Journal articles: 'TWO-QUDIT SYSTEM'

1

Mansour, M., M. Daoud, and L. Bouhouch. "Absolutely maximally entangled states from phase states." International Journal of Quantum Information 17, no. 01 (February 2019): 1950009. http://dx.doi.org/10.1142/s0219749919500096.

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We derive absolutely maximally entangled (AME) states from phase states for a multi-qudit system whose dynamics is governed by a two-qudit interaction Hamiltonian of Heisenberg type. AME states are characterized by being maximally entangled for all bipartitions of the multi-qudit system and present absolute multipartite entanglement. The key ingredient of this approach is the theory of phase states for finite-dimensional systems (qudits). We define further the unitary phase operators of [Formula: see text]-qudit systems and we give next the corresponding separable phase states. Using a qudit–qudit Hamiltonian acting as entangling operator on separable phase states, we generate entangled phase states. Finally, from the labeled entangled phase states, we derive the absolutely maximally entangled states.

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2

Man’ko, V. I., and L. A. Markovich. "Deformed Entropic and Information Inequalities forX-States of Two-Qubit and Single Qudit States." Advances in Mathematical Physics 2015 (2015): 1–4. http://dx.doi.org/10.1155/2015/717621.

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Theq-deformed entropies of quantum and classical systems are discussed. Standard andq-deformed entropic inequalities forX-states of the two-qubit system and the state of single qudit withj=3/2are presented.

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3

Man'ko, V. I., and L. A. Markovich. "New Minkowski type inequalities and entropic inequalities for quantum states of qudits." International Journal of Quantum Information 12, no. 07n08 (November 2014): 1560021. http://dx.doi.org/10.1142/s0219749915600217.

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The two-parameter Minkowski like inequality written for composite quantum system state is obtained for arbitrary Hermitian non-negative matrix with trace equal to unity. The inequality can be used as entropic and information inequality for density matrix of noncomposite finite quantum system, e.g. for a single qudit state. The analogs of strong subadditivity condition for the single qudit is discussed in context of obtained Minkowski like inequality.

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4

Ducuara, Andrés Felipe, Javier Madroñero, and John Henry Reina. "On the Activation of Quantum Nonlocality." Universitas Scientiarum 21, no. 2 (May 16, 2016): 129. http://dx.doi.org/10.11144/javeriana.sc21-2.otao.

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<p>We report on some quantum properties of physical systems, namely, entanglement, nonlocality, k-copy nonlocality (superactivation of nonlocality), hidden nonlocality (activation of nonlocality through local filtering) and the activation of nonlocality through tensoring and local filtering. The aim of this work is two-fold. First, we provide a review of the numerical procedures that must be followed in order to calculate the aforementioned properties, in particular, for any two-qubit system, and reproduce the bounds for two-qudit Werner states. Second, we use such numerical tools to calculate new bounds of these properties for two-qudit Isotropic states and two-qubit Hirsch states.</p>

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5

FUJII, KAZUYUKI, KUNIO FUNAHASHI, and TAKAYUKI KOBAYASHI. "JARLSKOG'S PARAMETRIZATION OF UNITARY MATRICES AND QUDIT THEORY." International Journal of Geometric Methods in Modern Physics 03, no. 02 (March 2006): 269–83. http://dx.doi.org/10.1142/s0219887806001144.

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In the paper (math–ph/0504049) Jarlskog gave an interesting simple parametrization to unitary matrices, which was essentially the canonical coordinate of the second kind in the Lie group theory (math–ph/0505047). In this paper we apply the method to a quantum computation based on multilevel system (qudit theory). Namely, by considering that the parametrization gives a complete set of modules in qudit theory, we construct the generalized Pauli matrices, which play a central role in the theory and also make a comment on the exchange gate of two–qudit systems. Moreover, we give an explicit construction to the generalized Walsh–Hadamard matrix in the case of n = 3, 4, and 5. For the case of n = 5, its calculation is relatively complicated. In general, a calculation to construct it tends to become more and more complicated as n becomes large. To perform a quantum computation the generalized Walsh–Hadamard matrix must be constructed in a quick and clean manner. From our construction it may be possible to say that a qudit theory with n ≥ 5 is not realistic. This paper is an introduction toward Quantum Engineering.

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6

Karakaş, Mikail Doğuş, and Azmi Gençten. "Construction of Two-Ququart Quantum Entanglement by Using Magnetic Resonance Selective Pulse Sequences." Zeitschrift für Naturforschung A 73, no. 10 (October 25, 2018): 911–18. http://dx.doi.org/10.1515/zna-2017-0441.

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AbstractA d-dimensional unit of information in quantum computing is called a qudit. For d = 4 there exist four magnetic quantum numbers of spin-3/2. These four levels can be called ququarts. Then, for the SI (S = 3/2, I = 3/2) spin system, 16 two-ququart states are obtained. In this study, first, two-ququart entangled states are constructed by using matrix representation of Hadamard and CNOT logic gates. Two-ququart entangled states are also constructed by using magnetic resonance selective pulse sequences of Hadamard and CNOT logic gates. Then, a generalised expression is obtained for the transformation of two-qudit entangled states between each other. This expression is applied for two-ququart entangled states.

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7

Mansour, Mostafa, and Mohammed Daoud. "Stabilizer codes and equientangled bases from phase states." International Journal of Modern Physics B 31, no. 20 (August 10, 2017): 1750132. http://dx.doi.org/10.1142/s0217979217501326.

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We develop a comprehensive approach of stabilizer codes and provide a scheme generating equientangled basis interpolating between the product basis and maximally entangled basis. The key ingredient is the theory of phase states for finite-dimensional systems (qudits). In this respect, we derive entangled phase states for a multiqudit system whose dynamics is governed by a two-qudit interaction Hamiltonian. We construct the stabilizer codes for this family of entangled phase states. The stabilizer phase states are defined as the common eigenvectors of the stabilizer group generators which are explicitly specified. Furthermore, we construct equally entangled bases from bipartite as well as multipartite entangled qudit phase states.

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8

TIAN, XIU-LAO, GUO-FANG SHI, and Yong ZHAO. "QUANTUM CHANNELS OF THE QUTRIT STATE TELEPORTATION." International Journal of Quantum Information 09, no. 03 (April 2011): 893–901. http://dx.doi.org/10.1142/s0219749911007502.

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Qudit quantum system can carry more information than that of qubit, the teleportation of qudit state has significance in quantum information task. We propose a method to teleport a general qutrit state (three-level state) and discuss the necessary and sufficient condition for realizing a successful and perfect teleportation, which is determined by the measurement matrix Tα and the quantum channel parameter matrix (CPM) X. By using this method, we study the channels of two-qutrit state and three-qutrit state teleportation.

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9

Achkir, O., M. Daoud, and M. Mansour. "Generalized graph states and mutually unbiased bases from multi-qudits phase states." Modern Physics Letters B 31, no. 17 (June 2017): 1750183. http://dx.doi.org/10.1142/s0217984917501834.

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The description of qudits in a formalism based on a generalized variant of Weyl–Heisenberg algebras is discussed. The unitary phase operators for a multi-qudit system and the corresponding phase states (the eigenstates of the phase operator) are constructed. We discuss the dynamics of multi-qudit phase states governed by a generalized Hamiltonian involving one- and two-body interactions which offer a remarkable connection between phase states, generalized graph states and the mutually unbiased bases. The entangled phase states are shown to possess the following properties simultaneously, namely the mutually unbiasedness of phase states resulting from the one-body generalized oscillator Hamiltonian and the entanglement properties of generalized graph states resulting from the two-body interaction Hamiltonian.

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10

Zhou, Tao, Jing Xin Cui, Xiaohua Wu, and Gui Lu Long. "Multicopy programmable discriminators between two unknown qudit states with group-theoretic approach." Quantum Information and Computation 12, no. 11&12 (November 2012): 1017–33. http://dx.doi.org/10.26421/qic12.11-12-9.

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The discrimination between two unknown states can be performed by a universal programmable discriminator, where the copies of the two possible states are stored in two program systems respectively and the copies of data, which we want to confirm, are provided in the data system. In the present paper, we propose a group-theretic approach to the multi-copy programmable state discrimination problem. By equivalence of unknown pure states to known mixed states and with the representation theory of $U(n)$ group, we construct the Jordan basis to derive the analytical results for both the optimal unambiguous discrimination and minimum-error discrimination. The POVM operators for unambiguous discrimination and orthogonal measurement operators for minimum-error discrimination are obtained. We find that the optimal failure probability and minimum-error probability for the discrimination between the mean input mixd states are dependent on the dimension of the unknown qudit states. We applied the approach to generalize the results of He and Bergou (2007) from qubit to qudit case, and we further solve the problem of programmable dicriminators with arbitrary copies of unknown states in both program and data systems.

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11

Zheng, Chao, Jin Tian, Daili Li, Jingwei Wen, Shijie Wei, and Yansong Li. "Efficient Quantum Simulation of an Anti-P-Pseudo-Hermitian Two-Level System." Entropy 22, no. 8 (July 24, 2020): 812. http://dx.doi.org/10.3390/e22080812.

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Besides Hermitian systems, quantum simulation has become a strong tool to investigate non-Hermitian systems, such as PT-symmetric, anti-PT-symmetric, and pseudo-Hermitian systems. In this work, we theoretically investigate quantum simulation of an anti-P-pseudo-Hermitian two-level system in different dimensional Hilbert spaces. In an arbitrary phase, we find that six dimensions are the minimum to construct the anti-P-pseudo-Hermitian two-level subsystem, and it has a higher success probability than using eight dimensions. We find that the dimensions can be reduced further to four or two when the system is in the anti-PT-symmetric or Hermitian phase, respectively. Both qubit-qudit hybrid and pure-qubit systems are able to realize the simulation, enabling experimental implementations in the near future.

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12

Duan, Junjun, Lin Zhang, Quan Qian, and Shao-Ming Fei. "A Characterization of Maximally Entangled Two-Qubit States." Entropy 24, no. 2 (February 7, 2022): 247. http://dx.doi.org/10.3390/e24020247.

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As already known by Rana’s result, all eigenvalues of any partial-transposed bipartite state fall within the closed interval [−12,1]. In this note, we study a family of bipartite quantum states where the minimal eigenvalues of partial-transposed states are −12. For a two-qubit system, we find that the minimal eigenvalue of its partial-transposed state is −12 if and only if such a two-qubit state is maximally entangled. However this result does not hold in general for a two-qudit system when the dimensions of the underlying space are larger than two.

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13

Jafarizadeh, M., A. Heshmati, and K. Aghayar. "Nonlinear and linear entanglement witnesses for bipartite systems via exact convex optimization." Quantum Information and Computation 10, no. 7&8 (July 2010): 562–79. http://dx.doi.org/10.26421/qic10.7-8-2.

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Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5.

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14

Nurdin, Hendra I., and John E. Gough. "Modular quantum memories using passive linear optics and coherent feedback." Quantum Information and Computation 15, no. 11&12 (September 2015): 1017–40. http://dx.doi.org/10.26421/qic15.11-12-9.

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In this paper, we show that quantum memory for qudit states encoded in a single photon pulsed optical field has a conceptually simple modular realization using only passive linear optics and coherent feedback. We exploit the idea that two decaying optical cavities can be coupled in a coherent feedback configuration to create an internal mode of the coupled system which is isolated and decoherence-free for the purpose of qubit storage. The qubit memory can then be switched between writing/read-out mode and storage mode simply by varying the routing of certain freely propagating optical fields in the network. It is then shown that the qubit memories can be interconnected with one another to form a qudit quantum memory. We explain each of the phase of writing, storage, and read-out for this modular quantum memory scheme. The results point a way towards modular architectures for complex compound quantum memories.

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15

PARASHAR, PREETI. "ON THE NON-EXISTENCE OF A UNIVERSAL HADAMARD GATE." International Journal of Quantum Information 05, no. 06 (December 2007): 845–55. http://dx.doi.org/10.1142/s0219749907003304.

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We establish the non-existence of a universal Hadamard gate for an arbitrary qubit, by considering two different principles; namely, no-superluminal signalling and non-increase of entanglement under LOCC. It is also shown that these principles are not violated if and only if the qubit states belong to the special ensemble obtained recently. We then extend the non-existence of the Hadamard operation to a multi-qubit system. In higher dimensions, the analog of the Hadamard gate is the quantum Fourier transform. We show that it is not possible to design this gate for an arbitrary qudit.

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16

Dally, Malak M., and Mohammad N. Abdulrahim. "On the Unitary Representations of the Braid Group B6." Mathematics 7, no. 11 (November 9, 2019): 1080. http://dx.doi.org/10.3390/math7111080.

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We consider a non-abelian leakage-free qudit system that consists of two qubits each composed of three anyons. For this system, we need to have a non-abelian four dimensional unitary representation of the braid group B 6 to obtain a totally leakage-free braiding. The obtained representation is denoted by ρ . We first prove that ρ is irreducible. Next, we find the points y ∈ C * at which the representation ρ is equivalent to the tensor product of a one dimensional representation χ ( y ) and μ ^ 6 ( ± i ) , an irreducible four dimensional representation of the braid group B 6 . The representation μ ^ 6 ( ± i ) was constructed by E. Formanek to classify the irreducible representations of the braid group B n of low degree. Finally, we prove that the representation χ ( y ) ⊗ μ ^ 6 ( ± i ) is a unitary relative to a hermitian positive definite matrix.

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17

Wang, Xin-Wen, Shi-Qing Tang, Li-Jun Xie, Deng-Yu Zhang, and Le-Man Kuang. "Many-to-one remote information concentration for qudits and multipartite entanglement." Quantum Information and Computation 14, no. 1&2 (January 2014): 122–36. http://dx.doi.org/10.26421/qic14.1-2-7.

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Telecloning and its reverse process, referred to as remote information concentration (RIC), have attracted considerable interest because of their potential applications in quantum-information processing. We here present a general scheme for RIC in $d$-level systems (qudits), in which the quantum information initially distributed in many spatially separated qudits can be remotely and deterministically concentrated to a single qudit via an entangled channel without performing any global operations. We show that the entangled channel of RIC can be different types of entangled states, including mixed states as well as pure ones. More interestingly, these mixed states include a bound entangled state which has a similar form to the generalized Smolin state but has different characteristics from it. We also show that there exists a multipartite entangled state which can be used to implement both telecloning and RIC in the two-level system. Our many-to-one RIC protocol could be slightly modified to perform some types of many-to-many RIC tasks.

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18

Dilley, Daniel, Alvin Gonzales, and Mark Byrd. "Guaranteeing completely positive quantum evolution." Journal of Physics A: Mathematical and Theoretical 54, no. 50 (November 23, 2021): 505302. http://dx.doi.org/10.1088/1751-8121/ac2e28.

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Abstract In open quantum systems, it is known that if the system and environment are in a product state, the evolution of the system is given by a linear completely positive (CP) Hermitian map. CP maps are a subset of general linear Hermitian maps, which also include non completely positive (NCP) maps. NCP maps can arise in evolutions such as non-Markovian evolution, where the CP divisibility of the map (writing the overall evolution as a composition of CP maps) usually fails. Positive but NCP maps are also useful as entanglement witnesses. In this paper, we focus on transforming an initial NCP map to a CP map through composition with the asymmetric depolarizing map. We use separate asymmetric depolarizing maps acting on the individual subsystems. Previous work have looked at structural physical approximation (SPA), which is a CP approximation of an NCP map using a mixture of the NCP map with a completely depolarizing map. We prove that the composition can always be made CP without completely depolarizing in any direction. It is possible to depolarize less in some directions. We give the general proof by using the Choi matrix and an isomorphism from a maximally entangled two qudit state to a set of qubits. We also give measures that describe the amount of disturbance the depolarization introduces to the original map. Given our measures, we show that asymmetric depolarization has many advantages over SPA in preserving the structure of the original NCP map. Finally, we give some examples. For some measures and examples, completely depolarizing (while not necessary) in some directions can give a better approximation than keeping the depolarizing parameters bounded by the required depolarization if symmetric depolarization is used.

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19

Wang, Hai, Asutosh Kumar, and Junde Wu. "Complementarity and nonlocality in two-qudit systems." Physics Letters A 382, no. 10 (March 2018): 685–89. http://dx.doi.org/10.1016/j.physleta.2017.12.053.

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20

IVANCHENKO, E. A. "ENTANGLEMENT DYNAMICS IN FINITE QUDIT CHAIN IN CONSISTENT MAGNETIC FIELD." International Journal of Quantum Information 10, no. 06 (September 2012): 1250068. http://dx.doi.org/10.1142/s0219749912500682.

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Based on the Liouville–von Neumann equation, we obtain a closed system of equations for the description of a qutrit or coupled qutrits in an arbitrary, time-dependent, external magnetic field. The dependence of the dynamics on the initial states and the magnetic field modulation is studied analytically and numerically. We compare the relative entanglement measure's dynamics in bi-qudits with permutation particle symmetry. We find the magnetic field modulation which retains the entanglement in the system of two coupled qutrits. Analytical formulae for the entanglement measures in finite chains from two to six qutrits or three quartits are presented.

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21

Nikolaeva, Anstasiia S., Evgeniy O. Kiktenko, and Aleksey K. Fedorov. "Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover’s Algorithm with Qudits." Entropy 25, no. 2 (February 20, 2023): 387. http://dx.doi.org/10.3390/e25020387.

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Qubits, which are the quantum counterparts of classical bits, are used as basic information units for quantum information processing, whereas underlying physical information carriers, e.g., (artificial) atoms or ions, admit encoding of more complex multilevel states—qudits. Recently, significant attention has been paid to the idea of using qudit encoding as a way for further scaling quantum processors. In this work, we present an efficient decomposition of the generalized Toffoli gate on five-level quantum systems—so-called ququints—that use ququints’ space as the space of two qubits with a joint ancillary state. The basic two-qubit operation we use is a version of the controlled-phase gate. The proposed N-qubit Toffoli gate decomposition has O(N) asymptotic depth and does not use ancillary qubits. We then apply our results for Grover’s algorithm, where we indicate on the sizable advantage of using the qudit-based approach with the proposed decomposition in comparison to the standard qubit case. We expect that our results are applicable for quantum processors based on various physical platforms, such as trapped ions, neutral atoms, protonic systems, superconducting circuits, and others.

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22

GUO, FENZHUO, TAILIN LIU, QIAOYAN WEN, and FUCHEN ZHU. "QUANTUM KEY DISTRIBUTION BASED ON ENTANGLEMENT SWAPPING BETWEEN TWO BELL STATES." International Journal of Quantum Information 04, no. 05 (October 2006): 769–79. http://dx.doi.org/10.1142/s0219749906002249.

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Based on entanglement swapping between two Bell states, two novel quantum key distribution protocols are proposed. One is for two-level systems, where there is no need for classical communication before each entanglement swapping. This feature is essential to its practical realization. Furthermore, to establish an arbitrarily long key, the protocol needs only two Bell states. The other is for d-level (d > 2) systems, in which higher security and higher source capacity are achieved. Using the theory of quadratic residue, we prove that in the two-qudit systems, each Bell state is a uniform superposition of all basis states in the dual basis, which is different to the situation in two-qubit systems. This difference means our two-level protocol cannot be generalized to the d-level situation directly. On the other hand, it results in higher security of our d-level protocol and is instructive to design quantum cryptography protocols.

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23

Nakajima, Y., Y. Kawano, H. Sekigawa, M. Nakanishi, S. Yamashita, and Y. Nakashima. "Synthesis of quantum circuits for $d$-level systems by using cosine-sine." Quantum Information and Computation 9, no. 5&6 (May 2009): 423–43. http://dx.doi.org/10.26421/qic9.5-6-6.

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We study the problem of designing minimal quantum circuits for any operations on $n$ qudits by means of the cosine-sine decomposition. Our method is based on a divide-and-conquer strategy. In that strategy, the size of the produced quantum circuit depends on whether the partitioning is balanced. We provide a new cosine-sine decomposition based on a balanced partitioning for $d$-level systems. The produced circuit is not asymptotically optimal except when $d$ is a power of two, but, when the number of qudits $n$ is small, our method can produce the smallest quantum circuit compared to the circuits produced by other synthesis methods. For example, when $d=3$ (three-level systems) and $n=2$ (two qudits), then the number of two-qudit operations called CINC, which is a generalized versions of CNOT, is 36 whereas the previous method needs 156 CINC gates. Moreover, we show that our method is useful for designing a polynomial-size quantum circuit for the radix-$d$ quantum Fourier transform.

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24

LIMA, G., LEONARDO NEVES, IVAN F. SANTOS, C. H. MONKEN, J. G. AGUIRRE GÓMEZ, CARLOS SAAVEDRA, and S. PÁDUA. "GENERATING AND DISTRIBUTING TWO-PHOTON ENTANGLED SPATIAL QUDITS." International Journal of Quantum Information 05, no. 01n02 (February 2007): 69–81. http://dx.doi.org/10.1142/s0219749907002815.

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We report an experiment to generate and propagate two entangled spatial qudits, [Formula: see text]-dimensional quantum systems, using spontaneous parametric down-conversion. The manipulation, via pump beam, of the transverse spatial correlation between the photon pairs is explored. Inserting apertures with [Formula: see text]-slits in the arms of the down-converted photons, we associate the qudit space with the [Formula: see text] possible paths followed by each photon. Experimental results for qudits with [Formula: see text] and 8 are shown. We demonstrate that the generated states cannot be classically correlated. We also show the propagation of entangled states of spatial qudits. Their free-space distribution is performed at the laboratory scale and the propagated states maintain a high fidelity with their original form.

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López-Saldívar, Julio A., Octavio Castaños, Margarita A. Man’ko, and Vladimir I. Man’ko. "A New Mechanism of Open System Evolution and Its Entropy Using Unitary Transformations in Noncomposite Qudit Systems." Entropy 21, no. 8 (July 27, 2019): 736. http://dx.doi.org/10.3390/e21080736.

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The evolution of an open system is usually associated with the interaction of the system with an environment. A new method to study the open-type system evolution of a qubit (two-level atom) state is established. This evolution is determined by a unitary transformation applied to the qutrit (three-level atom) state, which defines the qubit subsystems. This procedure can be used to obtain different qubit quantum channels employing unitary transformations into the qutrit system. In particular, we study the phase damping and spontaneous-emission quantum channels. In addition, we mention a proposal for quasiunitary transforms of qubits, in view of the unitary transform of the total qutrit system. The experimental realization is also addressed. The probability representation of the evolution and its information-entropic characteristics are considered.

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Sauer, A., J. Z. Bernád, H. J. Moreno, and G. Alber. "Entanglement in bipartite quantum systems: Euclidean volume ratios and detectability by Bell inequalities." Journal of Physics A: Mathematical and Theoretical 54, no. 49 (November 19, 2021): 495302. http://dx.doi.org/10.1088/1751-8121/ac3469.

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Abstract Euclidean volume ratios between quantum states with positive partial transpose and all quantum states in bipartite systems are investigated. These ratios allow a quantitative exploration of the typicality of entanglement and of its detectability by Bell inequalities. For this purpose a new numerical approach is developed. It is based on the Peres–Horodecki criterion, on a characterization of the convex set of quantum states by inequalities resulting from Newton identities and from Descartes’ rule of signs, and on a numerical approach involving the multiphase Monte Carlo method and the hit-and-run algorithm. This approach confirms not only recent analytical and numerical results on two-qubit, qubit-qutrit, and qubit-four-level qudit states but also allows for a numerically reliable numerical treatment of so far unexplored qutrit–qutrit states. Based on this numerical approach with the help of the Clauser–Horne–Shimony–Holt inequality and the Collins–Gisin inequality the degree of detectability of entanglement is investigated for two-qubit quantum states. It is investigated quantitatively to which extent a combined test of both Bell inequalities can increase the detectability of entanglement beyond what is achievable by each of these inequalities separately.

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Daoud, Mohammed, and Maurice R. Kibler. "Generalized Weyl-Heisenberg Algebra, Qudit Systems and Entanglement Measure of Symmetric States via Spin Coherent States. Part II: The Perma-Concurrence Parameter." Symmetry 11, no. 7 (July 3, 2019): 875. http://dx.doi.org/10.3390/sym11070875.

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This paper deals with separable and entangled qudits | ψ d ⟩ (quantum states in dimension d) constructed from Dicke states made of N = d - 1 qubits. Such qudits present the property to be totally symmetric under the interchange of the N qubits. We discuss the notion of perma-concurrence P d for the qudit | ψ d ⟩ , introduced by the authors (Entropy 2018, 20, 292), as a parameter for characterizing the entanglement degree of | ψ d ⟩ . For d = 3 , the perma-concurrence P 3 constitutes an alternative to the concurrence C for symmetric two-qubit states. We give several expressions of P d (in terms of matrix permanent and in terms of unit vectors of R 3 pointing on the Bloch sphere) and precise the range of variation of P d (going from separable to maximally entangled states). Numerous examples are presented for P d . Special attention is devoted to states of W type and to maximally entangled states of Bell and Greenberger–Horne–Zeilinger type.

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28

NEVES, LEONARDO, G. LIMA, J. G. AGUIRRE GÓMEZ, C. H. MONKEN, C. SAAVEDRA, and S. PÁDUA. "TWO-PHOTON HIGH-DIMENSIONAL SPATIAL ENTANGLEMENT: THEORY AND EXPERIMENT." Modern Physics Letters B 20, no. 01 (January 10, 2006): 1–23. http://dx.doi.org/10.1142/s0217984906009566.

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We review recent theoretical and experimental works where are proposed and demonstrated how to use photon pairs created by spontaneous parametric down-conversion to generate entangled states of D-dimensional quantum systems, or qudits. This is the first demonstration of high-dimensional entanglement based on the intrinsic transverse momentum entanglement of the type-II down-converted photons. The qudit space is defined by an aperture made up of an opaque screen with D slits (paths), placed in the arms of the twin photons. By manipulating the pump beam profile we can prepare different entangled states of these possible paths. We focus our attention on an important case for applications in quantum information: the maximally entangled states. Experimental results for qudits with D=4 and D=8 are shown and measuring a two-photon conditional interference, we also demonstrate the nonclassical character of the correlations.

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Candeloro, Alessandro, Matteo G. A. Paris, and Marco G. Genoni. "On the properties of the asymptotic incompatibility measure in multiparameter quantum estimation." Journal of Physics A: Mathematical and Theoretical 54, no. 48 (November 10, 2021): 485301. http://dx.doi.org/10.1088/1751-8121/ac331e.

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Abstract We address the use of asymptotic incompatibility (AI) to assess the quantumness of a multiparameter quantum statistical model. AI is a recently introduced measure which quantifies the difference between the Holevo and the symmetric logarithmic derivative (SLD) scalar bounds, and can be evaluated using only the SLD operators of the model. At first, we evaluate analytically the AI of the most general quantum statistical models involving two-level (qubit) and single-mode Gaussian continuous-variable quantum systems, and prove that AI is a simple monotonous function of the state purity. Then, we numerically investigate the same problem for qudits (d-dimensional quantum systems, with 2 < d ⩽ 4), showing that, while in general AI is not in general a function of purity, we have enough numerical evidence to conclude that the maximum amount of AI is attainable only for quantum statistical models characterized by a purity larger than μ min = 1 / ( d − 1 ) . In addition, by parametrizing qudit states as thermal (Gibbs) states, numerical results suggest that, once the spectrum of the Hamiltonian is fixed, the AI measure is in one-to-one correspondence with the fictitious temperature parameter β characterizing the family of density operators. Finally, by studying in detail the definition and properties of the AI measure we find that: (i) given a quantum statistical model, one can readily identify the maximum number of asymptotically compatible parameters; (ii) the AI of a quantum statistical model bounds from above the AI of any sub-model that can be defined by fixing one or more of the original unknown parameters (or functions thereof), leading to possibly useful bounds on the AI of models involving noisy quantum dynamics.

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30

Hrmo, Pavel, Benjamin Wilhelm, Lukas Gerster, Martin W. van Mourik, Marcus Huber, Rainer Blatt, Philipp Schindler, Thomas Monz, and Martin Ringbauer. "Native qudit entanglement in a trapped ion quantum processor." Nature Communications 14, no. 1 (April 19, 2023). http://dx.doi.org/10.1038/s41467-023-37375-2.

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AbstractQuantum information carriers, just like most physical systems, naturally occupy high-dimensional Hilbert spaces. Instead of restricting them to a two-level subspace, these high-dimensional (qudit) quantum systems are emerging as a powerful resource for the next generation of quantum processors. Yet harnessing the potential of these systems requires efficient ways of generating the desired interaction between them. Here, we experimentally demonstrate an implementation of a native two-qudit entangling gate up to dimension 5 in a trapped-ion system. This is achieved by generalizing a recently proposed light-shift gate mechanism to generate genuine qudit entanglement in a single application of the gate. The gate seamlessly adapts to the local dimension of the system with a calibration overhead that is independent of the dimension.

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31

Miyahara, Hideyuki, Yiyou Chen, Vwani Roychowdhury, and Louis-Serge Bouchard. "Decoherence mitigation by embedding a logical qubit in a qudit." Quantum Information Processing 22, no. 7 (July 11, 2023). http://dx.doi.org/10.1007/s11128-023-04035-9.

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AbstractQuantum information stored in a qubit is rapidly lost to the environment. The realization of robust qubits is one of the most important challenges in quantum computing. Herein, we propose to embed a logical qubit within the manifold of a qudit as a scheme to preserve quantum information over extended periods of time. Under identical conditions (e.g., decoherence channels), the submanifold of the logical qubit exhibits extended lifetimes compared to a pure two-level system (qubit). The retention of quantum information further improves with separation between the sublevels of the logical qubit. Lifetime enhancement can be understood in terms of entropy production of the encoding and nonencoding subspaces during evolution under a quantum map for a d-level system. The additional pathways for coherent evolution through intermediate sublevels within a d-level manifold provide an information-preserving mechanism: reversible alternative channels to the irreversible loss of information to the environment characteristic of open quantum systems.

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32

Özgüler, A. Barış, and Davide Venturelli. "Numerical gate synthesis for quantum heuristics on bosonic quantum processors." Frontiers in Physics 10 (October 20, 2022). http://dx.doi.org/10.3389/fphy.2022.900612.

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There is a recent surge of interest and insights regarding the interplay of quantum optimal control and variational quantum algorithms. We study the framework in the context of qudits which are, for instance, definable as controllable electromagnetic modes of a superconducting cavity system coupled to a transmon. By employing recent quantum optimal control approaches described in (Petersson and Garcia, 2021), we showcase control of single-qudit operations up to eight states, and two-qutrit operations, mapped respectively onto a single mode and two modes of the resonator. We discuss the results of numerical pulse engineering on the closed system for parametrized gates useful to implement Quantum Approximate Optimization Algorithm (QAOA) for qudits. The results show that high fidelity (> 0.99) is achievable with sufficient computational effort for most cases under study, and extensions to multiple modes and open, noisy systems are possible. The tailored pulses can be stored and used as calibrated primitives for a future compiler in circuit quantum electrodynamics (cQED) systems.

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33

Ghosh, Debadrita, Thomas Jennewein, and Urbasi Sinha. "Direct determination of arbitrary dimensional entanglement monotones using statistical correlators and minimal complementary measurements." Quantum Science and Technology, August 31, 2022. http://dx.doi.org/10.1088/2058-9565/ac8e28.

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Abstract Higher dimensional quantum systems (qudits) present a potentially more efficient means, compared to qubits, for implementing various information theoretic tasks. One of the ubiquitous resources in such explorations is entanglement. Entanglement Monotones (EMs) are of key importance, particularly for assessing the efficacy of a given entangled state as a resource for information theoretic tasks. Till date, investigations towards determination of EMs have focused on providing their tighter lower bounds. There is yet no general scheme available for direct determination of the EMs. Consequently, an empirical determination of any EM has not yet been achieved for entangled qudit states. The present paper fills this gap, both theoretically as well as experimentally. First, we derive analytical relations between statistical correlation measures i.e. Mutual Predictability ($\mathcal{MP}$), Mutual Information ($\mathcal{MI}$) and Pearson Correlation Coefficient ($\mathcal{PCC}$) and standard EMs i.e. Negativity ($\mathcal{N}$) and Entanglement of Formation ($\mathcal{EOF}$) in arbitrary dimensions. As a proof of concept, we then experimentally measure $\mathcal{MP}$, $\mathcal{MI}$ and $\mathcal{PCC}$ of two-qutrit pure states and determine their $\mathcal{N}$ and $\mathcal{EOF}$ using these derived relations. This is a useful addition to the experimenter's toolkit wherein by using a limited number of measurements (in this case 1 set of measurements), one can directly measure the EMs in a bipartite arbitrary dimensional system. We obtain the value of $\mathcal{N}$ for our bipartite qutrit to be 0.907 $\pm$ 0.013 and the $\mathcal{EOF}$ to be 1.323 $\pm$ 0.022. Since the present scheme enables determination of more than one entanglement monotone by the same limited number of measurements, we argue that it can serve as a unique experimental platform for quantitatively comparing and contrasting the operational implications of entanglement monotones as well as showing their non-monotonicity for a given bipartire pure qudit state.

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Gu, Tianren, Xiao Yuan, and Bujiao Wu. "Efficient measurement schemes for bosonic systems." Quantum Science and Technology, July 12, 2023. http://dx.doi.org/10.1088/2058-9565/ace6cd.

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Abstract Boson is one of the most basic types of particles and preserves the commutationrelation. An efficient way to measure a bosonic system is important not onlyfor simulating complex physics phenomena of bosons (such as nuclei) on a qubitbased quantum computer, but for extracting classical information from a quantumsimulator/computer that itself is built with bosons (such as a continuous variablequantum computer). Extending the recently proposed measurement schemes forqubits, such as shadow tomography and other local measurement schemes, here westudy efficient measurement approaches for bosonic systems. We consider truncatedqudit and continuous variable systems, corresponding to simulated bosons on a discretequantum computer and an inherent boson system, respectively, and propose differentmeasurement schemes with theoretical analyses of the variances for these two cases. Wenumerically test the schemes for measuring nuclei vibrations simulated using a discretequantum computer and a continuous variable Gaussian state, and the simulation resultsshow great improvement of the performance of the proposed method compared toconventional ones.

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35

Tabia, Gelo Noel M., Kai-Siang Chen, Chung-Yun Hsieh, Yu-Chun Yin, and Yeong-Cherng Liang. "Entanglement transitivity problems." npj Quantum Information 8, no. 1 (August 25, 2022). http://dx.doi.org/10.1038/s41534-022-00616-1.

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AbstractOne of the goals of science is to understand the relation between a whole and its parts, as exemplified by the problem of certifying the entanglement of a system from the knowledge of its reduced states. Here, we focus on a different but related question: can a collection of marginal information reveal new marginal information? We answer this affirmatively and show that (non-) entangled marginal states may exhibit (meta)transitivity of entanglement, i.e., implying that a different target marginal must be entangled. By showing that the global n-qubit state compatible with certain two-qubit marginals in a tree form is unique, we prove that transitivity exists for a system involving an arbitrarily large number of qubits. We also completely characterize—in the sense of providing both the necessary and sufficient conditions—when (meta)transitivity can occur in a tripartite scenario when the two-qudit marginals given are either the Werner states or the isotropic states. Our numerical results suggest that in the tripartite scenario, entanglement transitivity is generic among the marginals derived from pure states.

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Calixto, Manuel, Alberto Mayorgas, and Julio Guerrero. "Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models." Quantum Information Processing 20, no. 9 (September 2021). http://dx.doi.org/10.1007/s11128-021-03218-6.

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AbstractCollective spin operators for symmetric multi-quDit (namely identical D-level atom) systems generate a U(D) symmetry. We explore generalizations to arbitrary D of SU(2)-spin coherent states and their adaptation to parity (multi-component Schrödinger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric N-quDit states, expressed in the last two cases in terms of collective U(D)-spin operator expectation values. Then, we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin–Meshkov–Glick models of 3-level identical atoms. We also propose an extension of the concept of SU(2)-spin squeezing to SU(D) and relate it to pairwise D-level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting D-level atom models.

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Wach, Noah L., Manuel S. Rudolph, Fred Jendrzejewski, and Sebastian Schmitt. "Data re-uploading with a single qudit." Quantum Machine Intelligence 5, no. 2 (August 14, 2023). http://dx.doi.org/10.1007/s42484-023-00125-0.

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AbstractQuantum two-level systems, i.e., qubits, form the basis for most quantum machine learning approaches that have been proposed throughout the years. However, higher dimensional quantum systems constitute a promising alternative and are increasingly explored in theory and practice. Here, we explore the capabilities of multi-level quantum systems, so-called qudits, for their use in a quantum machine learning context. We formulate classification and regression problems with the data re-uploading approach and demonstrate that a quantum circuit operating on a single qudit is able to successfully learn highly non-linear decision boundaries of classification problems such as the MNIST digit recognition problem. We demonstrate that the performance strongly depends on the relation between the qudit states representing the labels and the structure of labels in the training data set. Such a bias can lead to substantial performance improvement over qubit-based circuits in cases where the labels, the qudit states, and the operators employed to encode the data are well-aligned. Furthermore, we elucidate the influence of the choice of the elementary operators and show that a squeezing operator is necessary to achieve good performances. We also show that there exists a trade-off for qudit systems between the number of circuit-generating operators in each processing layer and the total number of layers needed to achieve a given accuracy. Finally, we compare classification results from numerically exact simulations and their equivalent implementation on actual IBM quantum hardware. The findings of our work support the notion that qudit-based algorithms exhibit attractive traits and constitute a promising route to increasing the computational capabilities of quantum machine learning approaches.

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38

Guerrero, Julio, Alberto Mayorgas, and Manuel Calixto. "Information diagrams in the study of entanglement in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models." Quantum Information Processing 21, no. 6 (June 2022). http://dx.doi.org/10.1007/s11128-022-03524-7.

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AbstractIn this paper we pursue the use of information measures (in particular, information diagrams) for the study of entanglement in symmetric multi-quDit systems. We use generalizations to $${U}(D)$$ U ( D ) of spin $${U}(2)$$ U ( 2 ) coherent states and their adaptation to parity (multicomponent Schrödinger cats), and we analyse one- and two-quDit reduced density matrices. We use these correlation measures to characterize quantum phase transitions occurring in Lipkin–Meshkov–Glick models of $$D=3$$ D = 3 -level identical atoms, and we propose the rank of the corresponding reduced density matrix as a discrete order parameter.

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39

Zalivako, Ilia V., Alexander S. Borisenko, Ilya A. Semerikov, Andrey E. Korolkov, Pavel L. Sidorov, Kristina P. Galstyan, Nikita V. Semenin, et al. "Continuous dynamical decoupling of optical 171Yb+ qudits with radiofrequency fields." Frontiers in Quantum Science and Technology 2 (August 8, 2023). http://dx.doi.org/10.3389/frqst.2023.1228208.

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The use of multilevel quantum information carriers, also known as qudits, has attracted significant interest as a way of further scaling quantum computing devices. However, such multilevel systems usually express shorter coherence time than their two-level counterparts, which limits their computational potential. We thus propose and experimentally demonstrate two approaches for realizing the continuous dynamical decoupling of magnetic-sensitive states with mF = ±1 for qudits encoded in optical transition of trapped 171Yb+ ions. We improve the coherence time of qudit levels by an order of magnitude (more than 9 ms) without any magnetic shielding, revealing the potential advantage of the symmetry of the 171Yb+ ion energy structure for counteracting magnetic field noise. Our results are a step toward realizing qudit-based algorithms using trapped ions.

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40

Sarkar, Shubhayan, Debashis Saha, Jędrzej Kaniewski, and Remigiusz Augusiak. "Self-testing quantum systems of arbitrary local dimension with minimal number of measurements." npj Quantum Information 7, no. 1 (October 14, 2021). http://dx.doi.org/10.1038/s41534-021-00490-3.

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AbstractBell nonlocality as a resource for device-independent certification schemes has been studied extensively in recent years. The strongest form of device-independent certification is referred to as self-testing, which given a device, certifies the promised quantum state as well as quantum measurements performed on it without any knowledge of the internal workings of the device. In spite of various results on self-testing protocols, it remains a highly nontrivial problem to propose a certification scheme of qudit–qudit entangled states based on violation of a single d-outcome Bell inequality. Here we address this problem and propose a self-testing protocol for the maximally entangled state of any local dimension using the minimum number of measurements possible, i.e., two per subsystem. Our self-testing result can be used to establish unbounded randomness expansion, $${{{\mathrm{log}}}\,}_{2}d$$ log 2 d perfect random bits, while it requires only one random bit to encode the measurement choice.

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41

Mahanti, Soumik, Sagnik Dutta, and Prasanta K. Panigrahi. "Classification and quantification of entanglement through wedge product and geometry." Physica Scripta, June 9, 2023. http://dx.doi.org/10.1088/1402-4896/acdd31.

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Abstract The wedge product of post-measurement vectors of a two-qubit state gives rise to a parallelogram, whose ‘area’ has been shown to be equivalent to the generalized I-concurrence measure of entanglement. In multi-qudit systems, the wedge product of post-measurement vectors naturally leads to a higher dimensional \textit{parallelepiped} which yields a modified faithful entanglement measure. Our new measure fine grains the entanglement monotone, wherein different entangled classes manifest with different geometries. We present a complete analysis of the bipartite qutrit case considering all possible geometric structures where three entanglement classes of pure bipartite qutrit states can be identified with different geometries of post-measurement vectors: three planar vectors; three mutually orthogonal vectors; and three vectors that are neither planar and not all of them are mutually orthogonal. It is further demonstrated that the geometric condition of area and volume maximization naturally leads to the maximization of entanglement. The wedge product approach uncovers an inherent geometry of entanglement and is found to be very useful for the characterization and quantification of entanglement in higher dimensional systems.

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42

Lu, Hsuan-Hao, Karthik V. Myilswamy, Ryan S. Bennink, Suparna Seshadri, Mohammed S. Alshaykh, Junqiu Liu, Tobias J. Kippenberg, Daniel E. Leaird, Andrew M. Weiner, and Joseph M. Lukens. "Bayesian tomography of high-dimensional on-chip biphoton frequency combs with randomized measurements." Nature Communications 13, no. 1 (July 27, 2022). http://dx.doi.org/10.1038/s41467-022-31639-z.

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AbstractOwing in large part to the advent of integrated biphoton frequency combs, recent years have witnessed increased attention to quantum information processing in the frequency domain for its inherent high dimensionality and entanglement compatible with fiber-optic networks. Quantum state tomography of such states, however, has required complex and precise engineering of active frequency mixing operations, which are difficult to scale. To address these limitations, we propose a solution that employs a pulse shaper and electro-optic phase modulator to perform random operations instead of mixing in a prescribed manner. We successfully verify the entanglement and reconstruct the full density matrix of biphoton frequency combs generated from an on-chip Si3N4 microring resonator in up to an 8 × 8-dimensional two-qudit Hilbert space, the highest dimension to date for frequency bins. More generally, our employed Bayesian statistical model can be tailored to a variety of quantum systems with restricted measurement capabilities, forming an opportunistic tomographic framework that utilizes all available data in an optimal way.

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43

Das, Sreetama, and Filippo Caruso. "A hybrid-qudit representation of digital RGB images." Scientific Reports 13, no. 1 (August 22, 2023). http://dx.doi.org/10.1038/s41598-023-39906-9.

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AbstractQuantum image processing is an emerging topic in the field of quantum information and technology. In this paper, we propose a new quantum image representation of RGB images with deterministic image retrieval, which is an improvement over all the similar existing representations in terms of using minimum resource. We use two entangled quantum registers constituting of total 7 qutrits to encode the color channels and their intensities. Additionally, we generalize the existing encoding methods by using both qubits and qutrits to encode the pixel positions of a rectangular image. This hybrid-qudit approach aligns well with the current progress of NISQ devices in incorporating higher dimensional quantum systems than qubits. We then describe the image encoding method using higher-order qubit-qutrit gates, and demonstrate the decomposition of these gates in terms of simpler elementary gates. We use the Google Cirq’s quantum simulator to verify the image preparation in both the ideal noise-free scenario and in presence of realistic noise modelling. We show that the complexity of the image encoding process is linear in the number of pixels. Lastly, we discuss the image compression and some basic RGB image processing protocols using our representation.

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44

Sedlák, Michal. "Quantum theory of unambiguous measurements." Acta Physica Slovaca. Reviews and Tutorials 59, no. 6 (December 1, 2009). http://dx.doi.org/10.2478/v10155-010-0099-3.

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Quantum theory of unambiguous measurementsIn the present paper I formulate a framework that accommodates many unambiguous discrimination problems. I show that the prior information about any type of constituent (state, channel, or observable) allows us to reformulate the discrimination among finite number of alternatives as the discrimination among finite number of average constituents. Using this framework I solve several unambiguous tasks. I present a solution to optimal unambiguous comparison of two ensembles of unknown quantum states. I consider two cases: 1) The two unknown states are arbitrary pure states of qudits. 2) Alternatively, they are coherent states of single-mode optical fields. For this case I propose simple and optimal experimental setup composed of beam-splitters and a photodetector. As a second tasks I consider an unambiguous identification (UI) of coherent states. In this task identical quantum systems are prepared in coherent states and labeled as unknown and reference states, respectively. The promise is that one reference state is the same as the unknown state and the task is to find out unambiguously which one it is. The particular choice of the reference states is unknown to us, and only the probability distribution describing this choice is known. In a general case when multiple copies of unknown and reference states are available I propose a scheme consisting of beamsplitters and photodetectors that is optimal within linear optics. UI can be considered as a search in a quantum database, whose elements are the reference states and the query is represented by the unknown state. This perspective motivated me to show that reference states can be recovered after the measurement and might be used (with reduced success rate) in subsequent UI. Moreover, I analyze the influence of noise in preparation of coherent states on the performance of the proposed setup. Another problem I address is the unambiguous comparison of a pair of unknown qudit unitary channels. I characterize all solutions and identify the optimal ones. I prove that in optimal experiments for comparison of unitary channels the entanglement is necessary. The last task I studied is the unambiguous comparison of unknown non-degenerate projective measurements. I distinguish between measurement devices with apriori labeled and unlabeled outcomes. In both cases only the difference of the measurements can be concluded unambiguously. For the labeled case I derive the optimal strategy if each unknown measurement is used only once. However, if the apparatuses are not labeled, then each measurement device must be used (at least) twice. In particular, for qubit measurement apparatuses with unlabeled outcomes I derive the optimal test state in the two-shots scenario.

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Journal articles on the topic 'TWO-QUDIT SYSTEM'

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