scholarly journals Symmetries and Geometries of Qubits, and Their Uses

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1732
Author(s):  
A. R. P. Rau
Keyword(s):  
Quantum Information ◽  
Quantum Physics ◽  
Quantum Correlations ◽  
Combinatorial Designs ◽  
Finite Projective Geometries ◽  
Projective Geometries ◽  
Higher Dimensional ◽  
Erlangen Program ◽  
Higher Spin ◽  
Further Development

The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits.

scholarly journals FINITE GEOMETRIES WITH QUBIT OPERATORS

2009 ◽  
Vol 12 (02) ◽  
pp. 359-366 ◽  
Author(s):  
AMBAR N. SENGUPTA
Keyword(s):  
Clifford Algebras ◽  
Quantum Gate ◽  
Finite Geometries ◽  
Fano Plane ◽  
Finite Projective Geometries ◽  
Composite Systems ◽  
Projective Geometries ◽  
Higher Dimensional

Finite projective geometries, especially the Fano plane, have been observed to arise in the context of certain quantum gate operators. We use Clifford algebras to explain why these geometries, both planar and higher dimensional, appear in the context of multi-qubit composite systems.

scholarly journals Experimentally Demonstrate the Spin-1 Information Entropic Inequality Based on Simulated Photonic Qutrit States

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 219
Author(s):  
Lianzhen Cao ◽  
Xia Liu ◽  
Yang Yang ◽  
Qinwei Zhang ◽  
Jiaqiang Zhao ◽  
...  
Keyword(s):  
Quantum Information ◽  
Quantum Correlation ◽  
Quantum Information Processing ◽  
Quantum Correlations ◽  
Quantum Systems ◽  
Enhancement Effect ◽  
Nonlinear Optical Effect ◽  
Higher Dimensional ◽  
Entropic Inequality ◽  
First Time

Quantum correlations of higher-dimensional systems are an important content of quantum information theory and quantum information application. The quantification of quantum correlation of high-dimensional quantum systems is crucial, but difficult. In this paper, using the second-order nonlinear optical effect and multiphoton interference enhancement effect, we experimentally implement the photonic qutrit states and demonstrate the spin-1 information entropic inequality for the first time to quantitative quantum correlation. Our work shows that information entropy is an important way to quantify quantum correlation and quantum information processing.

scholarly journals Quantum Computation and Arrows of Time

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 49
Author(s):  
Nathan Argaman
Keyword(s):  
Quantum Computation ◽  
Quantum Physics ◽  
Bell’S Theorem ◽  
Arrow Of Time ◽  
Bell's Theorem ◽  
Classical Domain ◽  
Further Development

Quantum physics is surprising in many ways. One surprise is the threat to locality implied by Bell’s Theorem. Another surprise is the capacity of quantum computation, which poses a threat to the complexity-theoretic Church-Turing thesis. In both cases, the surprise may be due to taking for granted a strict arrow-of-time assumption whose applicability may be limited to the classical domain. This possibility has been noted repeatedly in the context of Bell’s Theorem. The argument concerning quantum computation is described here. Further development of models which violate this strong arrow-of-time assumption, replacing it by a weaker arrow which is yet to be identified, is called for.

Non-Markovianity leading to coherence revivals in an open quantum system

2020 ◽  
pp. 2050040
Author(s):  
Y. Yugra ◽  
F. De Zela
Keyword(s):  
Quantum Information ◽  
Quantum System ◽  
Open Quantum System ◽  
Open Systems ◽  
Quantum Correlations ◽  
Experimental Proof ◽  
Optical Setup ◽  
Open Quantum ◽  
All Optical ◽  
Proof Of Principle

Coherence and quantum correlations have been identified as fundamental resources for quantum information tasks. As recently shown, these resources can be interconverted. In multipartite systems, entanglement represents a prominent case among quantum correlations, one which can be activated from coherence. All this makes coherence a key resource for securing the operational advantage of quantum technologies. When dealing with open systems, decoherence hinders full exploitation of quantum resources. Here, we present a protocol that allows reaching the maximal achievable amount of coherence in an open quantum system. By implementing our protocol, or suitable variants of it, coherence losses might be fully compensated, thereby leading to coherence revivals. We provide an experimental proof of principle of our protocol through its implementation with an all-optical setup.

Finite Projective Geometries

1952 ◽  
Vol 4 ◽  
pp. 302-313 ◽  
Author(s):  
Gerald Berman
Keyword(s):  
Explicit Construction ◽  
Primitive Elements ◽  
Galois Fields ◽  
Finite Projective Geometries ◽  
Linear Subspaces ◽  
Projective Geometries

James Singer [12] has shown that there exists a collineation which is transitive on the (t - 1)-spaces, that is, (t - 1)-dimensional linear subspaces, of PG(t, pn). In this paper we shall generalize this result showing that there exist t - r collineations which together are transitive on the s-spaces of PG(t, pn). An explicit construction will be given for such a set of collineations with the aid of primitive elements of Galois fields. This leads to a calculus for the linear subspaces of finite projective geometries.

scholarly journals Reality, Indeterminacy, Probability, and Information in Quantum Theory

Entropy ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. 747
Author(s):  
Arkady Plotnitsky
Keyword(s):  
Information Theory ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Information ◽  
Quantum Correlations ◽  
Quantum Information Theory ◽  
Quantum Phenomena

Following the view of several leading quantum-information theorists, this paper argues that quantum phenomena, including those exhibiting quantum correlations (one of their most enigmatic features), and quantum mechanics may be best understood in quantum-informational terms. It also argues that this understanding is implicit already in the work of some among the founding figures of quantum mechanics, in particular W. Heisenberg and N. Bohr, half a century before quantum information theory emerged and confirmed, and gave a deeper meaning to, to their insights. These insights, I further argue, still help this understanding, which is the main reason for considering them here. My argument is grounded in a particular interpretation of quantum phenomena and quantum mechanics, in part arising from these insights as well. This interpretation is based on the concept of reality without realism, RWR (which places the reality considered beyond representation or even conception), introduced by this author previously, in turn, following Heisenberg and Bohr, and in response to quantum information theory.

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