Local Deterministic Transformations of Three-Qubit Pure States

Journal of Physics A: Mathematical and General, 2001

We study invariants of three-qubit states under local unitary transformations, i.e. functions on the space of entanglement types, which is known to have dimension 6. We show that there is no set of six algebraically independent polynomial invariants of degree ≤ 6, and find such a set with maximum degree 8. We describe an intrinsic definition of a canonical state on each orbit, and discuss the (non-polynomial) invariants associated with it.

The European Physical Journal D

Detection and classification of entanglement properties of a multi-qubit system is a topic of great interest. This topic has been studied extensively and thus we found different approaches for the detection and classification of multi-qubit entangled states. We have applied partial transposition operation on one of the qubit of the three-qubit system and then studied the entanglement properties of the three-qubit system, which is under investigation. Since the partial transposition operation is not a quantum operation so we have approximated partial transposition operation in such a way so that it represent a completely positive map. The approximated partial transposition operation is also known as structural physical approximation of partial transposition (SPA-PT). We have studied in detail the application of SPA-PT on a three qubit system and provided explicitly the matrix elements of the density matrix describing SPA-PT of a three qubit system. Moreover, we propose criterion to classify all possible stochastic local operations and classical communication(SLOCC) inequivalent classes of a pure as well as mixed three qubit state through SPA-PT map, which makes our criterion experimentally realizable. We have illustrated our criterion for detection and classification of three-qubit entangled states by considering few examples.

2012

Using techniques from symplectic geometry, we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one-particle reduced density matrices exactly when their ordered spectra belong to the boundary of the, so called, Kirwan polytope. Otherwise, the states with given reduced density matrices are parameterized, up to local unitary equivalence, by two real variables. Given inevitable experimental imprecisions, this means that already for three qubits a pure quantum state can never be reconstructed from single-particle tomography. We moreover show that knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the Greenberger-Horne-Zeilinger (GHZ) state by stochastic local operations and classical communication (SLOCC), and discuss generalizations of our results to an arbitary number of qubits.

arXiv: Quantum Physics, 2019

Agrawal and Pati \cite{agrawal} had shown that there exist a class of three-qubit W-states, which act as a resource state for the perfect teleportation of a single qubit. They have used three-qubit measurement basis to execute their teleportation protocol. In this work, we modify Agrawal-Pati teleportation protocol. Our modified teleportation protocol is based on two-qubit measurement basis for the perfect teleportation of a single qubit. We deduce the condition for the successful execution of modified teleportation protocol and this leads us to new class of three-qubit W-states, which is required as a resource state to achieve the modified perfect teleportation protocol. We have constructed operators that can be used to verify the condition of teleportation in experiment. This verification is necessary for the detection of whether the given three-qubit state is useful in modified teleportation protocol or not. Then we have shown that the set containing three-qubit W-states used in ...

2008

We look for local unitary operators $W_1 \otimes W_2$ which would rotate all equally entangled two-qubit pure states by the same but arbitrary amount. It is shown that all two-qubit maximally entangled states can be rotated through the same but arbitrary amount by local unitary operators. But there is no local unitary operator which can rotate all equally entangled non-maximally

Journal of Physics A: Mathematical and General, 2000

Entanglement types of pure states of three spin-1 2 particles are classified by means of their stabilisers in the group of local unitary transformations. It is shown that the stabiliser is generically discrete, and that a larger stabiliser indicates a stationary value for some local invariant. We describe all the exceptional states with enlarged stabilisers.

Entropy, 2018

We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those inva...

Chinese Physics B, 2013

We propose a novel deterministic protocol that two senders are capable of remotely preparing arbitrary two-and three-qubit states for a remote receiver using EPR pairs and GHZ state as the quantum channel. Compared with the existing deterministic protocols [An et al. 2011 Phys. Lett. A 375 3570 and Chen et al. 2012 J. Phys. A: Math. Theor. 45 055303], the quantum resources and classical information in our scheme are decreased, and the whole operation process is simplified.

Physical Review A, 2000

Invertible local transformations of a multipartite system are used to define equivalence classes in the set of entangled states. This classification concerns the entanglement properties of a single copy of the state. Accordingly, we say that two states have the same kind of entanglement if both of them can be obtained from the other by means of local operations and classical communication (LOCC) with nonzero probability. When applied to pure states of a three-qubit system, this approach reveals the existence of two inequivalent kinds of genuine tripartite entanglement, for which the GHZ state and a W state appear as remarkable representatives. In particular, we show that the W state retains maximally bipartite entanglement when any one of the three qubits is traced out. We generalize our results both to the case of higher dimensional subsystems and also to more than three subsystems, for all of which we show that, typically, two randomly chosen pure states cannot be converted into each other by means of LOCC, not even with a small probability of success.

Physical Review A, 2003

We study non-local two-qubit operations from a geometric perspective. By applying a Cartan decomposition to su(4), we find that the geometric structure of non-local gates is a 3-Torus. We derive the invariants for local transformations, and connect these local invariants to the coordinates of the 3-Torus. Since different points on the 3-Torus may correspond to the same local equivalence class, we use the Weyl group theory to reduce the symmetry. We show that the local equivalence classes of two-qubit gates are in one-to-one correspondence with the points in a tetrahedron except on the base. We then study the properties of perfect entanglers, that is, the two-qubit operations that can generate maximally entangled states from some initially separable states. We provide criteria to determine whether a given two-qubit gate is a perfect entangler and establish a geometric description of perfect entanglers by making use of the tetrahedral representation of non-local gates. We find that exactly half the non-local gates are perfect entanglers. We also investigate the nonlocal operations generated by a given Hamiltonian. We first study the gates that can be directly generated by a Hamiltonian. Then we explicitly construct a quantum circuit that contains at most three non-local gates generated by a two-body interaction Hamiltonian, together with at most four local gates generated by single qubit terms. We prove that such a quantum circuit can simulate any arbitrary two-qubit gate exactly, and hence it provides an efficient implementation of universal quantum computation and simulation.