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This note is intended to foster a discussion about the extent to which typical problems arising in quantum information theory are algorithmically decidable (in principle rather than in practice).
A decision method for elementary algebra and geometry
A. Tarski · 1951
Earlier work this paper cites.
A new decision method for elementary algebra
A. Seidenberg · 1954
Earlier work this paper cites.
Decidability of second-order theories and automata on infinite trees
Michael O. Rabin · 1969
Earlier work this paper cites.
Computable fields and arithmetically definable ordered fields
A.H. Lachlan and E.W. Madison · 1970
Earlier work this paper cites.
A note on computable real fields
E. W. Madison · 1970
Earlier work this paper cites.
Unsolvability in 3 × 3 3\times 3 matrices
M.S. Paterson · 1970
Earlier work this paper cites.
Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solution
M. Davis, Yu. Matijacevic, and J. Robinson · 1976
Earlier work this paper cites.
Handbook of mathematical logic
Jon Barwise, editor · 1977
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Computability, an introduction to recursive function theory
Nigel J. Cutland · 1980
Earlier work this paper cites.
Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model
R. F. Werner · 1989
Earlier work this paper cites.
On the combinatorial and algebraic complexity of quantifier elimination
S. Basu, R. Pollack, and M.-F. Roy · 1994
Earlier work this paper cites.
The elementary theory of restricted analytic fields with exponentiation
Lou van den Dries, Angus Macintyre, and David Marker · 1994
Earlier work this paper cites.
Complexity and real computation
Leonore Blum, Felipe Cucker, Michael Shub, and Steve Smale · 1997
Earlier work this paper cites.
Real quantifier elimination in practice
Andreas Dolzmann, Thomas Sturm, and Volker Weispfenning · 1998
Cited alongside, same era.
Zero-error information theory
J. Körner and A. Orlitsky · 1998
Cited alongside, same era.
Undecidable problems for probabilistic automata of fixed dimension
Vincent Blondel and Vincent Canterini · 2001
Cited alongside, same era.
Nonadditivity of bipartite distillable entanglement follows from conjecture on bound entangled Werner states
P. W. Shor, J. A. Smolin, and B. M. Terhal · 2001
Cited alongside, same era.
Bell inequalities and entanglement
R. F. Werner and M. M. Wolf · 2001
Cited alongside, same era.
Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a bell inequality
Jonathan Barrett · 2002
Cited alongside, same era.
Improved undecidability results on the emptiness problem of probabilistic and quantum cut-point languages
Mika Hirvensalo · 2007
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Bounding the set of quantum correlations
Miguel Navascués, Stefano Pironio, and Antonio Acín · 2007
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Various aspects of finite quantum automata
Mika Hirvensalo · 2008
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Positive polynomials and sums of squares
Murray Marshall · 2008
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Quantum communication with zero-capacity channels
Graeme Smith and Jon Yard · 2008
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Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1 p>1
A. J. Winter and P. Hayden · 2008
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Counterexample to an additivity conjecture for output purity of quantum channels
A. S. Holevo and R. F. Werner · 2002
Cited alongside, same era.
Activating distillation with an infinitesimal amount of bound entanglement
Karl Gerd H. Vollbrecht and Michael M. Wolf · 2002
Cited alongside, same era.
Many copies may be required for entanglement distillation
John Watrous · 2004
Cited alongside, same era.
Decidable and undecidable problems about quantum automata
Vincent D. Blondel, Emmanuel Jeandel, Pascal Koiran, and Natacha Portier · 2005
Cited alongside, same era.
Quantum automata and algebraic groups
Harm Derksen, Emmanuel Jeandel, and Pascal Koiran · 2005
Cited alongside, same era.
Decision problems for semi-Thue systems with a few rules
Yuri Matiyasevich and Géraud Sénizergues · 2005
Cited alongside, same era.
Post correspondence problem and small dimensional matrices
Tero Harju · 2009
Later among the works it cites.
A counterexample to additivity of minimum output entropy
M. B. Hastings · 2009
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Unital quantum channels: convex structure and revivals of Birkhoff’s theorem
Christian Mendl and Michael Wolf · 2009
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Computability Theory, Reverse Mathematics, and Ordered Fields
O.Levin · 2009
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Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems
Károly F. Pál and Tamás Vértesi · 2010
Later among the works it cites.
Factorization and dilation problems for completely positive maps on von neumann algebras
Uffe Haagerup and Magdalena Musat · 2011
Closest in time.
Connes’ embedding problem and Tsirelson’s problem
M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V. B. Scholz, and R. F. Werner · 2011
Closest in time.