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Birkhoff-von Neumann Teoremi bağlamında kuantum sihirli kareler

Sunay Doğan

Published Sep 18, 2024• 0 citations
OpenAlex

Abstract

Quantum magic squares are generalizations of doubly stochastic matrices to arbitrary, possibly non-commutative algebras, more specifically C∗-algebras. Birkhoff-von Neumann Theorem states that every doubly stochastic matrix is a convex combination of permutation matrices. One inevitable question is whether a quantum magic square is a convex combinations of quantum permutation matrices. It has been proven by Cuevas, Drescher, and Netzer that the generalization of the Birkhoff-von Neumann Theorem in the quantum setting is not true in general. A quantum magic square is a convex combination of Arveson extreme points but not necessarily a convex combination of quantum permutation matrices. Additionally, the relationships between different subsets of quantum magic squares, including their inclusion properties, have been studied in this context.

Related papers

Quantum magic squares: Dilations and their limitations

Gemma De las Cuevas, Tom Drescher, Tim Netzer

Quantum permutation matrices and quantum magic squares are generalizations of permutation matrices and magic squares, where the entries are no longer numbers but elements from arbitrary (non-commutative) algebras. The famous Birkhoff–von Neumann theorem characterizes magic squares as convex combinations of permutation matrices. In the non-commutative case, the corresponding question is as follows: Does every quantum magic square belong to the matrix convex hull of quantum permutation matrices? That is, does every quantum magic square dilate to a quantum permutation matrix? Here, we show that this is false even in the simplest non-commutative case. We also classify the quantum magic squares that dilate to a quantum permutation matrix with commuting entries and prove a quantitative lower bound on the diameter of this set. Finally, we conclude that not all Arveson extreme points of the free spectrahedron of quantum magic squares are quantum permutation matrices.

arXiv:1912.07332•Journal of Mathematical Physics•Nov 1, 2020•20 citations

Birkhoff–von Neumann's theorem, doubly normalized tensors, and joint measurability

Leonardo Guerini, Alexandre Baraviera

Quantum measurements can be interpreted as a generalization of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalization of doubly stochastic matrices that we call doubly normalized tensors (DNTs), and investigate a corresponding version of Birkhoff–von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability appears naturally as a mathematical feature of DNTs in this context and that this feature is necessary and sufficient for a characterization similar to Birkhoff–von Neumann's. Conversely, we also show that DNTs arise from a particular instance of a joint measurability problem, remarking the relevance of this quantum-theoretical property in general operator theory.

Linear and Multilinear Algebra•Dec 31, 2022•0 citations

Joint measurability meets Birkhoff-von Neumann's theorem

Leonardo Guerini, Alexandre Baraviera

Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.

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Graph Quantum Magic Squares and Free Spectrahedra

La Piana, Francesca

Recently De les Coves, Drescher and Netzer showed that an analogue of the Birkhoff--von Neumann theorem fails in the quantum setting. Motivated by this and questions arising in the study of quantum automorphisms of graphs, we introduce a graph-based variant of quantum magic squares and show that the analogue already fails for the cycle \(C_4\), via an explicit counterexample. We also show that they admit monic linear matrix inequality descriptions, hence form compact free spectrahedra.

arXiv:2512.08797•Dec 9, 2025•0 citations

Birkhoff violation in even-polygon general probabilistic theories: implications for state transformations and information processing

Ananya Chakraborty, Sahil Gopalkrishna Naik, Samrat Sen +4 more

Abstract The entropy principle asserts that spontaneous processes drive systems toward states of higher disorder. Operational origin of such randomness is a question of utmost importance. The seminal Birkhoff theorem (also known as Birkhoff–von Neumann theorem) identifies doubly stochastic transformations as convex mixtures of reversible transformations, thereby providing an epistemic account of disorder. Quantum theory, however, does not satisfy this equivalence in general. In this work, we investigate this question within the framework of Generalized Probabilistic Theories (GPTs). We prove a violation of the Birkhoff theorem in the square-bit model, a canonical non-quantum GPT, and further show that this violation persists asymptotically under tensor products of any finite, yet arbitrarily large, number of copies of even-polygon models. This provides evidence that the failure of the Birkhoff theorem is an intrinsic operational feature of a wide class of GPTs, rather than a peculiarity of quantum theory. Furthermore, we show that, beyond quantum theory, the preorder over states induced by mixtures of reversible transformations can differ from that induced by doubly stochastic transformations, leading to inequivalent notions of state convertibility and distinct resource theories of purity. Beyond its foundational significance, we show that the violation of the Birkhoff theorem admits a concrete operational interpretation: in the square-bit theory, it serves as a genuine resource for enhancing the performance of Random Access Codes. Finally, we discuss the construction of entropy functions in GPTs that remain consistent with thermodynamic principles, in the spirit of von Neumann’s formulation.

Journal of Physics A Mathematical and Theoretical•Oct 14, 2026•0 citations

Magic squares: Latin, semiclassical, and quantum

Gemma De las Cuevas, Tim Netzer, Inga Valentiner-Branth

Quantum magic squares have recently been introduced as a “magical” combination of quantum measurements. In contrast to quantum measurements, they cannot be purified (i.e., dilated to a quantum permutation matrix)—only the so-called semiclassical ones can. Purifying establishes a relation to an ideal world of fundamental theoretical and practical importance; the opposite of purifying is described by the matrix convex hull. In this paper, we prove that semiclassical magic squares can be purified to quantum Latin squares, which are “magical” combinations of orthonormal bases. Conversely, we prove that the matrix convex hull of quantum Latin squares is larger compared to the semiclassical ones. This tension is resolved by our third result: we prove that the quantum Latin squares that are semiclassical are precisely those constructed from a classical Latin square. Our work sheds light on the internal structure of quantum magic squares, on how this is affected by the matrix convex hull, and, more generally, on the nature of the “magical” composition rule, both at the semiclassical and at the quantum level.

arXiv:2209.10230•Journal of Mathematical Physics•Feb 1, 2023•5 citations
Quantum magic squares are generalizations of doubly stochastic matrices to arbitrary, possibly non-commutative algebras, more specifically C∗-algebras. Birkhoff-von Neumann Theorem states that every doubly stochastic matrix is a convex combination of permutation matrices. One inevitable question is whether a quantum magic square is a convex combinations of quantum permutation matrices. It has been proven by Cuevas, Drescher, and Netzer that the generalization of the Birkhoff-von Neumann Theorem in the quantum setting is not true in general. A quantum magic square is a convex combination of Arveson extreme points but not necessarily a convex combination of quantum permutation matrices. Additionally, the relationships between different subsets of quantum magic squares, including their inclusion properties, have been studied in this context.
Quantum permutation matrices and quantum magic squares are generalizations of permutation matrices and magic squares, where the entries are no longer numbers but elements from arbitrary (non-commutative) algebras. The famous Birkhoff–von Neumann theorem characterizes magic squares as convex combinations of permutation matrices. In the non-commutative case, the corresponding question is as follows: Does every quantum magic square belong to the matrix convex hull of quantum permutation matrices? That is, does every quantum magic square dilate to a quantum permutation matrix? Here, we show that this is false even in the simplest non-commutative case. We also classify the quantum magic squares that dilate to a quantum permutation matrix with commuting entries and prove a quantitative lower bound on the diameter of this set. Finally, we conclude that not all Arveson extreme points of the free spectrahedron of quantum magic squares are quantum permutation matrices.
Quantum measurements can be interpreted as a generalization of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalization of doubly stochastic matrices that we call doubly normalized tensors (DNTs), and investigate a corresponding version of Birkhoff–von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability appears naturally as a mathematical feature of DNTs in this context and that this feature is necessary and sufficient for a characterization similar to Birkhoff–von Neumann's. Conversely, we also show that DNTs arise from a particular instance of a joint measurability problem, remarking the relevance of this quantum-theoretical property in general operator theory.
Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
Recently De les Coves, Drescher and Netzer showed that an analogue of the Birkhoff--von Neumann theorem fails in the quantum setting. Motivated by this and questions arising in the study of quantum automorphisms of graphs, we introduce a graph-based variant of quantum magic squares and show that the analogue already fails for the cycle C4C_4C4​, via an explicit counterexample. We also show that they admit monic linear matrix inequality descriptions, hence form compact free spectrahedra.
Abstract The entropy principle asserts that spontaneous processes drive systems toward states of higher disorder. Operational origin of such randomness is a question of utmost importance. The seminal Birkhoff theorem (also known as Birkhoff–von Neumann theorem) identifies doubly stochastic transformations as convex mixtures of reversible transformations, thereby providing an epistemic account of disorder. Quantum theory, however, does not satisfy this equivalence in general. In this work, we investigate this question within the framework of Generalized Probabilistic Theories (GPTs). We prove a violation of the Birkhoff theorem in the square-bit model, a canonical non-quantum GPT, and further show that this violation persists asymptotically under tensor products of any finite, yet arbitrarily large, number of copies of even-polygon models. This provides evidence that the failure of the Birkhoff theorem is an intrinsic operational feature of a wide class of GPTs, rather than a peculiarity of quantum theory. Furthermore, we show that, beyond quantum theory, the preorder over states induced by mixtures of reversible transformations can differ from that induced by doubly stochastic transformations, leading to inequivalent notions of state convertibility and distinct resource theories of purity. Beyond its foundational significance, we show that the violation of the Birkhoff theorem admits a concrete operational interpretation: in the square-bit theory, it serves as a genuine resource for enhancing the performance of Random Access Codes. Finally, we discuss the construction of entropy functions in GPTs that remain consistent with thermodynamic principles, in the spirit of von Neumann’s formulation.
Quantum magic squares have recently been introduced as a “magical” combination of quantum measurements. In contrast to quantum measurements, they cannot be purified (i.e., dilated to a quantum permutation matrix)—only the so-called semiclassical ones can. Purifying establishes a relation to an ideal world of fundamental theoretical and practical importance; the opposite of purifying is described by the matrix convex hull. In this paper, we prove that semiclassical magic squares can be purified to quantum Latin squares, which are “magical” combinations of orthonormal bases. Conversely, we prove that the matrix convex hull of quantum Latin squares is larger compared to the semiclassical ones. This tension is resolved by our third result: we prove that the quantum Latin squares that are semiclassical are precisely those constructed from a classical Latin square. Our work sheds light on the internal structure of quantum magic squares, on how this is affected by the matrix convex hull, and, more generally, on the nature of the “magical” composition rule, both at the semiclassical and at the quantum level.