than those previously reported with two bases. The quantum eigenvalue of the Mermin operator grows as the dimension of the Hilbert space, $3^N$, rather than $2^N$, as obtained with two measurement bases. The number of distinct GHZ contradictions also increases as $3^N$.
We realise a quantum three-level system with photons distributed among three different spatial and polarization modes. Ambiguous measurement of the state of the qutrit are realised by blocking one out for the three modes at any one time. Using these measurements we construct a test of a Leggett-Garg inequality as well as tests of no-signalling-in-time for the measurements. We observe violations of the Leggett-Garg inequality that can not be accounted for in terms of signalling. Moreover, we tailor the qutrit dynamics such that both ambiguous and unambiguous measurements are simultaneously non-…
Qutrit state discrimination with mid-circuit measurements
本文提出利用量子比特二值判别器实现基于线路中测量的量子三态判别，性能与IQ判别器相当，并可用于泄漏随机基准测试等表征实验。
Numerous quantum information protocols make use of maximally entangled two-particle states, or Bell states, in which information is stored in the correlations between the two particles rather than their individual properties. Retrieving information stored in this way means distinguishing between different Bell states, yet the well known no-go theorem establishes that projective linear evolution and local measurement (LELM) detection schemes can only reliably distinguish three of the four qubit Bell states. We establish maximum distinguishability of the qutrit Bell states of bosons via projecti…
Quantum Teleportation of a Single Qutrit using Two-Qutrit Entangled States
本研究通过最小修改Bennett隐形传态协议，利用SU(3)群表示论的两-qutrit纠缠态实现单个qutrit的量子隐形传态，证实高维系统需非幺正测量算符。
Nonviolation of the CHSH inequality under local spin-1 measurements on two spin qutrits
本文基于解析表达式和大量数值验证，发现任意两自旋三态量子比特态在局域自旋-1测量下均不违反CHSH不等式，并比较了并发度与CHSH参数，揭示了与自旋-1/2情形的差异。
Bell state measurements, in which two quantum bits are projected onto a maximally entangled state, are an essential component of quantum information science. We propose and experimentally demonstrate the projection of two quantum systems with three states (qutrits) onto a generalized maximally entangled state. Each qutrit is represented by the polarization of a pair of indistinguishable photons - a biphoton. The projection is a joint measurement on both biphotons using standard linear optics elements. This demonstration enables the realization of quantum information protocols with qutrits, suc…
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…
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…
The power of qutrits for non-adaptive measurement-based quantum computing
本研究将非自适应测量型量子计算（NMQC）从量子比特扩展到qutrit，证明量子关联可用最多$3^n-1$个qutrit计算所有三元函数，并给出高效示例及反例，展示了qutrit在NMQC中的优势与局限。
Generalized quantum measurements (also known as POVMs) are of great importance in quantum information and quantum foundations, but often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting POVMs, the SIC-POVM, is the most compact, set of measurements that can be used to fully describe a quantum state. We use our technique to carry out the first experimental characterization of the state of a qutrit using SIC-POVMs. Because of the highly symmetric nature of this measurement, …
It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose is symmetric informationally complete positive operator-valued measurements (SIC-POVMs). SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliar…
Two-qutrit Werner state is always local
本文证明双qutrit Werner态在每站点两个投影测量、三个结果的场景下对所有参数值均具有局域隐变量模型，通过推导充分条件并验证其满足CGLMP不等式。
CHSH inequality always holds in bipartite qutrits with spin-1 observables
该研究证明在双粒子三能级系统中，所有态在自旋-1测量下均满足CHSH不等式，解决了关于非纠缠态不违反该不等式的猜想。
Bell-Clauser-Horne-Shimony-Holt inequality (in terms of correlation functions) of two qutrits is studied in detail by employing tritter measurements. A uniform formula for the maximum value of this inequality for tritter measurements is obtained. Based on this formula, we show that non-maximally entangled states violate the Bell-CHSH inequality more strongly than the maximally entangled one. This result is consistent with what was obtained by Ac{í}n {\it et al} [Phys. Rev. A {\bf 65}, 052325 (2002)] using the Bell-Clauser-Horne inequality (in terms of probabilities).
Teleporting an unknown qutrit state via a 2-qudit entangled channel
本文提出了一种利用2-量子态纠缠态作为量子信道来传输单个三能级态的量子teleportation方案，通过系统性地构造测量基，设计集体单元变换，实现高维纠缠通道下的qutrit teleportation。
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…
Contextuality is a foundational phenomenon underlying key differences between quantum theory and classical realistic descriptions of the world. Here we propose an experimental test which is capable of revealing contextuality in all qutrit systems, except the completely mixed state, provided we choose the measurement basis appropriately. The 3-level system is furnished by the polarization and spatial degrees of freedom of a single photon, which encompass three orthogonal modes. Projective measurements along rays in the 3-dimensional Hilbert space are made by linear optical elements and detector…
We propose a protocol that allows both the creation and distribution of entanglement, resulting in two distant parties (Alice and Bob) conclusively sharing a bipartite Bell State. The system considered is a graph of three-level objects ("qutrits") coupled by SU(3) exchange operators. The protocol begins with a third party (Charlie) encoding two lattice sites in unentangled states, and allowing unitary evolution under time. Alice and Bob perform a projective measurement on their respective qutrits at a given time, and obtain a maximally-entangled Bell state with a certain probablility. We also …
An ability to describe quantum states directly by average values of measurement outcomes is provided by the Bloch vector. For an informationally complete set of measurements one can construct unique Bloch vector for any quantum state. However, not every Bloch vector corresponds to a quantum state. It seems that only for two dimensional quantum systems it is easy to distinguish proper Bloch vectors from improper ones, i.e. the ones corresponding to quantum states from the other ones. I propose an alternative approach to the problem in which more than one vector is used. In particular, I show th…
Based on mutually unbiased measurements, an optimal tomographic scheme for the multiqutrit states is presented explicitly. Because the reconstruction process of states based on mutually unbiased states is free of information waste, we refer to our scheme as the optimal scheme. By optimal we mean that the number of the required conditional operations reaches the minimum in this tomographic scheme for the states of qutrit systems. Special attention will be paid to how those different mutually unbiased measurements are realized; that is, how to decompose each transformation that connects each mut…
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 …
Qutrits (i.e., three-level quantum systems) can be used to achieve many quantum information and communication tasks due to their large Hilbert spaces. In this work, we propose a scheme to transfer an unknown quantum state between two flux qutrits coupled to two superconducting coplanar waveguide resonators. The quantum state transfer can be deterministically achieved without measurements. Because resonator photons are virtually excited during the operation time, the decoherences caused by the resonator decay and the unwanted inter-resonator crosstalk are greatly suppressed. Moreover, our appro…
Knot your average qutrit: Measurement-induced entanglement splitting and the cabling dictionary for GHZ and W States
该研究将Aravind的纠缠拓扑对应关系扩展到三量子位，发现三量子位纠缠分裂的分类依据是索引集合结构而非传统的GHZ/W类，推导了两类三量子位态在两种基下的纠缠分裂结果。
We present a way to create entangled states of two superconducting transmon qutrits based on circuit QED. Here, a qutrit refers to a three-level quantum system. Since only resonant interaction is employed, the entanglement creation can be completed within a short time. The degree of entanglement for the prepared entangled state can be controlled by varying the weight factors of the initial state of one qutrit, which allows the prepared entangled state to change from a partially entangled state to a maximally entangled state. Because a single cavity is used, only resonant interaction is employe…
We propose local strategies to protect global quantum information. The protocols, which are quantum error correcting codes for dissipative systems, are based on environment measurements, direct feedback control and simple encoding of the logical qubits into physical qutrits whose decaying transitions are indistinguishable and equally probable. The simple addition of one extra level in the description of the subsystems allows for local actions to fully and deterministically protect global resources, such as entanglement. We present codes for both quantum jump and quantum state diffusion measure…
Realisation of a Protected Cat-Qutrit Manifold via Engineered Quantum Tunnelling
本文通过工程三光子KPO实现保护的玻色子三态，展示了量子相干性和猫态形成，揭示了三光子KPO的基本量子特性。
Information Geometry of Four-Parameter Single-Qutrit States: From Quantum to Semiclassical Geometric Tensors
本研究针对四参数单量子比特态，推导了经典与量子Fisher信息矩阵及半经典几何张量的闭式表达式，揭示了测量在投影极限下恢复量子几何信息，并量化了测量导致的几何信息损失。
In this article, we introduce a framework for quantum state tomography of qutrits by projective measurements. The framework is based on photon-counting with measurement results distorted due to the Poisson noise and dark counts. Two different measurement schemes are investigated numerically and compared in terms of their efficiency for distinct numbers of photons per measurement. The accuracy of state reconstruction is quantified by figures of merit which are presented on graphs versus the amount of noise.
Compared with a qubit, a qudit (i.e., $d$-level or $d$-state quantum system) provides a larger Hilbert space to store and process information. On the other hand, qudit-based hybrid quantum computing usually requires performing hybrid quantum gates with qudits different in their nature or in their encoding format. In this work, we consider the qutrit case, i.e., the case for a qudit with $d$=3. We propose a simple method to realize a hybrid quantum controlled-SUM gate with one superconducting (SC) qutrit and a cat-state qutrit. This gate plus single-qutrit gates form a universal set of ternary …
State of a $d$-dimensional quantum system can only be inferred by performing an informationally complete measurement with $m\geqslant d^2$ outcomes. However, an experimentally accessible measurement can be informationally incomplete. Here we show that a single informationally incomplete measuring apparatus is still able to provide all the information about the quantum system if applied several times in a row. We derive a necessary and sufficient condition for such a measuring apparatus and give illustrative examples for qubits, qutrits, general $d$-level systems, and composite systems of $n$ q…
We investigate the notion of uncertainty region using the variance based sum uncertainty relation for qubits and qutrits.We compare uncertainty region of the qubit (a 2-level system) with that of the qutrit (3-level system) by considering sum uncertainty relation for two non-commuting Pauli-like observables, acting on the two dimensional qubit Hilbert space. We identify that physically valid uncertainty region of a qubit is smaller than that of a qutrit. This implies that an enhanced precision can be achieved in the measurement of incompatible Pauli-like observables acting on the 2-dimensional…
The Aharon-Vaidman (AV) game exemplifies the advantage of using simple quantum systems to outperform classical strategies. We present an experimental test of this advantage by using a three-state quantum system (qutrit) encoded in a spatial mode of a single photon passing through three slits. The preparation of a particular state is controlled as the photon propagates through the slits by varying the number of open slits and their respective phases. The measurements are achieved by placing detectors in the specific positions in the near and far fields after the slits. This set of tools allowed…
We investigate the practicality of the method proposed by Maciel et al. [Phys. Rev. A. 80, 032325(2009)] for detecting the entanglement of two spatial qutrits (3-dimensional quantum systems), which are encoded in the discrete transverse momentum of single photons transmitted through a multi-slit aperture. The method is based on the acquisition of partial information of the quantum state through projective measurements, and a data processing analysis done with semi-definite programs. This analysis relies on generating gradually an optimal entanglement witness operator, and numerical investigati…
Making use of coherence and entanglement as metrological quantum resources allows to improve the measurement precision from the shot-noise- or quantum limit to the Heisenberg limit. Quantum metrology then relies on the availability of quantum engineered systems that involve controllable quantum degrees of freedom which are sensitive to the measured quantity. Sensors operating in the qubit mode and exploiting their coherence in a phase-sensitive measurement have been shown to approach the Heisenberg scaling in precision. Here, we show that this result can be further improved by operating the qu…
We propose a probabilistic quantum protocol to realize a nonlinear transformation of qutrit states, which by iterative applications on ensembles can be used to distinguish two types of pure states. The protocol involves single-qutrit and two-qutrit unitary operations as well as post-selection according to the results obtained in intermediate measurements. We utilize the nonlinear transformation in an algorithm to identify a quantum state provided it belongs to an arbitrary known finite set. The algorithm is based on dividing the known set of states into two appropriately designed subsets which…
We classify the Markov chains that can be generated on the set of quantum states by a unitarily evolving 3-dim quantum system (qutrit) that is repeatedly measured with a projective measurement (PVM) consisting of one rank-2 projection and one rank-1 projection. The dynamics of such a system can be visualized as taking place on the union of a Bloch ball and a single point, which correspond to the respective projections. The classification is given in terms of the eigenvalues of the 2x2 matrix that describes the dynamics arising on the Bloch ball, i.e., on the 2-dim subsystem. We also express th…
We study the measurement-induced geometric discord based on the trace norm and generalize some properties known for qutrits to qudits. Previous preliminary results for bipartite qutrit systems (i.e. d=3 systems) are here strictly proved for arbitrary d. Present study supports observations, coming also from other approaches, that systems with $d \geq 3$ show similar behaviour when quantum correlations are concerned, but there is pronounced difference between d=2 and d=3. Qubit systems are exceptionally simple. Underlying geometry of state spaces and related Lie groups are responsible for that.
Uncountably many inequivalent maximally entangled measurements for two qutrits
研究两个三量子比特最大纠缠测量，给出单参数族测量基，证明其在局域酉操作下不等价，由三维对称信息完备态集构建，还构造了野生误差基首个例子，并讨论了不同测量在量子信息场景下的性能差异。
Enhanced high-dimensional teleportation in correlated amplitude damping noise by weak measurement and environment-assisted measurement
本文提出利用弱测量和环境辅助测量两种策略，增强相关振幅阻尼噪声下的qutrit隐形传态保真度，并发现噪声相关性可提高成功概率，EAM方案在保真度上优于WM方案。
Protecting three-dimensional entanglement from correlated amplitude damping channel
针对关联振幅阻尼噪声下的三维 qutrit-qutrit 纠缠保护问题，论文比较 WM+QMR 与 EAM+QMR 两种策略，证明后者更能有效保护纠缠并提高成功概率。
Enhancing the quantum correlations in realistic quantum systems interacting with the environment of finite temperature is an important subject in quantum information processing. In this paper, we use weak measurement and measurement reversal to enhance the quantum correlations in a quantum system consisting of two particles. The transitions of the quantum correlations measured by the local quantum uncertainty of qubit-qubit and qutrit-qutrit quantum systems under generalized amplitude damping channels are investigated. We show that, after the weak measurement and measurement reversal, the join…
Spin Squeezing Enhanced Quantum Magnetometry with Nitrogen-Vacancy Center Qutrits
本文研究利用标准 Ramsey 协议和 Lindblad 方程，探讨三能级 NV 中心在退相干下的自旋压缩增强量子磁力测量，发现合理调度压缩与自由演化时间可在无动力学解耦时维持精度优势。
We present a simultaneous-cooling protocol for a double-resonator system via projective measurements on an ancillary $V$-type qutrit. Through repeated measurements on the ground state of the ancillary system, the two resonators can be cooled down to their respective ground states from thermal states. With respect to the measurement-based cooling, an optimized measurement-interval $τ_{\rm opt}$ is analytically obtained for the first time, which is inversely proportional to the collective thermal Rabi frequency $Ω_{\rm th}$ as a function of the resonators' average population of the last round. U…
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 …
We propose a way to prepare Greenberger-Horne-Zeilinger (GHZ) entangled photon Fock states of three cavities, by using a superconducting flux qutrit coupled to the cavities. This proposal does not require the use of classical microwave pulses and measurement during the entire operation. Thus, the operation is greatly simplified and the circuit engineering complexity and cost is much reduced. The proposal is quite general and can be applied to generate three-cavity GHZ entangled photon Fock states when the three cavities are coupled by a different three-level physical system such as a supercond…
Provided that cavities are initially in a Greenberger-Horne-Zeilinger (GHZ) entangled state, we show that GHZ states of N-group qubits distributed in N cavities can be created via a 3-step operation. The GHZ states of the N-group qubits are generated by using N-group qutrits placed in the N cavities. Here, "qutrit" refers to a three-level quantum system with the two lowest levels representing a qubit while the third level acting as an intermediate state necessary for the GHZ state creation. This proposal does not depend on the architecture of the cavity-based quantum network and the way for co…
Engineered dynamical maps combining coherent and dissipative transformations of quantum states with quantum measurements, have demonstrated a number of technological applications, and promise to be a crucial tool in quantum thermodynamic processes. Here, we exploit the control on the effective open spin qutrit dynamics of an NV center, to experimentally realize an autonomous feedback process (Maxwell demon) with tunable dissipative strength. The feedback is enabled by random measurement events that condition the subsequent dissipative evolution of the qutrit. The efficacy of the autonomous Max…
We present an efficient method to generate a Greenberger-Horne-Zeilinger (GHZ) entangled state of three cat-state qubits (cqubits) via circuit QED. The GHZ state is prepared with three microwave cavities coupled to a superconducting transmon qutrit. Because the qutrit remains in the ground state during the operation, decoherence caused by the energy relaxation and dephasing of the qutrit is greatly suppressed. The GHZ state is created deterministically because no measurement is involved. Numerical simulations show that high-fidelity generation of a three-cqubit GHZ state is feasible with prese…
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