scholarly journals Large violations in Kochen Specker contextuality and their applications

2021 ◽  
Author(s):  
Ravishankar Ramanathan ◽  
Yuan Liu ◽  
Pawel Horodecki
Keyword(s):  
Quantum Mechanics ◽  
Present State ◽  
Hilbert Spaces ◽  
Independent Interest ◽  
Entangled State ◽  
Maximally Entangled State ◽  
Technical Result ◽  
Orthogonal Representation ◽  
State Dependent ◽  
Minimum Dimension

Abstract It is of interest to study how contextual quantum mechanics is, in terms of the violation of Kochen Specker state-independent and state-dependent non-contextuality inequalities. We present state-independent non-contextuality inequalities with large violations, in particular, we exploit a connection between Kochen-Specker proofs and pseudo-telepathy games to show KS proofs in Hilbert spaces of dimension $d \geq 2^{17}$ with the ratio of quantum value to classical bias being $O(\sqrt{d}/\log d)$. We study the properties of this KS set and show applications of the large violation. It has been recently shown that Kochen-Specker proofs always consist of substructures of state-dependent contextuality proofs called $01$-gadgets or bugs. We show a one-to-one connection between $01$-gadgets in $\mathbb{C}^d$ and Hardy paradoxes for the maximally entangled state in $\mathbb{C}^d \otimes \mathbb{C}^d$. We use this connection to construct large violation $01$-gadgets between arbitrary vectors in $\mathbb{C}^d$, as well as novel Hardy paradoxes for the maximally entangled state in $\mathbb{C}^d \otimes \mathbb{C}^d$, and give applications of these constructions. As a technical result, we show that the minimum dimension of the faithful orthogonal representation of a graph in $\mathbb{R}^d$ is not a graph monotone, a result that may be of independent interest.

scholarly journals Optimal teleportation via noisy quantum channels without additional qubit resources

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Dong-Gil Im ◽  
Chung-Hyun Lee ◽  
Yosep Kim ◽  
Hyunchul Nha ◽  
M. S. Kim ◽  
...  
Keyword(s):  
Quantum Computing ◽  
Quantum Teleportation ◽  
Quantum Communication ◽  
Quantum Noise ◽  
Single Copy ◽  
Entangled State ◽  
Quantum Network ◽  
Quantum Channels ◽  
Maximally Entangled State ◽  
Near Term

AbstractQuantum teleportation exemplifies how the transmission of quantum information starkly differs from that of classical information and serves as a key protocol for quantum communication and quantum computing. While an ideal teleportation protocol requires noiseless quantum channels to share a pure maximally entangled state, the reality is that shared entanglement is often severely degraded due to various decoherence mechanisms. Although the quantum noise induced by the decoherence is indeed a major obstacle to realizing a near-term quantum network or processor with a limited number of qubits, the methodologies considered thus far to address this issue are resource-intensive. Here, we demonstrate a protocol that allows optimal quantum teleportation via noisy quantum channels without additional qubit resources. By analyzing teleportation in the framework of generalized quantum measurement, we optimize the teleportation protocol for noisy quantum channels. In particular, we experimentally demonstrate that our protocol enables to teleport an unknown qubit even via a single copy of an entangled state under strong decoherence that would otherwise preclude any quantum operation. Our work provides a useful methodology for practically coping with decoherence with a limited number of qubits and paves the way for realizing noisy intermediate-scale quantum computing and quantum communication.

Infinitely entangled states

2003 ◽  
Vol 3 (4) ◽  
pp. 281-306
Author(s):  
M. Keyl ◽  
D. Schlingemann ◽  
R.F. Werner
Keyword(s):  
Hilbert Spaces ◽  
Degrees Of Freedom ◽  
Entangled States ◽  
Entangled State ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Density Operators ◽  
Fundamental Paper ◽  
Extended Notion

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

Bell inequality for quNits with binary measurements

2003 ◽  
Vol 3 (2) ◽  
pp. 157-164
Author(s):  
H. Bechmann-Pasquinucci ◽  
N. Gisin
Keyword(s):  
Quantum Cryptography ◽  
Bell Inequality ◽  
The Other ◽  
Entangled State ◽  
Maximally Entangled State ◽  
Von Neumann ◽  
Chsh Inequality ◽  
Intermediate States ◽  
Von Neumann Measurements

We present a generalized Bell inequality for two entangled quNits. On one quNit the choice is between two standard von Neumann measurements, whereas for the other quNit there are N^2 different binary measurements. These binary measurements are related to the intermediate states known from eavesdropping in quantum cryptography. The maximum violation by \sqrt{N} is reached for the maximally entangled state. Moreover, for N=2 it coincides with the familiar CHSH-inequality.

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260005 ◽  
Author(s):  
GIANNI CASSINELLI ◽  
PEKKA LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Problem ◽  
Partial Solution ◽  
Space Realization ◽  
Symmetry Transformations ◽  
Axiomatic Quantum Mechanics

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

Linear Algebra and the Dirac Notation

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Vector Spaces ◽  
Complex Vector ◽  
Dual Vector ◽  
Finite Dimensional ◽  
Basic Facts

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.

scholarly journals Entanglement dynamics for static two-level atoms in cosmic string spacetime

2020 ◽  
Vol 80 (2) ◽  
Author(s):  
Pingyang He ◽  
Hongwei Yu ◽  
Jiawei Hu
Keyword(s):  
Cosmic String ◽  
Minkowski Spacetime ◽  
Scalar Fields ◽  
Entangled State ◽  
Entanglement Dynamics ◽  
Entanglement Generation ◽  
Maximally Entangled State ◽  
Interatomic Separation ◽  
Antisymmetric State ◽  
String Distance

Abstract We study the entanglement dynamics of two static atoms coupled with a bath of fluctuating scalar fields in vacuum in the cosmic string spacetime. Three different alignments of atoms, i.e. parallel, vertical, and symmetric alignments with respect to the cosmic string are considered. We focus on how entanglement degradation and generation are influenced by the cosmic string, and find that they are crucially dependent on the atom-string distance r, the interatomic separation L, and the parameter $$\nu $$ν that characterizes the nontrivial topology of the cosmic string. For two atoms initially in a maximally entangled state, the destroyed entanglement can be revived when the atoms are aligned vertically to the string, which cannot happen in the Minkowski spacetime. When the symmetrically aligned two-atom system is initially in the antisymmetric state, the lifetime of entanglement can be significantly enhanced as $$\nu $$ν increases. For two atoms which are initially in the excited state, when the interatomic separation is large compared to the transition wavelength, entanglement generation cannot happen in the Minkowski spacetime, while it can be achieved in the cosmic string spacetime when the position of the two atoms is appropriate with respect to the cosmic string and $$\nu $$ν is large enough.

QUANTUM TELEPORTATION OF AN ARBITRARY N-QUBIT STATE VIA NON-MAXIMALLY ENTANGLED STATE

2007 ◽  
Vol 05 (05) ◽  
pp. 673-683 ◽  
Author(s):  
YU-LING LIU ◽  
ZHONG-XIAO MAN ◽  
YUN-JIE XIA
Keyword(s):  
Quantum Teleportation ◽  
Success Probability ◽  
Qubit State ◽  
Bell States ◽  
Entangled State ◽  
Quantum Channels ◽  
Maximally Entangled State ◽  
Ghz States ◽  
Measurement Basis ◽  
Unit Probability

We explicitly present two schemes for quantum teleportation of an arbitrary N-qubit entangled state using, respectively, non-maximally entangled Bell states and GHZ states as the quantum channels, and generalized Bell states as the measurement basis. The scheme succeeds with unit fidelity but less than unit probability. By introducing additional qubit and unitary operations, the success probability of these two schemes can be increased.

scholarly journals MAXIMALLY ENTANGLED STATES AND BELL'S INEQUALITY IN RELATIVISTIC REGIME

2009 ◽  
Vol 07 (01) ◽  
pp. 395-401 ◽  
Author(s):  
SHAHPOOR MORADI
Keyword(s):  
Ghz State ◽  
Entangled States ◽  
Entangled State ◽  
Maximally Entangled State ◽  
Expectation Value ◽  
Spin Observables ◽  
Chsh Inequality ◽  
Relativistic Regime ◽  
Maximal Violation ◽  
Maximally Entangled States

In this letter we show that in the relativistic regime, maximally entangled state of two spin-1/2 particles not only gives maximal violation of the Bell-CHSH inequality but also gives the largest violation attainable for any pairs of four spin observables that are noncommuting for both systems. Also, we extend our results to three spin-1/2 particles. We obtain the largest eigenvalue of Bell operator and show that this value is equal to the expectation value of Bell operator on GHZ state.

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