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.
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 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…
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 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…
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 …
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…
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…
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 …
Based on the fact that the entanglement can not be created locally, we proposed a quantum bit commitment protocol, in which entangled states and quantum algorithms is used. The bit is not encoded with the form of the quantum states, and delaying the measurement is required. Therefore the protocol will not be denied by the Mayers-Lo-Chau no-go theorem, and unconditional security is achieved.
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…
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).
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…
We propose a way to prepare Greenberger-Horne-Zeilinger (GHZ) entangled photon Fock states of three cavities, by using a superconducting flux qutrit coupled to the cavities. This proposal does not require the use of classical microwave pulses and measurement during the entire operation. Thus, the operation is greatly simplified and the circuit engineering complexity and cost is much reduced. The proposal is quite general and can be applied to generate three-cavity GHZ entangled photon Fock states when the three cavities are coupled by a different three-level physical system such as a supercond…
We construct $3\otimes 3$ PPT entangled edge states with maximal ranks, to complete the classification of $3\otimes 3$ PPT entangled edge states by their types. The ranks of the states and their partial transposes are 8 and 6, respectively. These examples also disprove claims in the literature.
We show how one can prepare three-qubit entangled states like W states, Greenberger-Horne-Zeilinger states as well as two-qutrit entangled states using the multiatom two-mode entanglement. We propose a technique of preparing such a multi-particle entanglement using stimulated Raman adiabatic passage. We consider a collection of three-level atoms in $Λ$ configuration simultaneously interacting with a resonant two-mode cavity for this purpose. Our approach permits a variety of multiparticle extensions.
The possibility of recovering and protecting the entanglement of accelerated 2-qubit and 2-qutrit systems is discussed using weak-reverse measurements. The accelerated partial entangled states are more responsive to be protected than the accelerated maximum entangled states. The accelerated coded local information in qutrit system is more robust than that encoded in a 2-qubit system, and it can be conserved even for larger values of accelerations. Meanwhile, the non-accelerated information in qubit system is not affected by the local operation compared with that depicted on qutrit system. For …
A complete orthonormal basis of N-qutrit unitary operators drawn from the Pauli Group consists of the identity and 9^N-1 traceless operators. The traceless ones partition into 3^N+1 maximally commuting subsets (MCS's) of 3^N-1 operators each, whose joint eigenbases are mutually unbiased. We prove that Pauli factor groups of order 3^N are isomorphic to all MCS's, and show how this result applies in specific cases. For two qutrits, the 80 traceless operators partition into 10 MCS's. We prove that 4 of the corresponding basis sets must be separable, while 6 must be totally entangled (and Bell-lik…
We construct entanglement witnesses with regard to the geometric structure of the Hilbert--Schmidt space and investigate the geometry of entanglement. In particular, for a two--parameter family of two--qutrit states that are part of the magic simplex we calculate the Hilbert--Schmidt measure of entanglement. We present a method to detect bound entanglement which is illustrated for a three--parameter family of states. In this way we discover new regions of bound entangled states. Furthermore we outline how to use our method to distinguish entangled from separable states.
The insight due to John Bell that the joint behavior of individually measured entangled quantum systems cannot be explained by shared information remains a mystery to this day. We describe an experiment, and its analysis, displaying non-locality of entangled qutrit pairs. The non-locality of such systems, as compared to qubit pairs, is of particular interest since it potentially opens the door for tests of bipartite non-local behavior independent of probabilistic Bell inequalities, but of deterministic nature.
