Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1709.02785 · International Mathematics Research Notices

Quantum Teleportation and Super-Dense Coding in Operator Algebras

Li Gao, Samuel J Harris, Marius Junge

Published Apr 16, 2019• 8 citations
arXivHTMLPDF

Abstract

Abstract We show for any $d,m\ge 2$ with $(d,m)\neq (2,2)$, the matrix-valued generalization of the (tensor product) quantum correlation set of $d$ inputs and $m$ outputs is not closed. Our argument uses a reformulation of super-dense coding and teleportation in terms of $C^*$-algebra isomorphisms. Namely, we prove that for certain actions of cyclic group ${{\mathbb{Z}}}_d$, $$\begin{equation*}M_d(C^*({{\mathbb{F}}}_{d^2}))\cong{{\mathcal{B}}}_d\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d , M_d({{\mathcal{B}}}_d)\cong C^*({{\mathbb{F}}}_{d^2})\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d,\end{equation*}$$where ${{\mathcal{B}}}_d$ is the universal unital $C^*$-algebra generated by the elements $u_{jk}, \, 0 \le i, j \le d-1$, satisfying the relations that $[u_{j,k}]$ is a unitary operator, and $C^*({{\mathbb{F}}}_{d^2})$ is the universal $C^*$-algebra of $d^2$ unitaries. These isomorphisms provide a nice connection between the embezzlement of entanglement and the non-closedness of quantum correlation sets.

DOI
10.1093/imrn/rnz095

Related papers

Unitary Correlation Sets and their Applications

S.J. Harris

We relate Connes' embedding problem in operator algebras to the Brown algebra Unc(n), which is defined as the universal unital C*-algebra generated by the entries of an n x n unitary matrix. In particular, we show that the embedding problem is equivalent to determining whether or not the algebra Unc(n) has the weak expectation property for any (equivalently, all) n at least 2. From this perspective, we develop a theory of what we call unitary correlation sets. These sets are analogous to the usual sets of bipartite probabilistic correlations arising from the typical models in quantum information theory. We show that the analogue of the weak Tsirelson problem for unitary correlation sets is again equivalent to Connes' embedding problem. Moreover, we show that as long as Alice and Bob's unitaries are of size at least 2, the set of spatial unitary correlations is never a closed set. This result is analogous to a recent theorem of Slofstra, which states that the set of quantum probabilistic correlations is not closed, so long as the input and output sets are large enough. \n \nWe also discuss several applications of the theory of unitary correlation sets. First, we show that the class of extended non-local games known as quantum XOR games is a rich enough class to detect the validity of Connes' embedding problem. That is, the embedding problem has a positive answer if and only if the value of every quantum XOR game in the commuting model agrees with the value of the game in the approximate finite-dimensional model. Second, we use a C*-algebraic analogue of the quantum teleportation and super-dense coding maps from quantum information theory to obtain separations between the tensor product model (or "quantum spatial" model) and the approximate finite-dimensional model (or "quantum approximate" model), for matrix-valued generalizations of the usual Tsirelson corrleation sets. We use some of the intermediate results to also obtain separations between the matrix versions of the finite-dimensional model (or "quantum model") and the tensor product model.

•Jul 4, 2019•3 citations

Quantum Correlations on Quantum Spaces

Piotr Mikołaj Sołtan, Paweł Kasprzak, Arkadiusz Bochniak

Abstract For given quantum (non-commutative) spaces ${\mathbb {P}}$ and ${\mathbb {O}}$, we study the quantum space of maps ${\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}}$ from ${\mathbb {P}}$ to ${\mathbb {O}}$. In case of finite quantum spaces, these objects turn out to be behind a large class of maps which generalize the classical $\textrm {qc}$-correlations known from quantum information theory to the setting of quantum input and output sets. We prove various operator algebraic properties of the ${\textrm C}^\ast$-algebras ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ such as the lifting property and residual finite dimensionality. Inside ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ we construct a universal operator system ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}$ related to ${\mathbb {P}}$ and ${\mathbb {O}}$, and show, among other things, that the embedding ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\subset {\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ is hyperrigid and has another interesting property, which we call the strong extension property. Furthermore, ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ is the ${\textrm C}^\ast$-envelope of ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}$ and a large class of non-signalling correlations on the quantum sets ${\mathbb {P}}$ and ${\mathbb {O}}$ arise from states on ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})\otimes _{\text {\tiny {\textrm {max}}}}{\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ as well as states on the commuting tensor product ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\otimes _{\text {\tiny {\textrm {c}}}}{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}$. Finally, we introduce and study the notion of a synchronous correlation with quantum input and output sets and prove several characterizations of such correlations and their relation to traces on ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$.

•Jan 1, 2022•6 citations

Quantum Correlations on Quantum Spaces

Arkadiusz Bochniak, Paweł Kasprzak, Piotr Mikołaj Sołtan

Abstract For given quantum (non-commutative) spaces ${\mathbb {P}}$ and ${\mathbb {O}}$, we study the quantum space of maps ${\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}}$ from ${\mathbb {P}}$ to ${\mathbb {O}}$. In case of finite quantum spaces, these objects turn out to be behind a large class of maps which generalize the classical $\textrm {qc}$-correlations known from quantum information theory to the setting of quantum input and output sets. We prove various operator algebraic properties of the ${\textrm C}^\ast$-algebras ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ such as the lifting property and residual finite dimensionality. Inside ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ we construct a universal operator system ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}$ related to ${\mathbb {P}}$ and ${\mathbb {O}}$, and show, among other things, that the embedding ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\subset {\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ is hyperrigid and has another interesting property, which we call the strong extension property. Furthermore, ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ is the ${\textrm C}^\ast$-envelope of ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}$ and a large class of non-signalling correlations on the quantum sets ${\mathbb {P}}$ and ${\mathbb {O}}$ arise from states on ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})\otimes _{\text {\tiny {\textrm {max}}}}{\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ as well as states on the commuting tensor product ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\otimes _{\text {\tiny {\textrm {c}}}}{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}$. Finally, we introduce and study the notion of a synchronous correlation with quantum input and output sets and prove several characterizations of such correlations and their relation to traces on ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$.

arXiv:2105.07820•International Mathematics Research Notices•May 9, 2022•1 citation

Entanglement-invertible channels

Dominic Verdon

In a well-known result [R. Werner, J. Phys. A: Math. Gen. 34(35), 7081 (2001)], Werner classified all tight quantum teleportation and dense coding schemes, showing that they correspond to unitary error bases. Here tightness is a certain dimensional restriction: the quantum system to be teleported and the entangled resource must be of dimension d, and the measurement must have d2 outcomes. Here we generalise this classification so as to remove the dimensional restriction altogether, thereby resolving an open problem raised in that work. In fact, we classify not just teleportation and dense coding schemes, but entanglement-reversible channels. These are channels between finite-dimensional C*-algebras which are reversible with the aid of an entangled resource state, generalising ordinary reversibility of a channel. We show that such channels correspond to families of linear maps which are bi-isometric with respect to a duality defined by the resource state. In particular, in Werner’s classification, a bijective correspondence between tight teleportation and dense coding schemes was shown: swapping Alice and Bob’s operations turns a teleportation scheme into a dense coding scheme and vice versa. We observe that this property generalises ordinary invertibility of a channel; we call it entanglement-invertibility. We show that entanglement-invertible channels are precisely the quantum bijections previously studied in noncommutative topology [B. Musto et al., J. Math. Phys. 59(8), 081706 (2018)], and therefore admit a classification in terms of Wang’s quantum permutation group [S. Wang, Commun. Math. Phys. 195, 195–211 (1998)].

arXiv:2204.04493•Journal of Mathematical Physics•Jun 1, 2024•0 citations

Algebraic and Topological Study of Bell States and Quantum Teleportation

Yong Zhang, Wei Zeng, Ming Lian

Bell states and quantum teleportation play crucial roles in quantum information and computation. However, a comprehensive theoretical study of both topics remains to be carried out. This work aims to investigate key algebraic properties of generalized Bell states and explore the topological features of quantum teleportation. First, the basis theorem and the basis group are introduced to show that the extension of a generalized Bell basis by a unitary matrix still forms an orthonormal basis. Then, a twist operator is defined to establish a connection between a generalized multi-qubit Bell state and a tensor product of two-qubit Bell states. In addition, the Temperley--Lieb algebra, the braid group relations, and the Yang--Baxter equation are employed to provide a topological description of generalized Bell states and quantum teleportation. The results demonstrate that our approach not only offers a clear illustration of relevant quantum information protocols but also reveals the topological nature of quantum entanglement and teleportation.

quant-pharXiv:2602.11218•Feb 12, 2026•0 citations

Universal entanglement-inspired correlations

Elizabeth Agudelo, Laura Ares, Jan Sperling

Quantum correlations, crucial for the advantage and advancement of quantum science and technology, arise from the impossibility of expressing a quantum state as a tensor product over a given set of parties. In this work, a generalized notion of correlations via arbitrary products is formulated. Remarkably, as a universal property, the connection between such general products and tensor products is established, allowing one to relate generic non-product states to the common notion of entangled states. We construct the set of free operations for general types of products by extending the local-operation-and-classical-communication paradigm, familiar from standard entanglement theory, thereby establishing a resource theory of correlations for general products. A generalization is provided beyond two factors that can be universally related to multipartite entanglement. Applications that highlight the usefulness of the approach are discussed, such as the factorization of fermionic states, the non-local factorization of multi-photon states into single-photon states, and the interesting possibility of understanding prime numbers as a form of single-party entanglement.

quant-pharXiv:2602.15520•Feb 18, 2026•0 citations
Abstract We show for any d,m≥2d,m\ge 2d,m≥2 with (d,m)≠(2,2)(d,m)\neq (2,2)(d,m)=(2,2), the matrix-valued generalization of the (tensor product) quantum correlation set of ddd inputs and mmm outputs is not closed. Our argument uses a reformulation of super-dense coding and teleportation in terms of C∗C^*C∗-algebra isomorphisms. Namely, we prove that for certain actions of cyclic group Zd{{\mathbb{Z}}}_dZd​, \begin{equation*}M_d(C^*({{\mathbb{F}}}_{d^2}))\cong{{\mathcal{B}}}_d\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d , M_d({{\mathcal{B}}}_d)\cong C^*({{\mathbb{F}}}_{d^2})\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d,\end{equation*}where Bd{{\mathcal{B}}}_dBd​ is the universal unital C∗C^*C∗-algebra generated by the elements ujk, 0≤i,j≤d−1u_{jk}, \, 0 \le i, j \le d-1ujk​,0≤i,j≤d−1, satisfying the relations that [uj,k][u_{j,k}][uj,k​] is a unitary operator, and C∗(Fd2)C^*({{\mathbb{F}}}_{d^2})C∗(Fd2​) is the universal C∗C^*C∗-algebra of d2d^2d2 unitaries. These isomorphisms provide a nice connection between the embezzlement of entanglement and the non-closedness of quantum correlation sets.
We relate Connes' embedding problem in operator algebras to the Brown algebra Unc(n), which is defined as the universal unital C*-algebra generated by the entries of an n x n unitary matrix. In particular, we show that the embedding problem is equivalent to determining whether or not the algebra Unc(n) has the weak expectation property for any (equivalently, all) n at least 2. From this perspective, we develop a theory of what we call unitary correlation sets. These sets are analogous to the usual sets of bipartite probabilistic correlations arising from the typical models in quantum information theory. We show that the analogue of the weak Tsirelson problem for unitary correlation sets is again equivalent to Connes' embedding problem. Moreover, we show that as long as Alice and Bob's unitaries are of size at least 2, the set of spatial unitary correlations is never a closed set. This result is analogous to a recent theorem of Slofstra, which states that the set of quantum probabilistic correlations is not closed, so long as the input and output sets are large enough. \n \nWe also discuss several applications of the theory of unitary correlation sets. First, we show that the class of extended non-local games known as quantum XOR games is a rich enough class to detect the validity of Connes' embedding problem. That is, the embedding problem has a positive answer if and only if the value of every quantum XOR game in the commuting model agrees with the value of the game in the approximate finite-dimensional model. Second, we use a C*-algebraic analogue of the quantum teleportation and super-dense coding maps from quantum information theory to obtain separations between the tensor product model (or "quantum spatial" model) and the approximate finite-dimensional model (or "quantum approximate" model), for matrix-valued generalizations of the usual Tsirelson corrleation sets. We use some of the intermediate results to also obtain separations between the matrix versions of the finite-dimensional model (or "quantum model") and the tensor product model.
Abstract For given quantum (non-commutative) spaces P{\mathbb {P}}P and O{\mathbb {O}}O, we study the quantum space of maps MP,O{\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}}MP,O​ from P{\mathbb {P}}P to O{\mathbb {O}}O. In case of finite quantum spaces, these objects turn out to be behind a large class of maps which generalize the classical qc\textrm {qc}qc-correlations known from quantum information theory to the setting of quantum input and output sets. We prove various operator algebraic properties of the C∗{\textrm C}^\astC∗-algebras C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​) such as the lifting property and residual finite dimensionality. Inside C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​) we construct a universal operator system SP,O{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}SP,O​ related to P{\mathbb {P}}P and O{\mathbb {O}}O, and show, among other things, that the embedding SP,O⊂C⁡(MP,O){\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\subset {\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})SP,O​⊂C(MP,O​) is hyperrigid and has another interesting property, which we call the strong extension property. Furthermore, C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​) is the C∗{\textrm C}^\astC∗-envelope of SP,O{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}SP,O​ and a large class of non-signalling correlations on the quantum sets P{\mathbb {P}}P and O{\mathbb {O}}O arise from states on C⁡(MP,O)⊗maxC⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})\otimes _{\text {\tiny {\textrm {max}}}}{\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​)⊗max​C(MP,O​) as well as states on the commuting tensor product SP,O⊗cSP,O{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\otimes _{\text {\tiny {\textrm {c}}}}{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}SP,O​⊗c​SP,O​. Finally, we introduce and study the notion of a synchronous correlation with quantum input and output sets and prove several characterizations of such correlations and their relation to traces on C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​).
Abstract For given quantum (non-commutative) spaces P{\mathbb {P}}P and O{\mathbb {O}}O, we study the quantum space of maps MP,O{\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}}MP,O​ from P{\mathbb {P}}P to O{\mathbb {O}}O. In case of finite quantum spaces, these objects turn out to be behind a large class of maps which generalize the classical qc\textrm {qc}qc-correlations known from quantum information theory to the setting of quantum input and output sets. We prove various operator algebraic properties of the C∗{\textrm C}^\astC∗-algebras C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​) such as the lifting property and residual finite dimensionality. Inside C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​) we construct a universal operator system SP,O{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}SP,O​ related to P{\mathbb {P}}P and O{\mathbb {O}}O, and show, among other things, that the embedding SP,O⊂C⁡(MP,O){\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\subset {\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})SP,O​⊂C(MP,O​) is hyperrigid and has another interesting property, which we call the strong extension property. Furthermore, C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​) is the C∗{\textrm C}^\astC∗-envelope of SP,O{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}SP,O​ and a large class of non-signalling correlations on the quantum sets P{\mathbb {P}}P and O{\mathbb {O}}O arise from states on C⁡(MP,O)⊗maxC⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})\otimes _{\text {\tiny {\textrm {max}}}}{\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​)⊗max​C(MP,O​) as well as states on the commuting tensor product SP,O⊗cSP,O{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\otimes _{\text {\tiny {\textrm {c}}}}{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}SP,O​⊗c​SP,O​. Finally, we introduce and study the notion of a synchronous correlation with quantum input and output sets and prove several characterizations of such correlations and their relation to traces on C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​).
In a well-known result [R. Werner, J. Phys. A: Math. Gen. 34(35), 7081 (2001)], Werner classified all tight quantum teleportation and dense coding schemes, showing that they correspond to unitary error bases. Here tightness is a certain dimensional restriction: the quantum system to be teleported and the entangled resource must be of dimension d, and the measurement must have d2 outcomes. Here we generalise this classification so as to remove the dimensional restriction altogether, thereby resolving an open problem raised in that work. In fact, we classify not just teleportation and dense coding schemes, but entanglement-reversible channels. These are channels between finite-dimensional C*-algebras which are reversible with the aid of an entangled resource state, generalising ordinary reversibility of a channel. We show that such channels correspond to families of linear maps which are bi-isometric with respect to a duality defined by the resource state. In particular, in Werner’s classification, a bijective correspondence between tight teleportation and dense coding schemes was shown: swapping Alice and Bob’s operations turns a teleportation scheme into a dense coding scheme and vice versa. We observe that this property generalises ordinary invertibility of a channel; we call it entanglement-invertibility. We show that entanglement-invertible channels are precisely the quantum bijections previously studied in noncommutative topology [B. Musto et al., J. Math. Phys. 59(8), 081706 (2018)], and therefore admit a classification in terms of Wang’s quantum permutation group [S. Wang, Commun. Math. Phys. 195, 195–211 (1998)].
Bell states and quantum teleportation play crucial roles in quantum information and computation. However, a comprehensive theoretical study of both topics remains to be carried out. This work aims to investigate key algebraic properties of generalized Bell states and explore the topological features of quantum teleportation. First, the basis theorem and the basis group are introduced to show that the extension of a generalized Bell basis by a unitary matrix still forms an orthonormal basis. Then, a twist operator is defined to establish a connection between a generalized multi-qubit Bell state and a tensor product of two-qubit Bell states. In addition, the Temperley--Lieb algebra, the braid group relations, and the Yang--Baxter equation are employed to provide a topological description of generalized Bell states and quantum teleportation. The results demonstrate that our approach not only offers a clear illustration of relevant quantum information protocols but also reveals the topological nature of quantum entanglement and teleportation.
Quantum correlations, crucial for the advantage and advancement of quantum science and technology, arise from the impossibility of expressing a quantum state as a tensor product over a given set of parties. In this work, a generalized notion of correlations via arbitrary products is formulated. Remarkably, as a universal property, the connection between such general products and tensor products is established, allowing one to relate generic non-product states to the common notion of entangled states. We construct the set of free operations for general types of products by extending the local-operation-and-classical-communication paradigm, familiar from standard entanglement theory, thereby establishing a resource theory of correlations for general products. A generalization is provided beyond two factors that can be universally related to multipartite entanglement. Applications that highlight the usefulness of the approach are discussed, such as the factorization of fermionic states, the non-local factorization of multi-photon states into single-photon states, and the interesting possibility of understanding prime numbers as a form of single-party entanglement.