Clifford 3-qubit states

Let us denote by [Formula: see text] the Clifford group (the circuit or operations generated by Hadamard, [Formula: see text] phase and the controlled-NOT gates) and by [Formula: see text] the set of qubit states that can be prepared by circuits from the Clifford group. In other words, a state [Formula: see text] if [Formula: see text] where [Formula: see text]. We will refer to states in [Formula: see text] as Clifford states. This paper studies the set of all three-qubit Clifford states. We prove that [Formula: see text] has 8640 states and if we define two states [Formula: see text] and [Formula: see text] in [Formula: see text] to be equivalent if [Formula: see text], with [Formula: see text] a local transformation in [Formula: see text], then the resulting quotient space has five orbits. More exactly, [Formula: see text] where the orbit [Formula: see text] is made up of states with entanglement entropy [Formula: see text]. For example, the first orbit [Formula: see text] contains the state [Formula: see text] and corresponds to the unentangled Clifford states. We say that [Formula: see text] is a real state if all its amplitudes [Formula: see text] are real numbers. We also say that an operator is real if all the entries of its matrix representation with respect to the computational basis are real numbers. In this paper, we also study the set of real Clifford 3 qubits and the way this set splits when we identify two real Clifford states [Formula: see text] and [Formula: see text] to be equivalent if [Formula: see text] where [Formula: see text] is a local real Clifford operator. An interesting aspect that follows from this study of Clifford states is the existence of two real Clifford states [Formula: see text] and [Formula: see text] that can be connected with a Clifford local transformation but they cannot be connected with a real Clifford local transformation. This is, the equation [Formula: see text] for [Formula: see text], does have a solution in the set of local transformations from [Formula: see text] but it does not have a solution among the local transformations from [Formula: see text] that are real. We go a little deeper and show that the equation [Formula: see text] does not have a solution for any local operation (not necessarily Clifford) whose entries are real numbers. Finally, we show how the CNOT gates act on the set of Clifford states and also in the set of real Clifford states.

Download Full-text

Measuring the Tangle of Three-Qubit States

Entropy ◽

10.3390/e22040436 ◽

2020 ◽

Vol 22 (4) ◽

pp. 436 ◽

Cited By ~ 2

Author(s):

Adrián Pérez-Salinas ◽

Diego García-Martín ◽

Carlos Bravo-Prieto ◽

José Latorre

Keyword(s):

Direct Measurement ◽

Canonical Form ◽

Quantum Circuit ◽

Quantum Gates ◽

Tripartite Entanglement ◽

Single Qubit ◽

Optimal Values ◽

Random States ◽

Qubit States ◽

Computational Basis

We present a quantum circuit that transforms an unknown three-qubit state into its canonical form, up to relative phases, given many copies of the original state. The circuit is made of three single-qubit parametrized quantum gates, and the optimal values for the parameters are learned in a variational fashion. Once this transformation is achieved, direct measurement of outcome probabilities in the computational basis provides an estimate of the tangle, which quantifies genuine tripartite entanglement. We perform simulations on a set of random states under different noise conditions to asses the validity of the method.

Download Full-text

THREE-QUBIT QUANTUM ENTANGLEMENT FOR SI (S = 3/2, I = 1/2) SPIN SYSTEM

International Journal of Quantum Information ◽

10.1142/s0219749911008313 ◽

2011 ◽

Vol 09 (07n08) ◽

pp. 1635-1642 ◽

Cited By ~ 1

Author(s):

A. GÜN ◽

A. GENÇTEN

Keyword(s):

Quantum Information ◽

Spin System ◽

Quantum Information Processing ◽

Matrix Representation ◽

Pulse Sequences ◽

Logic Gates ◽

Entangled States ◽

Spin States ◽

The Matrix ◽

Qubit States

In quantum information processing, spin-3/2 electron or nuclear spin states are known as two-qubit states. For SI (S = 3/2, I = 1/2) spin system, there are eight three-qubit states. In this study, first, three-qubit CNOT logic gates are obtained. Then three-qubit entangled states are obtained by using the matrix representation of Hadamard and three-qubit CNOT logic gates. By considering single 31P@C60 molecule as SI (S = 3/2, I = 1/2) spin system, three-qubit entangled states are also obtained using the magnetic resonance pulse sequences of Hadamard and CNOT logic gates.

Download Full-text

Entanglement entropy in quasi-symmetric multi-qubit states

International Journal of Quantum Information ◽

10.1142/s0219749915500070 ◽

2015 ◽

Vol 13 (02) ◽

pp. 1550007 ◽

Cited By ~ 1

Author(s):

Zhi-Hua Li ◽

An-Min Wang

Keyword(s):

Phase Transition ◽

Quantum Phase Transition ◽

Entanglement Entropy ◽

Ground States ◽

Quantum Phase ◽

Transition Points ◽

Qubit States ◽

Basis Vectors ◽

Dicke States

We generalize the symmetric multi-qubit states to their q-analogs, whose basis vectors are identified with the q-Dicke states. We study the entanglement entropy in these states and find that entanglement is extruded towards certain regions of the system due to the inhomogeneity aroused by q-deformation. We also calculate entanglement entropy in ground states of a related q-deformed Lipkin–Meshkov–Glick (LMG) model and show that the singularities of entanglement can correctly signify the quantum phase transition points for different strengths of q-deformation.

Download Full-text

Improving the Robustness of Entangled States by Basis Transformation

Entropy ◽

10.3390/e21010059 ◽

2019 ◽

Vol 21 (1) ◽

pp. 59 ◽

Cited By ~ 1

Author(s):

Xin-Wen Wang ◽

Shi-Qing Tang ◽

Yan Liu ◽

Ji-Bing Yuan

Keyword(s):

Probability Distribution ◽

Quantum Entanglement ◽

Error Distribution ◽

Entangled States ◽

Noise Strength ◽

Practical Application ◽

Distribution Parameters ◽

The Right ◽

Qubit States ◽

Computational Basis

In the practical application of quantum entanglement, entangled particles usually need to be distributed to many distant parties or stored in different quantum memories. In these processes, entangled particles unavoidably interact with their surrounding environments, respectively. We here systematically investigate the entanglement-decay laws of cat-like states under independent Pauli noises with unbalanced probability distribution of three kinds of errors. We show that the robustness of cat-like entangled states is not only related to the overall noise strength and error distribution parameters, but also to the basis of qubits. Moreover, we find that whether a multi-qubit state is more robust in the computational basis or transversal basis depends on the initial entanglement and number of qubits of the state as well as the overall noise strength and error distribution parameters of the environment. However, which qubit basis is conductive to enhancing the robustness of two-qubit states is only dependent on the error distribution parameters. These results imply that one could improve the intrinsic robustness of entangled states by simply transforming the qubit basis at the right moment. This robustness-improving method does not introduce extra particles and works in a deterministic manner.

Download Full-text

The Prime state and its quantum relatives

Quantum ◽

10.22331/q-2020-12-11-371 ◽

2020 ◽

Vol 4 ◽

pp. 371

Author(s):

D. García-Martín ◽

E. Ribas ◽

S. Carrazza ◽

J.I. Latorre ◽

G. Sierra

Keyword(s):

Entanglement Entropy ◽

Prime Numbers ◽

Quantum States ◽

Direct Access ◽

Distribution Of Primes ◽

Quantum Fourier Transform ◽

Arithmetic Properties ◽

Primes In Arithmetic Progressions ◽

Arbitrary Precision ◽

Computational Basis

The Prime state of n qubits, |Pn⟩, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.

Download Full-text

Bell’s inequality, generalized concurrence and entanglement in qubits

International Journal of Modern Physics A ◽

10.1142/s0217751x19500325 ◽

2019 ◽

Vol 34 (06n07) ◽

pp. 1950032 ◽

Cited By ~ 1

Author(s):

Po-Yao Chang ◽

Su-Kuan Chu ◽

Chen-Te Ma

Keyword(s):

Upper Bound ◽

Entanglement Entropy ◽

Qubit State ◽

Bell’S Inequality ◽

Bell's Inequality ◽

Qubit System ◽

Maximal Violation ◽

Topological Entanglement Entropy ◽

Qubit States

It is well known that the maximal violation of the Bell’s inequality for a two-qubit system is related to the entanglement formation in terms of a concurrence. However, a generalization of this relation to an [Formula: see text]-qubit state has not been found. In this paper, we demonstrate some extensions of the relation between the upper bound of the Bell’s violation and a generalized concurrence in several [Formula: see text]-qubit states. In particular, we show the upper bound of the Bell’s violation can be expressed as a function of the generalized concurrence, if a state can be expressed in terms of two variables. We apply the relation to the Wen-Plaquette model and show that the topological entanglement entropy can be extracted from the maximal Bell’s violation.

Download Full-text