We present an analysis of noisy atomic channels involving qutrits. We choose a three-level atom with V-configuration to be the qutrit state. Gell-Mann matrices and a generalized Bloch vector (8-dimensional) are used to describe the qutrit density operator. We introduce quantum quasi-distributions for qutrits that provide a simple description of entanglement. Studying the time-evolution for the atomic variables we find the Kraus representation of spontaneous emission quantum channel (SE channel). Furthermore, we consider a generalized Werner state of two qutrits and investigate the separability condition in the presence of spontaneous emission noise. The influence of spontaneous emission on the separability of Werner states for qutrit and qubit states is compared.
We present an extension of the Wootters concurrence for the case of two qutrits in mixed states. The reduction of our extension to the case of two levels shows complete agreement with Wootters concurrence for two qubits. As an explicit example, we compute the concurrence for a family of symmetric states and we obtain the bounds on the limit for separability. Our results are compared with those of the negativity.
The question of the generation of random mixed states is discussed, aiming for the computation of probabilistic characteristics of composite finite dimensional quantum systems. Particularly, we consider the generation of the random Hilbert-Schmidt and Bures ensembles of qubit and qutrit pairs and compute the corresponding probabilities to find a separable state among the states of a fixed rank.
We define the separability and entanglement notion for particle with spin $s=1$. We consider two cases. In the first the particle is composed of two fermions with $s_1=1/2$ and $s_2=1/2$. In the second case the state is the qutrit state which is not composed system. The notion of negativity and concurrence is defined for the qutrit state. The concurrence and negativity of entangled and separable qutrit states determined by the parameters of the density matrix are explicitly calculated. The maximum entanglement of the qutrit state is observed for maximum values of non diagonal matrix elements o…
We consider the separability of various joint states for N qutrits. We derive two results: (i) the separability condition for a two-qutrit state that is a mixture of the maximally mixed state and a maximally entangled state (such a state is a generalization of the Werner state for two qubits); (ii) upper and lower bounds on the size of the neighborhood of separable states surrounding the maximally mixed state for N qutrits.
With a probability of success of $95 \%$ we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres-Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be sepa…
We present an analysis of spontaneous emission in a 3-level atom as an example of a qutrit state under the action of noisy quantum channels. We choose a 3-level atom with V-configuration to be the qutrit state. Gell-Mann matrices and a generalized Bloch vector (8-dimensional) are used to describe the qutrit density operator. Using the time-evolution equations of atomic variables we find the Kraus representation of spontaneous emission quantum channel (SE channel). Furthermore, we consider a generalized Werner state of two qutrits and investigate the separability condition. We give similar anal…
Based on a recent proposal [Phys. Rev. A 71, 062337 (2005)], we have experimentally realized two photon polarization qutrits by using non-maximally entangled states and linear optical transformations. By this technique high fidelity mutually unbiased qutrits are generated at a high brilliance level.
A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems.
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 formulate a two-party communication complexity problem and present its quantum solution that exploits the entanglement between two qutrits. We prove that for a broad class of protocols the entangled state can enhance the efficiency of solving the problem in the quantum protocol over any classical one if and only if the state violates Bell's inequality for two qutrits.
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…
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…
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.
Three player quantum Kolkata restaurant problem is modeled using three entangled qutrits. This first use of three level quantum states in this context is a step towards a $N$-choice generalization of the $N$-player quantum minority game. It is shown that a better than classical payoff is achieved by a Nash equilibrium solution where the space of available strategies is spanned by subsets of SU(3) and the players share a tripartite entangled initial state.
Squeezed state of light is one of the important subjects in quantum optics which is generated by optical nonlinear interactions. In this paper, we especially focus on qubit like entangled squeezed states (ESS's) generated by beam splitters, phase-shifter and cross Kerr nonlinearity. Moreover the Wigner function of two-mode qubit and qutrit like ESS are investigated. We will show that the distances of peaks of Wigner functions for two-mode ESS are entanglement sensitive and can be a witness for entanglement. Like the qubit cases, monogamy inequality is fulfilled for qutrit like ESS. These trend…
Recently it has been discovered that certain two-qutrit entangled states interacting with global and/or multi-local decoherence undergo distillability sudden death (DSD). We investigate this phenomenon for qutrit-qutrit systems interacting with statistically independent zero-temperature reservoirs. We show that certain initially prepared free-entangled states become bound-entangled in a finite time due to the action of Markovian dissipative environment. Moreover, in contrast with local dephasing, simple local unitary transformations can completely avoid distillability sudden death under amplit…
We propose an efficient method to generate a GHZ entangled state of n photons in n microwave cavities (or resonators) via resonant interaction to a single superconducting qutrit. The deployment of a qutrit, instead of a qubit, as the coupler enables us to use resonant interactions exclusively for all qutrit-cavity and qutrit-pulse operations. This unique approach significantly shortens the time of operation which is advantageous to reducing the adverse effects of qutrit decoherence and cavity decay on fidelity of the protocol. Furthermore, the protocol involves no measurement on either the sta…
We propose a scheme for building a heralded two-qutrit entangled state from polarized photons. An optical circuit is presented to build the maximally entangled two-qutrit state from two heralded Bell pairs and ideal threshold detectors. Several schemes are discussed for constructing the two Bell pairs. We also show how one can produce an unbalanced two-qutrit state that could be of general purpose use in some protocols. In terms of applications of the maximally entangled qutrit state, we mainly focus on how to use the state to demonstrate a violation of the Collins-Gisin-Linden-Massar-Popescu …
We report an exhaustive numerical analysis of violations of local realism by two qutrits in all possible pure entangled states. In Bell type experiments we allow any pairs of local unitary U(3) transformations to define the measurement bases. Surprisingly, Schmidt rank-2 states, resembling pairs of maximally entangled qubits, lead to the most noise-robust violations of local realism. The phenomenon seems to be even more pronounced for four and five dimensional systems, for which we tested a few interesting examples.
In Kaszlikowski [Phys. Rev. Lett. {\bf 85}, 4418 (2000)], it has been shown numerically that the violation of local realism for two maximally entangled $N$-dimensional ($3 \leq N$) quantum objects is stronger than for two maximally entangled qubits and grows with $N$. In this paper we present the analytical proof of this fact for N=3.
We list in increasing order -- 1/3, 3/8, 2/5, 135 pi/1024, 16/(3 pi^2), 3 pi/16, 5/8, 105 pi/512, 2 - 435 pi/1024, 11/16, 1 -- a number of exact two-qubit Hilbert-Schmidt (HS) separability probabilities, we are able to compute. Each probability corresponds to a specific scenario -- a class of 4 x 4 density matrices (rho) with an m-subset (m<6) of its six off-diagonal pairs of symmetrically located entries set to zero. Initially, we consider only scenarios in which the (6-m) non-nullified pairs are real, but then permit them to be complex (as well as quaternionic) in nature also. The general an…
We develop a systematic method to construct the Bell states of a qubit bipartite system while taking $SU(2)$ group as the basis group. An alternative formulation of fidelity, called $SU(2)$ fidelity, is proposed which gives the Bell-CHSH inequality. Taking analogy from the octet theory governed by the $SU(3)$ group, it is shown that the method can be easily generalized to derive all possible entangled states and the $SU(3)$ fidelity of the higher dimensional bipartite qutrit system. The properties of these states are studied which reveal the existence of some inequalities for the qutrit bipart…
We introduce the concept of distillability sudden death, i.e., free entangled states can evolve into non-distillable (bound entangled or separable) states in finite time under local noise. We describe the phenomenon through a specific model of local dephasing noise and compare the behavior of states in terms of the Bures fidelity. Then we propose a few methods to avoid distillability sudden death of states under (general) local dephasing noise, so that free entangled states can be robust against decoherence. Moreover, we find that bound entangled states are unstable in the limit of infinite ti…
We consider the problems of maximizing the entanglement negativity of X-form qubit-qutrit density matrices with (i) a fixed spectrum and (ii) a fixed purity. In the first case, the problem is solved in full generality whereas, in the latter, partial solutions are obtained by imposing extra spectral constraints such as rank-deficiency and degeneracy, which enable a semidefinite programming treatment for the optimization problem at hand. Despite the technically-motivated assumptions, we provide strong numerical evidence that three-fold degenerate X states of purity $P$ reach the highest entangle…
By modifying the method of Bruss and Peres, we construct two new families of entangled two qutrit states. For all density matrices in these families the (i,j)th entry is 0 for i+j odd. The first family depends on 27 independent real parameters and includes both PPT and NPT states. The second family consists of PPT entangled states. The number of independent real parameters of this family is at least 11.
We have performed a Bell-type test for energy-time entangled qutrits. A method of inferring the Bell violation in terms of an associated interference visibility is derived. Using this scheme we obtained a Bell value of $2.784 \pm 0.023$, representing a violation of $34 σ$ above the limit for local variables. The scheme has been developed for use at telecom wavelengths and using proven long distance quantum communication architecture to optimize the utility of this high dimensional entanglement resource.
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…
We have developed a scheme to generate, control, transmit and measure entangled photonic qutrits (two photons each of dimension d = 3). A Bell test of this source has previously been reported elsewhere [1], therefore, here we focus on how the control of the system is realized. Motivated by these results, we outline how the scheme can be used for two specific quantum protocols, namely key distribution and coin tossing and discuss some of their advantages and disadvantages.
The general individual (non-coherent) attack on the ping-pong protocol with completely entangled pairs of three-dimensional quantum systems (qutrits) is analyzed. The expression for amount of the eavesdropper's information as functions from probability of attack detection is derived. It is shown, that the security of the ping-pong protocol with pairs of qutrits is higher the security of the protocol with pairs of qubits. It is also shown, that with the use by legitimate users in a control mode two mutually unbiased measuring bases the ping-pong protocol with pairs of qutrits, similar to the pr…
We have proposed a simple scheme to entangle two distant qutrits trapped in separate optical cavities. The quantum information of each qutrit is skillfully encoded on the degenerate ground states of a pair of atoms, hence the entanglement between them is relatively stable against spontaneous emission. In Lamb-Dicke limits, it is not necessary to require coincidence detections, which will relax the conditions for the experimental realization. The scheme is robust against the inefficient detections.
Distillability sudden death and sudden birth in a two-qutrit system under decoherence of finite temperature is studied in detail. By using of the negativity and realignment criterion, it is shown that certain initial prepared free entangled states may become bound entangled or separable states in a finite time. Moreover, initial prepared bound entangled or separable states also may become distillabile entangled states in a finite time.
A new scheme of quantum coding is presented. The scheme concerns the quantum states to which Schumacher's compression does not apply. It is shown that two qubits can be encoded in a single qutrit in such a way that one can faithfully reconstruct the state of one qubit. Decision on which of these two qubits is to be reconstructed can be made long after the encoding took place. The scheme succeeds with the average probability of 2/3.
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 …
We have experimentally realized a technique to generate, control and measure entangled qutrits, 3-dimensional quantum systems. This scheme uses spontaneous parametric down converted photons and unbalanced 3-arm fiber optic interferometers in a scheme analogous to the Franson interferometric arrangement for qubits. The results reveal a source capable of generating maximally entangled states with a net state fidelity, F = 0.985 $\pm$ 0.018. Further the control over the system reveals a high, net, 2-photon interference fringe visibility, V = 0.919 $\pm$ 0.026. This has all been done at telecom wa…
A multipartite quantum state is entangled if it is not separable. Quantum entanglement plays a fundamental role in many applications of quantum information theory, such as quantum teleportation. Stochastic local quantum operations and classical communication (SLOCC) cannot essentially change quantum entanglement without destroying it. Therefore, entanglement can be classified by dividing quantum states into equivalence classes, where two states are equivalent if each can be converted into the other by SLOCC. Properties of this classification, especially in the case of non two-dimensional quant…
The study of quantum cryptography and quantum non-locality have traditionnally been based on two-level quantum systems (qubits). In this paper we consider a generalisation of Ekert's cryptographic protocol [Ekert] where qubits are replaced by qutrits. The security of this protocol is related to non-locality, in analogy with Ekert's protocol. In order to study its robustness against the optimal individual attacks, we derive the information gained by a potential eavesdropper applying a cloning-based attack.
We investigate the entanglement properties of a three--parameter family of states that are part of the magic simplex of two qutrits, which is a simplex of states that are mixtures of maximally entangled two--qutrit Bell states. Using entanglement witnesses we reveal large regions of bound entangled and separable states.
The current work shows the efficacy of Dzyaloshinshkii-Moriya (DM) interaction to free the bound entanglement. Based on the work [Sharma, K.K., Pandey, S.N., Quant. Info. Proc. 15, 1539 (2016)], we present further results in two qutrits bound entangled state proposed by Jurkovaski et al. We consider a closed system of two qutrits and an auxiliary qutrit which interacts with either one of the two qutrits in a closed system. We erase the auxiliary qutrit from the system by doing partial trace operation and open system dynamics has been studied. We have found, the probability amplitude of auxilia…
The bound entangled state carries noisy entanglement and it is very hard to distill but the usefulness of bound entangled states has been depicted in different applications. This article represents a comparative dynamical study of an open quantum system for one of the bound entangled states proposed by Bennett et al. The study is conducted under the influence of Heisenberg, bi-linear bi-quadratic and Dzyaloshinskii-Moriya (DM) interaction. During the study, an auxiliary qutrit interacts with one of the qutrits of the selected two qutrit bound entangled state through different interactions. The…
Paralleling our recent computationally-intensive (quasi-Monte Carlo) work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric -- and also their analogous formulas (quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same seven billion well-distributed (``low-discrepancy'') sample points, w…
Classifying states as entangled or separable is a highly challenging task, while it is also one of the foundations of quantum information processing theory. This task is higly nontrivial even for relatively simple cases, such as two-qutrit Bell-diagonal states, i.e., mixture of nine mutually orthogonal maximally entangled states. In this article we apply the Gilbert algorithm to revise previously obtained results for this class. In particular we use ``cartography of entanglement'' to argue that most states left in [Hiesmayr, B. C. {\em Scientific Reports} {\bf 11}, 19739 (2021)] as unknown to …
Geometric relations between separable and entangled two-qubit and two-qutrit quantum information states are studied. To characterize entanglement of two qubit states, we establish a relation between reduced density matrix and the concurrence. For the rebit states, the geometrical meaning of concurrence as double area of a parallelogram is found and for generic qubit states it is expressed by determinant of the complex Hermitian inner product metric, where reduced density matrix coincides with the inner product metric. In the case of generic two-qutrit state, for reduced density matrix we find …
In this paper we present the results of numerical calculations about the minimal value of detection efficiency for violating the Clauser - Horne inequality for qutrits. Our results show how the use of non-maximally entangled states largely improves this limit respect to maximally entangled ones. A stronger resistance to noise is also found.
We use the linear programming algorithm introduced by Akulin et al. [V. M. Akulin, G. A. Kabatiansky, and A. Mandilara, Phys. Rev. A 92, 042322 (2015)] to perform best separable approximation on two-qutrit random density matrices. We combine the numerical results with theoretical methods in order to generate random representative families of positive partial transposed bound entangled (BE) states and analyze their properties. Our results are disclosing that for the two-qutrit system the BE states have negligible volume and that these form tiny `islands' sporadically distributed over the surfac…
There is no doubt about the fact that entanglement and nonlocality are distinct resources. It is acknowledged that a clear illustration of this point is the difference between maximally entangled states and states that maximally violate a Bell inequality. We give strong evidence that this anomaly may be an artefact of the measures that have been used to quantify nonlocality. By reasoning that the numeric value of a Bell function is akin to a witness rather than a quantifier, we define a measure of nonlocality and show that, for pairs of qutrits and of four-level systems, maximal entanglement d…
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…
Milz and Strunz ({\it J. Phys. A}: {\bf{48}} [2015] 035306) recently studied the probabilities that two-qubit and qubit-qutrit states, randomly generated with respect to Hilbert-Schmidt (Euclidean/flat) measure, are separable. They concluded that in both cases, the separability probabilities (apparently exactly $\frac{8}{33}$ in the two-qubit scenario) hold {\it constant} over the Bloch radii ($r$) of the single-qubit subsystems, jumping to 1 at the pure state boundaries ($r=1$). Here, firstly, we present evidence that in the qubit-qutrit case, the separability probability is uniformly distrib…
In this brief report we show the new Bell-Clauser-Horne inequality for two entangled three dimensional quantum systems (so called qutrits). This inequality is violated by a maximally entangled state of two qutrits observed via symmetric three input and three output port beamsplitter only if the amount of noise in the system equals ${11-6\sqrt3 \over 2}\approx 0.308$. This is in perfect agreement with the previous numerical calculations presented in Kaszlikowski {\it et. al.} Phys. Rev. Lett. {\bf 85}, 4418 (2000).
The entangled quantum state of a photon pair propagating through atmospheric turbulence suffers decay of entanglement due to the scintillation it experiences. Here we investigate the robustness against this decay for different qutrit states. We use an infinitesimal-propagation equation to obtain the density matrix as a function of the propagation distance and we use the tangle to quantify the entanglement between a pair of qutrits. The evolution of various initial states as they propagate through turbulence is considered. Using optimization of the initial parameters, we obtain expressions for …
We compare the classification as entangled or separable of Bell diagonal bipartite qudits with positive partial transposition (PPT) and their properties for different dimensions. For dimension $d \geq 3$, a form of entanglement exists that is hard to detect and called bound entanglement due to the fact that such entangled states cannot be used for entanglement distillation. Up to this date, no efficient solution is known to differentiate bound entangled from separable states. We address and compare this problem named separability problem for a family of bipartite Bell diagonal qudits with spec…
Bell's Theorem started with two qubits, spins $1/2$. It is a no-go statement on classical (local causal) models of quantum correlations. Only after 25 years, it turned out that for three qubits the situation is even more mind boggling. General statements concerning higher dimensional systems, qutrits etc., started to appear even later, once the spin (higher than $1/2$) picture was replaced by a more broad one, allowing all possible observables. This work is continuation of the Gdansk effort to take advantage of the fact that Bell Theorem can be put in form of a linear programming problem, whic…
