Entanglement types for two-qubit states with real amplitudes

Perdomo, Oscar; Leyton Ortega, Vicente; Perdomo-Ortiz, Alejandro

doi:10.1007/s11128-021-03025-z

Title: Entanglement types for two-qubit states with real amplitudes

Journal Article · Wed Mar 03 00:00:00 EST 2021 · Quantum Information Processing

DOI:https://doi.org/10.1007/s11128-021-03025-z· OSTI ID:1777764

Perdomo, Oscar ^[1]; Leyton Ortega, Vicente ^[2]; Perdomo-Ortiz, Alejandro ^[3]

Central Connecticut State Univ., New Britain, CT (United States); Rigetti Computing, Berkeley, CA (United States)
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Engineering Div.
Zapata Computing Canada, Inc., Toronto, ON (Canada); Univ. College London (United Kingdom). Dept. of Computer Science; Rigetti Computing, Berkeley, CA (United States)

We study the set of two-qubit pure states with real amplitudes and their geometrical representation in the three-dimensional sphere. In this representation, we show that the maximally entangled states—those locally equivalent to the Bell states—form two disjoint circles perpendicular to each other. We also show that taking the natural Riemannian metric on the sphere, the set of states connected by local gates are equidistant to this pair of circles. Moreover, the unentangled or so-called product states are π/4 units away to the maximally entangled states. This is, the unentangled states are the farthest away to the maximally entangled states. In this way, if we define two states to be equivalent if they are connected by local gates, we have that there are as many equivalent classes as points in the interval [0,π/4] with the point 0 corresponding to the maximally entangled states. The point π/4 corresponds to the unentangled states which geometrically are described by a torus. Finally, for every 0<π/4 the point d corresponds to a disjoint pair of torus. Finally, we also show how this geometrical interpretation allows to clearly see that any pair of two-qubit states with real amplitudes can be connected with a circuit that only has single-qubit gates and one controlled-Z gate.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1777764

Journal Information:: Quantum Information Processing, Vol. 20, Issue 3; ISSN 1570-0755

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

References (20)

Topology of the three-qubit space of entanglement types Walck, Scott N.; Glasbrenner, James K.; Lochman, Matthew H. Physical Review A, Vol. 72, Issue 5 https://doi.org/10.1103/PhysRevA.72.052324	journal	November 2005
Embedded minimal tori in S3 and the Lawson conjecture Brendle, Simon Acta Mathematica, Vol. 211, Issue 2 https://doi.org/10.1007/s11511-013-0101-2	journal	January 2013
Limited Holism and Real-Vector-Space Quantum Theory Hardy, Lucien; Wootters, William K. Foundations of Physics, Vol. 42, Issue 3 https://doi.org/10.1007/s10701-011-9616-6	journal	December 2011
Quantum circuits for isometries Iten, Raban; Colbeck, Roger; Kukuljan, Ivan Physical Review A, Vol. 93, Issue 3 https://doi.org/10.1103/PhysRevA.93.032318	journal	March 2016
Clifford 3-qubit states Perdomo, Oscar International Journal of Quantum Information, Vol. 18, Issue 03 https://doi.org/10.1142/S0219749920500045	journal	April 2020
Geometry of entanglement in the Bloch sphere Boyer, Michel; Liss, Rotem; Mor, Tal Physical Review A, Vol. 95, Issue 3 https://doi.org/10.1103/PhysRevA.95.032308	journal	March 2017
Quantum Steering Ellipsoids Jevtic, Sania; Pusey, Matthew; Jennings, David Physical Review Letters, Vol. 113, Issue 2 https://doi.org/10.1103/PhysRevLett.113.020402	journal	July 2014
A generative modeling approach for benchmarking and training shallow quantum circuits Benedetti, Marcello; Garcia-Pintos, Delfina; Perdomo, Oscar npj Quantum Information, Vol. 5, Issue 1 https://doi.org/10.1038/s41534-019-0157-8	journal	May 2019
On Multi-Particle Entanglement Linden, N.; Popescu, S.; Popescu, S. Fortschritte der Physik, Vol. 46, Issue 4-5 https://doi.org/10.1002/(SICI)1521-3978(199806)46:4/5<567::AID-PROP567>3.0.CO;2-H	journal	June 1998
Entanglement monotones Vidal, Guifré Journal of Modern Optics, Vol. 47, Issue 2-3 https://doi.org/10.1080/09500340008244048	journal	February 2000
Synthesis of quantum-logic circuits Shende, V. V.; Bullock, S. S.; Markov, I. L. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 25, Issue 6 https://doi.org/10.1109/TCAD.2005.855930	journal	June 2006
Optimal number of controlled-NOT gates to generate a three-qubit state Žnidarič, Marko; Giraud, Olivier; Georgeot, Bertrand Physical Review A, Vol. 77, Issue 3 https://doi.org/10.1103/PhysRevA.77.032320	journal	March 2008
Conditions for a Class of Entanglement Transformations Nielsen, M. A. Physical Review Letters, Vol. 83, Issue 2 https://doi.org/10.1103/PhysRevLett.83.436	journal	July 1999
Quantum steering ellipsoids, extremal physical states and monogamy Milne, Antony; Jevtic, Sania; Jennings, David New Journal of Physics, Vol. 16, Issue 8 https://doi.org/10.1088/1367-2630/16/8/083017	journal	August 2014
Geometry of Quantum States Bengtsson, Ingemar; Zyczkowski, Karol https://doi.org/10.1017/CBO9780511535048	book	January 2006
An elementary introduction to the geometry of quantum states with pictures Avron, J.; Kenneth, O. Reviews in Mathematical Physics, Vol. 32, Issue 02 https://doi.org/10.1142/S0129055X20300010	journal	August 2019
Exact and asymptotic measures of multipartite pure-state entanglement Bennett, Charles H.; Popescu, Sandu; Rohrlich, Daniel Physical Review A, Vol. 63, Issue 1 https://doi.org/10.1103/PhysRevA.63.012307	journal	December 2000
Quantum-state preparation with universal gate decompositions Plesch, Martin; Brukner, Časlav Physical Review A, Vol. 83, Issue 3 https://doi.org/10.1103/PhysRevA.83.032302	journal	March 2011
Geometric quantum mechanics Brody, Dorje C.; Hughston, Lane P. Journal of Geometry and Physics, Vol. 38, Issue 1 https://doi.org/10.1016/S0393-0440(00)00052-8	journal	April 2001
The unknottedness of minimal embeddings Lawson, H. Blaine Inventiones mathematicae, Vol. 11, Issue 3 https://doi.org/10.1007/bf01404649	journal	September 1970

Similar Records

Geometric theory of nonlocal two-qubit operations

Journal Article · Tue Apr 01 00:00:00 EST 2003 · Physical Review. A · OSTI ID:1777764

Jun, Zhang; Sastry, Shankar; Vala, Jiri; +1 more

Tripartite entanglement in qudit stabilizer states and application in quantum error correction

Journal Article · Tue Nov 15 00:00:00 EST 2011 · Physical Review. A · OSTI ID:1777764

Looi, Shiang Yong; Griffiths, Robert B.

Resource-efficient generation of linear cluster states by linear optics with postselection

Journal Article · Fri Jan 30 00:00:00 EST 2015 · Journal of Physics. B, Atomic, Molecular and Optical Physics · OSTI ID:1777764

Uskov, D. B.; Alsing, P. M.; Fanto, M. L.; +4 more

Related Subjects

97 MATHEMATICS AND COMPUTING
quantum state preparation
geometry of entanglement
quantum entanglement

Title: Entanglement types for two-qubit states with real amplitudes

Citation Formats

References (20)

Similar Records

Related Subjects