Access the full text.
Sign up today, get DeepDyve free for 14 days.
E. Størmer (2018)
Extension of positive mapsMATHEMATICA SCANDINAVICA
A. Sanpera, D. Bruß, M. Lewenstein (2000)
Schmidt number witnesses and bound entanglementPhysical Review A, 63
(2008)
Linear Algebra Appl
P. Horodecki (1997)
Separability criterion and inseparable mixed states with positive partial transpositionPhysics Letters A, 232
W. Arveson (1969)
On subalgebras of $C^*$-algebrasBulletin of the American Mathematical Society, 75
Lukasz Skowronek, E. Størmer, K. Życzkowski (2009)
Cones of positive maps and their duality relationsJournal of Mathematical Physics, 50
M. Rahaman, Samuel Jaques, V. Paulsen (2018)
Eventually entanglement breaking mapsJournal of Mathematical Physics
Mathew Kennedy, Nicholas Manor, V. Paulsen (2017)
Composition of PPT mapsQuantum Inf. Comput., 18
Kang-Da Wu, Tulja Kondra, S. Rana, Carlo Scandolo, G. Xiang, Chuan‐Feng Li, G. Guo, A. Streltsov (2020)
Operational Resource Theory of Imaginarity.Physical review letters, 126 9
A. Peres (1996)
Separability Criterion for Density Matrices.Physical review letters, 77 8
W. Wootters (2014)
The rebit three-tangle and its relation to two-qubit entanglementJournal of Physics A: Mathematical and Theoretical, 47
Jonathan Rosenberg (2015)
Structure and applications of real C*-algebrasarXiv: Operator Algebras
S. Popescu (1995)
Bell's Inequalities and Density Matrices: Revealing "Hidden" Nonlocality.Physical review letters, 74 14
G. Chiribella, G. D’Ariano, P. Perinotti (2010)
Informational derivation of quantum theoryPhysical Review A, 84
V. Paulsen (2003)
Completely Bounded Maps and Operator Algebras
Man-Duen Choi (1975)
Positive semidefinite biquadratic formsLinear Algebra and its Applications, 12
E. Størmer (2012)
Positive linear maps of operator algebrasActa Mathematica, 110
Shengjun Wu, J. Anandan (2003)
What is quantum entanglement
R. Werner (1989)
Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model.Physical review. A, General physics, 40 8
M. Ruskai, M. Junge, D. Kribs, P. Hayden (2012)
Operator structures in quantum information theory
v) there exists k ≥ 1 , a real C*-subalgebra C ⊆ (cid:10) ∞ k ( C ) ⊗ M p ( C ) and a factorization, Φ = Δ ◦ Γ where Γ : M n ( R ) → C and Δ : C → M m ( R ) are real linear completely positive maps,
C. Palazuelos (2012)
Superactivation of quantum nonlocality.Physical review letters, 109 19
M. Renou, D. Trillo, M. Weilenmann, T. Le, A. Tavakoli, N. Gisin, A. Acín, M. Navascués (2021)
Quantum theory based on real numbers can be experimentally falsifiedNature, 600
B. Collins, P. Hayden, I. Nechita (2015)
Random and free positive maps with applications to entanglement detection
Antoniya Aleksandrova, V. Borish, W. Wootters (2012)
Real-vector-space quantum theory with a universal quantum bitPhysical Review A, 87
J. Pillis (1967)
Linear transformations which preserve hermitian and positive semidefinite operators.Pacific Journal of Mathematics, 23
Valter Moretti, Marco Oppio (2016)
Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincar\'e symmetryarXiv: Mathematical Physics
L. Hardy (2011)
Reformulating and Reconstructing Quantum TheoryarXiv: Quantum Physics
G. Chiribella, G. D’Ariano, P. Perinotti (2009)
Probabilistic theories with purificationPhysical Review A, 81
John Watrous (2018)
The Theory of Quantum Information
Man-Duen Choi (1972)
Positive Linear Maps on C*-AlgebrasCanadian Journal of Mathematics, 24
G. Chiribella, K. Davidson, V. Paulsen, M. Rahaman (2022)
Counterexamples to the extendibility of positive unital norm-one mapsLinear Algebra and its Applications
MD Choi (1972)
Positive linear maps on C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{*}$$\end{document}-algebrasCan. J. Math., 24
M. Christandl, Alexander Müller-Hermes, M. Wolf (2018)
When Do Composed Maps Become Entanglement Breaking?Annales Henri Poincaré, 20
Man-Duen Choi (1975)
Completely positive linear maps on complex matricesLinear Algebra and its Applications, 10
(ii) p -positive if and only if C Φ is Hermitian and (cid:6) V | C Φ V (cid:7) ≥ 0 for every vector V ∈ K n ⊗ K m with Schmidt rank at most
L. Gurvits, H. Barnum (2002)
Largest separable balls around the maximally mixed bipartite quantum statePhysical Review A, 66
v) for every r and every p -positive map Ψ : M m ( K ) → M r ( K ) the map Ψ ◦ Φ is completely positive,
N. Johnston, D. Kribs, V. Paulsen, Rajesh Pereira (2010)
Minimal and maximal operator spaces and operator systems in entanglement theoryJournal of Functional Analysis, 260
A. Jamiołkowski (1972)
Linear transformations which preserve trace and positive semidefiniteness of operatorsReports on Mathematical Physics, 3
G. Chiribella, Yuxiang Yang, A. Yao (2013)
Quantum replication at the Heisenberg limitNature Communications, 4
W. Wootters (2010)
Entanglement Sharing in Real-Vector-Space Quantum TheoryFoundations of Physics, 42
WK Wootters (1990)
Local accessibility of quantum statesComplex. Entropy Phys. Inf., 8
C. Caves, C. Fuchs, P. Rungta (2000)
Entanglement of Formation of an Arbitrary State of Two RebitsFoundations of Physics Letters, 14
R. Hildebrand (2008)
Semidefinite descriptions of low-dimensional separable matrix conesLinear Algebra and its Applications, 429
M. Horodecki, P. Shor, M. Ruskai (2003)
Entanglement Breaking ChannelsReviews in Mathematical Physics, 15
L. Masanes, Markus Müller, R. Augusiak, D. Pérez-García (2012)
Existence of an information unit as a postulate of quantum theoryProceedings of the National Academy of Sciences, 110
Charles Bennett, D. DiVincenzo, T. Mor, P. Shor, J. Smolin, B. Terhal (1998)
Unextendible product bases and bound entanglementPhysical Review Letters, 82
B. Dakić, Č. Brukner (2009)
Quantum Theory and Beyond: Is Entanglement Special?arXiv: Quantum Physics
B. Terhal (1999)
Bell inequalities and the separability criterionPhysics Letters A, 271
JS Bell (1964)
On the Einstein Podolsky Rosen paradoxPhys. Phys. Fiz., 1
Z. Ruan (2003)
Complexifications of real operator spacesIllinois Journal of Mathematics, 47
(2018)
Einstein-Podolsky-Rosen Paradox
D. Chruściński, A. Kossakowski (2008)
Spectral Conditions for Positive MapsCommunications in Mathematical Physics, 290
K. Audenaert, K. Audenaert, B. Moor (2001)
Optimizing completely positive maps using semidefinite programmingPhysical Review A, 65
H. Araki (1980)
A Characterization of the State Space of Quantum Mechanics, 28
M. Navascués, T. Vértesi (2010)
Activation of nonlocal quantum resources.Physical review letters, 106 6
L. Hardy, W. Wootters (2010)
Limited Holism and Real-Vector-Space Quantum TheoryFoundations of Physics, 42
R Horodecki, P Horodecki, M Horodecki, K Horodecki (2009)
Quantum entanglementRev. Mod. Phys., 81
N. Brunner, D. Cavalcanti, Stefano Pironio, V. Scarani, S. Wehner (2013)
Bell Nonlocality
J Rosenberg (2016)
Structure and applications of real C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{*}$$\end{document}-algebrasOper. Algebras Appl., 671
G. Chiribella (2021)
Process tomography in general physical theoriesSymmetry, 13
D. Blecher, Worawit Tepsan (2020)
Real Operator Algebras and Real Positive MapsIntegral Equations and Operator Theory, 93
B. Terhal, P. Horodecki (1999)
Schmidt number for density matricesPhysical Review A, 61
U. Beyer (2016)
Complexity Entropy And The Physics Of Information
L. Masanes, Markus Mueller (2010)
A derivation of quantum theory from physical requirementsNew Journal of Physics, 13
Kedar Ranade, Mazhar Ali (2007)
The Jamiołkowski Isomorphism and a Simplified Proof for the Correspondence Between Vectors Having Schmidt Number k and k-Positive MapsOpen Systems and Information Dynamics, 14
for K = R , P is C - p -separable if and only if Tr( C Φ P ) ≥ 0 for all Φ : M n ( R ) → M m ( R ) such that (cid:11) Φ is p -positive
L. Hardy (2001)
Quantum Theory From Five Reasonable AxiomsarXiv: Quantum Physics
C Palazuelos (2012)
Superactivation of quantum nonlocalityPhys. Rev Ŀett., 109
Φ = Δ ◦ Γ where Γ : M n ( K ) → l ∞ k ( K ) ⊗ M p ( K ) and Δ : l ∞ k ( K ) ⊗ M p ( K ) → M m ( K ) are completely positive maps for some k ≥ 1
I. Klep, S. McCullough, Klemen vSivic, A. Zalar (2016)
There are many more positive maps than completely positive mapsarXiv: Functional Analysis
Toshiyuki Takasaki, J. Tomiyama (1983)
On the geometry of positive maps in matrix algebrasMathematische Zeitschrift, 184
Lin Chen, D. Ðokovic (2016)
Length filtration of the separable statesProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472
L. Clarisse (2004)
Characterization of distillability of entanglement in terms of positive mapsPhysical Review A, 71
(1996)
Separability of Mixed States: Necessary and Sufficient Conditions
Eric Hanson, C. Rouzé, Daniel França (2019)
Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic TimesAnnales Henri Poincaré, 21
Yazhen Wang (2012)
Quantum Computation and Quantum InformationStatistical Science, 27
(2019)
The positive partial transpose conjecture for n=3
Z. Ruan (2003)
On Real Operator SpacesActa Mathematica Sinica, 19
ECG Stueckelberg (1960)
Quantum theory in real Hilbert spaceHelv. Phys. Acta, 33
The theory of positive maps plays a central role in operator algebras and functional analysis and has countless applications in quantum information science. The theory was originally developed for operators acting on complex Hilbert spaces and less is known about its variant on real Hilbert spaces. In this article, we study positive maps acting on a full matrix algebra over the reals, pointing out a number of fundamental differences with the complex case and discussing their implications in quantum information. We provide a necessary and sufficient condition for a real map to admit a positive complexification and connect the existence of positive maps with non-positive complexification with the existence of mixed states that are entangled in real Hilbert space quantum mechanics, but separable in the complex version, providing explicit examples both for the maps and for the states. Finally, we discuss entanglement breaking and PPT maps, and we show that a straightforward real version of the PPT-squared conjecture is false even in dimension 2. Nevertheless, we show that the original PPT-squared conjecture implies a different conjecture for real maps, in which the PPT property is replaced by a stronger property of invariance under partial transposition (IPT). When the IPT property is assumed, we prove an asymptotic version of the conjecture.
Annales Henri Poincaré – Springer Journals
Published: Dec 1, 2023
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.