Violation of the CHSH inequality by a bipartite quantum state is now used in many quantum applications. However, the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-$s$ measurements is still known only for $s=1/2$. In the present article, for an arbitrary state of two spin-$s$ qudits, each of dimension $d=2s+1\geq 2$, we introduce the notion of the spin-$s$ correlation matrix, which has dimension $3\times 3$ for all $s\geq \frac{1}{2}$; establish its relation to the general correlation $(d^{2}-1)\times (d^{2}-1)$ matrix of this state within the generalized Pauli representation and derive in terms of the spin-$s$ correlation matrix the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-$s$ measurements in this state. Specifying this general expression for the two-qudit GHZ state, the nonlocal two-qudit Werner state, and some nonseparable pure two-qudit states, we find that, under local Alice and Bob high-spin ($s\geq1$) measurements in each of these nonseparable states, including the maximally entangled one, the CHSH inequality is not violated. Moreover, unlike the case of spin-$1/2$ measurements, where each pure nonseparable two-qubit state violates the CHSH inequality and the maximal value of its CHSH expectation increases monotonically with a growth of its entanglement, the situation under high-spin measurements is quite different -- for a pure two-qudit state with a higher degree of entanglement, the maximal value of the CHSH expectation turns out to be less than for a pure two-qudit state with lower entanglement and even for a separable one.
As a continuation of our previous work (arXiv:0708.4333) an algebraic geometrical study of a single $d$-dimensional qudit is made, with $d$ being {\it any} positive integer. The study is based on an intricate relation between the symplectic module of the generalized Pauli group of the qudit and the fine structure of the projective line over the (modular) ring $\bZ_{d}$. Explicit formulae are given for both the number of generalized Pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of $\bZ^{2}_{d}$. We find, remarkably…
We present a general approach for quantifying tolerance of a nonlocal N-partite state to any local noise under different classes of quantum correlation scenarios with arbitrary numbers of settings and outcomes at each site. This allows us to derive new precise bounds in d and N on noise tolerances for: (i) an arbitrary nonlocal N-qudit state; (ii) the N-qudit Greenberger-Horne-Zeilinger (GHZ) state; (iii) the N-qubit W state and the N-qubit Dicke states, and to analyse asymptotics of these precise bounds for large N and d.
New quantum entropic inequality for states of system of n >_ 1 qudits is obtained. The inequality has the form of quantum subadditivity condition of bipartite qudit system and coincides with this subadditivity condition for the system of two qudits. The general statement on existence of the subadditivity condition for arbitrary probability distribution and arbitrary qudit-system tomogram is formulated. The nonlinear quantum channels creating the entangled states from separable ones are discussed.
We propose a protocol ${\cal D}_n$ for faithfully teleporting an arbitrary $n$-qudit state with the tensor product state (TPS) of $n$ generalized Bell states (GBSs) as the quantum channel. We also put forward explicit protocol ${\cal D}'_n$ and ${\cal D}''_n$ for faithfully teleporting an arbitrary $n$-qudit state with two classes of $2n$-qudit GESs as the quantum channel, where the GESs are a kind of genuine entangled states we construct and can not be reducible to the TPS of $n$ GBSs.
Collective 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 (multicomponent 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 …
A general scheme for controlled teleportation of an arbitrary multi-qudit state with d-dimensional Greenberger-Horne-Zeilinger (GHZ) states is proposed. For an arbitrary m-qudit state, the sender Alice performs m generalized Bell-state projective measurements on her 2m qudits and the controllers need only take some single-particle measurements. The receiver Charlie can reconstruct the unknown m-qudit state by performing some single-qudit unitary operations on her particles if she cooperates with all the controllers. As the quantum channel is a sequence of maximally entangled GHZ states, the in…
Two measures of entanglement, negativity and concurrence are studied for two arbitrary qudits. We obtain negativity as an expectation value of an operator. The differences of the squares of negativity and concurrence are invariants of multilevel entanglement. Explicit results for qutrits and quadrits are obtained.
We report interferometric schemes to prepare arbitrary states of four-dimensional qudits (ququarts) based on biphoton states of ultrafast-pumped frequency-nondegenerate spontaneous parametric down-conversion. Preparation and tomographic characterization of a few examples of general single-ququart states, a pure state, a completely mixed state, and a partially-mixed state, are experimentally demonstrated.
A qudit ($d$-level quantum systems) has a large Hilbert space and thus can be used to achieve many quantum information and communication tasks. Here, we propose a method to transfer arbitrary $d$-dimensional quantum states (known or unknown) between two superconducting qudits coupled to a single cavity. The state transfer can be performed fast because of employing resonant interactions only. In addition, quantum states can be deterministically transferred without measurement. Numerical simulations show that high-fidelity transfer of quantum states between two superconducting transmon qudits ($…
A relation is established in the present paper between Dicke states in a d-dimensional space and vectors in the representation space of a generalized Weyl-Heisenberg algebra of finite dimension d. This provides a natural way to deal with the separable and entangled states of a system of N = d-1 symmetric qubit states. Using the decomposition property of Dicke states, it is shown that the separable states coincide with the Perelomov coherent states associated with the generalized Weyl-Heisenberg algebra considered in this paper. In the so-called Majorana scheme, the qudit (d-level) states are r…
New kind of matrix inequality known for bipartite system density matrix is obtained for arbitrary density matrix of composite or noncomposite qudit systems including the single qudit state. The examples of two qubit system and qudit with j=3/2 are discussed.
We consider information characteristics of single qudit state (spin j=9/2), such as von Neumann entropy, von Neumann mutual information. We review different mathematical properties of these information characteristics: subadditivity and strong subadditivity conditions, Araki-Lieb inequality. The inequalities are entropic inequalities for composite systems (bipartite, tripartite), but they can be written for noncomposite systems. Using the density matrix, describing the noncomposite qudit system state in explicit matrix form we proved new entropic inequalities for single qudit state (spin j=9/2…
We present a general scheme for multiparty-controlled teleportation of an arbitrary m-qudit ($d$-dimensional quantum system) state by using non-maximally entangled states as the quantum channel. The sender performs m generalized Bell-state measurements on her 2$m$ particles, the controllers take some single-particle measurements with the measuring basis $X_{d}$ and the receiver only need to introduce one auxiliary two-level particle to extract quantum information probabilistically with the fidelity unit if he cooperates with all the controllers. All the parties can use some decoy photons to se…
Within the general context of the architecture in quantum computer design, this paper aims is to provide a general strategy to obtain a block-matrix representation of quantum gates applied to qubits placed in arbitrary positions over an arbitrary dimensional input state. The model is also extended to the framework of quantum computation with qudits. An application in the context of the quantum computational logic is provided.
We study entanglement properties of hypergraph states in arbitrary finite dimension. We compute multipartite entanglement of elementary qudit hypergraph states, namely those endowed with a single maximum-cardinality hyperedge. We show that, analogously to the qubit case, also for arbitrary dimension there exists a lower bound for multipartite entanglement of connected qudit hypergraph states; this is given by the multipartite entanglement of an equal-dimension elementary hypergraph state featuring the same number of qudits as the largest-cardinality hyperedge. We highlight interesting differen…
Spatial qudits are D-dimensional ($D\geq 2$) quantum systems carrying information encoded in the discretized transverse momentum and position of single photons. We present a proof-of-principle demonstration of a method for preparing arbitrary pure states of such systems by using a single phase-only spatial light modulator (SLM). The method relies on the encoding of the complex transmission function corresponding to a given spatial qudit state onto a preset diffraction order of a phase-only grating function addressed at the SLM. Fidelities of preparation above 94% were obtained with this method…
Quantum information masking is a protocol that hides the original quantum information from subsystems and spreads it over quantum correlation, which is available to multipartite except bipartite systems. In this work, we explicitly study the quantum information masking in multipartite scenario and prove that all the k-level quantum states can be masked into a m-qudit systems (m > 4) whose local dimension d < k and the upper bound of k is tighter than the quantum Singleton bound. In order to observe the masking process intuitively, explicitly controlled operations are provided. Our scheme well …
In this paper, we prove that it is impossible to construct a convariant quantum feedback control instruments which aim to correct the channel noise imposed on an unknown qubit. The proof is based on the searching for the optimal quantum control protocol and it turns out there exist no better %\textit{Complete Positive Trace %Preserving}(CPTP) maps complete positive and covariant quantum operations which provide a higher fidelity than the trivial Identity operators. The generalization of the investigation to bipartite entangled pure state is also included.
The capability to generate and manipulate quantum states in high-dimensional Hilbert spaces is a crucial step for the development of quantum technologies, from quantum communication to quantum computation. One-dimensional quantum walk dynamics represents a valid tool in the task of engineering arbitrary quantum states. Here we affirm such potential in a linear-optics platform that realizes discrete-time quantum walks in the orbital angular momentum degree of freedom of photons. Different classes of relevant qudit states in a six-dimensional space are prepared and measured, confirming the feasi…
For an even qudit dimension $d\geq 2,$ we introduce a class of two-qudit states exhibiting perfect correlations/anticorrelations and prove via the generalized Gell-Mann representation that, for each two-qudit state from this class, the maximal violation of the original Bell inequality is bounded from above by the value $3/2$ - the upper bound attained on some two-qubit states. We show that the two-qudit Greenberger-Horne-Zeilinger (GHZ) state with an arbitrary even $d\geq 2$ exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results ar…
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 $r$ …
High-dimensional degrees of freedom of photons can encode more quantum information than their two-dimensional counterparts. While the increased information capacity has advantages in quantum applications (such as quantum communication), controlling and manipulating these systems has been challenging. Here we show a method to perform lossless arbitrary high-dimensional Pauli-X gates for single photon. The X-gate consists of a cyclic permutation of qudit basis vectors, and, together with the Z gate, forms the basis for performing arbitrary transformations. We propose an implementation of such ga…
We study photon-mediated interactions between molecular spin qudits in the dispersive regime of operation. We derive from a microscopic model the effective interaction between molecular spins, including their crystal field anisotropy (i.e., the presence of non-linear spin terms) and their multi-level structure. Finally, we calculate the long time dynamics for a pair of interacting molecular spins using the method of multiple scales analysis. This allows to find the set of 2-qudit gates that can be realized for a specific choice of molecular spins and to determine the time required for their im…
We formulate and prove the main properties of the generalized Gell-Mann representation for traceless qudit observables with eigenvalues in $[-1,1]$ and analyze via this representation violation of the CHSH inequality by a general two-qudit state. For the maximal value of the CHSH expectation in a two-qudit state with an arbitrary qudit dimension $d\geq2$, this allows us to find two new bounds, lower and upper, expressed via the spectral properties of the correlation matrix for a two-qudit state. We have not yet been able to specify if the new upper bound improves the Tsirelson upper bound for …
The achievement of quantum supremacy boosted the need for a robust medium of quantum information. In this task, higher-dimensional qudits show remarkable noise tolerance and enhanced security for quantum key distribution applications. However, to exploit the advantages of such states, we need a thorough characterisation of their entanglement. Here, we propose a measure of entanglement which can be computed either for pure and mixed states of a $M$-qudit hybrid system. The entanglement measure is based on a distance deriving from an adapted application of the Fubini-Study metric. This measure i…
We generalize the entanglement swapping scheme originally proposed for two pairs of qubits to an arbitrary number $q$ of systems composed from an arbitrary number $m_j$ of qudits. Each of the system is supposed to be prepared in a maximally entangled state of $m_j$ qudits, while different systems are not correlated at all. We show that when a set $\sum_{j=1}^q a_j$ particles (from each of the $q$ systems $a_j$ particles are measured) are subjected to a generalized Bell-type measurement, the resulting set of $\sum_{j=1}^q (m_j-a_j)$ particles will collapse into a maximally entangled state.
We show that entangling capacities based on the Jamiolkowski isomorphism may be used to place lower bounds on the communication capacities of arbitrary bipartite unitaries. Therefore, for these definitions, the relations which have been previously shown for two-qubit unitaries also hold for arbitrary dimensions. These results are closely related to the theory of the entanglement-assisted capacity of channels. We also present more general methods for producing ensembles for communication from initial states for entanglement creation.
Spin bases of relevance for quantum systems with cyclic symmetry as well as for quantum information and quantum computation are constructed from the theory of angular momentum. This approach is connected to the use of generalized Pauli matrices (in dimension d) arising from a polar decomposition of the group SU(2). Numerous examples are given for d=2, 3 and 4.
We investigate chains of 'd' dimensional quantum spins (qudits) on a line with generic nearest neighbor interactions without translational invariance. We find the conditions under which these systems are not frustrated, i.e. when the ground states are also the common ground states of all the local terms in the Hamiltonians. The states of a quantum spin chain are naturally represented in the Matrix Product States (MPS) framework. Using imaginary time evolution in the MPS ansatz, we numerically investigate the range of parameters in which we expect the ground states to be highly entangled and fi…
Using the tomographic probability representation of qudit states and the inverse spin-portrait method, we suggest a bijective map of the qudit density operator onto a single probability distribution. Within the framework of the approach proposed, any quantum spin-j state is associated with the (2j+1)(4j+1)-dimensional probability vector whose components are labeled by spin projections and points on the sphere. Such a vector has a clear physical meaning and can be relatively easily measured. Quantum states form a convex subset of the 2j(4j+3) simplex, with the boundary being illustrated for qub…
Unitary control of qudits can improve the collective spin squeezing of an atomic ensemble. Preparing the atoms in a state with large quantum fluctuations in magnetization strengthens the entangling Faraday interaction. The resulting increase in interatomic entanglement can be converted into metrologically useful spin squeezing. Further control can squeeze the internal atomic spin without compromising entanglement, providing an overall multiplicative factor in the collective squeezing. We model the effects of optical pumping and study the tradeoffs between enhanced entanglement and decoherence.…
We obtain a new inequality for arbitrary Hermitian matrices. We describe particular linear maps called the matrix portrait of arbitrary NxN matrices. The maps are obtained as analogs of partial tracing of density matrices of multipartite qudit systems. The structure of the maps is inspired by "portrait" map of the probability vectors corresponding to the action on the vectors by stochastic matrices containing either unity or zero matrix elements. We obtain new entropic inequalities for arbitrary qudit states including a single qudit and an discuss entangled single qudit state. We consider in d…
We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the maximal violation of the original Bell inequality is upper bounded by 3/2 and specify the two-qubit states where this quantum upper bound is attained. The case of two-qutrit states is more complicated. Here, for all two-qutrit states, we obtain the same upper bound 3/2 for violation of the original Bell inequality under Alice and Bob spin measurem…
We generalize Greenberger-Horne-Zeilinger (GHZ) theorem to an arbitrary number of D-dimensional systems. Contrary to conventional approaches using compatible composite observables, we employ incompatible and concurrent observables, whose common eigenstate is still a generalized GHZ state. It is these concurrent observables which enable to prove a genuinely N-partite and D-dimensional GHZ theorem. Our principal idea is illustrated for a four-partite system with D which is an arbitrary multiple of 3. By extending to N qudits, we show that GHZ theorem holds as long as N is not divisible by all no…
We develop a procedure of generalized continuous dynamical decoupling (GCDD) for an ensemble of $d$-level systems (qudits), allowing one to protect the action of an arbitrary multi-qudit gate from general noise. We first present our GCDD procedure for the case of an arbitrary qudit and apply it to the case of a Hadamard gate acting on a qutrit. This is done using a model that, in principle, could be implemented using the three magnetic hyperfine states of the ground energy level of $^{87}\mathrm{Rb}$ and laser beams whose intensities and phases are modulated according to our prescription. We s…
Quantum bit or qubit is a two-level system, which builds the foundation for quantum computation, simulation, communication and sensing. Quantum states of higher dimension, i.e., qutrits (D = 3) and especially qudits (D = 4 or higher), offer significant advantages. Particularly, they can provide noise-resistant quantum cryptography, simplify quantum logic and improve quantum metrology. Flying and solid-state qudits have been implemented on the basis of photonic chips and superconducting circuits, respectively. However, there is still a lack of room-temperature qudits with long coherence time an…
New entropic inequality for quantum and tomographic Shannon information for system of two qubits is obtained. The inequality relating quantum information and spin-tomographic information for particle with spin j = 3=2 is found. The method to extend the obtained new entropic and information inequalities for one qudit and arbitrary composite system of qudits is suggested.
In this work the quantum random walk search algorithm with walk coin constructed by generalized Householder reflection and phase multiplier has been studied. The coin register is one qudit with arbitrary dimension. Monte Carlo simulations, in combination with supervised machine learning, are used to find walk coins making the quantum algorithm more robust to deviations in the coin's parameters. By applying deep neural network we make prediction for the parameters of an optimal coin with arbitrary size and estimate the stability for such coin.
We study the physics of a magnetic molecule described by a "giant" spin with multiple ($d > 2$) spin states interacting with the quantized cavity field produced by a superconducting resonator. By means of the input-output formalism, we derive an expression for the output modes in the dispersive regime of operation. It includes the effect of magnetic anisotropy, which makes different spin transitions addressable. We find that the measurement of the cavity transmission allows to uniquely determine the spin state of the qudits. We discuss, from an effective Hamiltonian perspective, the conditions…
The steering property known for two qubit-state in terms of specific inequalities for correlation function is translated for the state of qudit with the spin $j=3/2$. Since most steering detection inequalities are based on the correlation functions we introduce analogs of such functions for the single qudit systems. The tomographic probability representation for the qudit states is applied. The connection between the correlation function in two qubit system and the single qudit is presented in an integral form with a intertwining kernel calculated explicitly in tomographic probability terms.
We show that molecular spin qudits provide an ideal platform to simulate the quantum dynamics of photon fields strongly interacting with matter. The basic unit of the proposed molecular quantum simulator can be realized by a simple dimer of a spin 1/2 and a spin $S$ transition metal ion, solely controlled by microwave pulses. The spin $S$ ion is exploited to encode the photon field in a flexible architecture, which enables the digital simulation of a wide range of spin-boson models much more efficiently than by using a multi-qubit register. The effectiveness of our proposal is demonstrated by …
We demonstrate, numerically, the possibility of manipulating the spin states of molecular nanomagnets with shaped microwave pulses designed with quantum optimal control theory techniques. The state-to-state or full gate transformations can be performed in this way in shorter times than using simple monochromatic resonant pulses. This enhancement in the operation rates can therefore mitigate the effect of decoherence. The optimization protocols and their potential for practical implementations are illustrated by simulations performed for a simple molecular cluster hosting a single Gd$^{3+}$ ion…
We propose a modified Quantum Energy Teleportation (QET) scheme that uses arbitrary-dimensional qudit probes and polynomially localized Hamiltonians. We find that with an appropriate scaling of parameters, the teleported energy scales with the teleportation distance more favourably than the non-local tails of the Hamiltonians. We show that by allowing the exchange of arbitrary amounts of information between agents and in a suitable limit, an arbitrarily large amount of energy can be teleported through a massless quantum field.
