# Zero-error quantum information and operator theory: emerging links

Lead Research Organisation:
Queen's University of Belfast

Department Name: Sch of Mathematics and Physics

### Abstract

Graphs are among the simplest mathematical objects - by definition, a graph is a set of points called vertices, and a set of edges, each edge connecting a pair of given vertices. If the vertices are labelled by the numbers 1,2,...,n, then each edge is a pair (i,j), where i and j are between 1 and n. Graphs have a large number of applications in computer science, engineering and mathematics itself. It was Shannon in the 1950's who realised that they can be used successfully in the theory of information. If a symbol, say i, is sent through a communication channel, then due to an error that may occur during this transfer, it may be confused by the receiver with another symbol, say j. The pairs (i,j) of symbols that can be confused in such way form the set of edges of a graph, called the confusability graph of the channel. Shannon defined an asymptotic parameter, called the zero-error capacity of the channel (or, equivalently, of the corresponding confusability graph), which measures the extent to which information can be sent through the channel with zero error.

Given a graph G on n vertices, in the 1980's, Pauslen, Power and Smith studied the linear subspace S(G) of the space M_n of all n by n matrices consisting of those elements that have zero i,j-entry when (i,j) is not an edge of G. The space S(G) is an operator system - that is, it is closed under the operation of conjugate transpose and contains the identity matrix - which fully identifies the underlying graph G. Operator systems have an illustrious history and a large number of applications within Analysis, and the aforementioned paper thus opened up an analytical avenue for Graph Theory. We note that not all operator systems in M_n can be obtained from graphs in the described way, and this is a crucial point for the approach used in our proposal.

At present, there are ongoing efforts for the construction of quantum computers. The theoretical science that lies behind these efforts is Quantum Information Theory, a part of the general field of quantum physics. The main feature of quantum, as opposed to classical, physics, is non-commutativity: the mathematical tools behind it use the space M_n and its infinite dimensional generalisations where the commutation rule ab = ba does not hold in general. Non-commutativity features prominently in the study of quantum channels, that is, channels used to transfer quantum information. Recently, Duan, Severini and Winter defined the confusability graph of a quantum channel as a certain operator system in M_n, and showed that every operator system in M_n arises in this way. It is thus natural to call operator systems in M_n non-commutative graphs. The class of operator systems S(G) can be identified in an easy and elegant way as a natural subclass of the class of all non-commutative graphs. Quantum zero-error capacities were introduced, but a number of important questions were left open.

The aim of the present research project is to study of quantum zero-error capacities using methods from Operator Theory - the general branch where the study of operator systems belongs. We plan to obtain new estimates on the quantum version of a parameter known as Lovasz number of a graph (a quantity that provides an easier computable bound for the zero-error capacity), study other parameters such as non-commutative chromatic numbers, and lay the foundations of a "non-commutative graph theory", where basic operations with classical graphs such as passing to a complement, a subgraph and a homomorphic image can be carried out successfully in the context of operator systems. We plan to address a number of questions regarding asymptotic versions of the introduced parameters that are expected to shed light on open problems and conjectures in Graph Theory and Quantum Information.

Given a graph G on n vertices, in the 1980's, Pauslen, Power and Smith studied the linear subspace S(G) of the space M_n of all n by n matrices consisting of those elements that have zero i,j-entry when (i,j) is not an edge of G. The space S(G) is an operator system - that is, it is closed under the operation of conjugate transpose and contains the identity matrix - which fully identifies the underlying graph G. Operator systems have an illustrious history and a large number of applications within Analysis, and the aforementioned paper thus opened up an analytical avenue for Graph Theory. We note that not all operator systems in M_n can be obtained from graphs in the described way, and this is a crucial point for the approach used in our proposal.

At present, there are ongoing efforts for the construction of quantum computers. The theoretical science that lies behind these efforts is Quantum Information Theory, a part of the general field of quantum physics. The main feature of quantum, as opposed to classical, physics, is non-commutativity: the mathematical tools behind it use the space M_n and its infinite dimensional generalisations where the commutation rule ab = ba does not hold in general. Non-commutativity features prominently in the study of quantum channels, that is, channels used to transfer quantum information. Recently, Duan, Severini and Winter defined the confusability graph of a quantum channel as a certain operator system in M_n, and showed that every operator system in M_n arises in this way. It is thus natural to call operator systems in M_n non-commutative graphs. The class of operator systems S(G) can be identified in an easy and elegant way as a natural subclass of the class of all non-commutative graphs. Quantum zero-error capacities were introduced, but a number of important questions were left open.

The aim of the present research project is to study of quantum zero-error capacities using methods from Operator Theory - the general branch where the study of operator systems belongs. We plan to obtain new estimates on the quantum version of a parameter known as Lovasz number of a graph (a quantity that provides an easier computable bound for the zero-error capacity), study other parameters such as non-commutative chromatic numbers, and lay the foundations of a "non-commutative graph theory", where basic operations with classical graphs such as passing to a complement, a subgraph and a homomorphic image can be carried out successfully in the context of operator systems. We plan to address a number of questions regarding asymptotic versions of the introduced parameters that are expected to shed light on open problems and conjectures in Graph Theory and Quantum Information.

### Planned Impact

Since the proposed research addresses questions of fundamental science and is couched in a mathematical setting, its economic and societal impact are naturally indirect. However, some primary pathways to impact may be identified as follows:

(a) Quantum information processing is on the threshold of becoming a viable technology. While a full-blown quantum computer is still some years away, quantum communication, metrology and cryptography are already beyond proof-of-principle stage. This development has the potential of a radical transformation of information technology, down to the very concept of information, eventually affecting everybody's daily experiences. The present project is concerned with the theoretical underpinnings of quantum communication, contributing thus to the quantum information revolution. This will inform future developments and should also be of benefit to the emerging industry sector.

(b) The subject matter of the present proposal is also suitable for reaching out to the general public. Namely, while the scientific problems with which it is concerned reach to the current frontiers of research, zero-error questions are elementary to state and their conceptual import easily grasped. We believe that this offers new ways of making quantum mechanics accessible to the public.

(c) The successful completion of the proposed research will fuel a number of areas with new directions for further development. The proposal is a meeting point of three distinct disciplines, namely, Quantum Information Theory, Graph Theory and Operator Theory, and aims at utilising ideas from each one of them in order to advance the general quantisation programme in Mathematics which has been recognised as an important step forward by a number of leading researchers.

(a) Quantum information processing is on the threshold of becoming a viable technology. While a full-blown quantum computer is still some years away, quantum communication, metrology and cryptography are already beyond proof-of-principle stage. This development has the potential of a radical transformation of information technology, down to the very concept of information, eventually affecting everybody's daily experiences. The present project is concerned with the theoretical underpinnings of quantum communication, contributing thus to the quantum information revolution. This will inform future developments and should also be of benefit to the emerging industry sector.

(b) The subject matter of the present proposal is also suitable for reaching out to the general public. Namely, while the scientific problems with which it is concerned reach to the current frontiers of research, zero-error questions are elementary to state and their conceptual import easily grasped. We believe that this offers new ways of making quantum mechanics accessible to the public.

(c) The successful completion of the proposed research will fuel a number of areas with new directions for further development. The proposal is a meeting point of three distinct disciplines, namely, Quantum Information Theory, Graph Theory and Operator Theory, and aims at utilising ideas from each one of them in order to advance the general quantisation programme in Mathematics which has been recognised as an important step forward by a number of leading researchers.

### Organisations

### Publications

Paulsen V
(2016)

*Estimating quantum chromatic numbers*in Journal of Functional Analysis
Paulsen V
(2015)

*QUANTUM CHROMATIC NUMBERS VIA OPERATOR SYSTEMS*in The Quarterly Journal of MathematicsDescription | One of the main discoveries was a way to link quantum chromatic numbers studied previously in the literature, to tensor products of operator systems. Thus, we were able to provide a direct link between the areas of Quantum Information Theory on one hand, and Operator Algebras, on the other, which was enhanced in the second output of the grant. |

Exploitation Route | Researchers with expertise in Operator Theory and Quantum Information Theory, interested in quantum parameters of graphs and capacities of quantum channel will be able to use our results for further advances and formulation of quantum versions of other graph parameters. |

Sectors | Digital/Communication/Information Technologies (including Software) |

Description | International Exchanges Scheme |

Amount | £10,800 (GBP) |

Organisation | The Royal Society |

Sector | Charity/Non Profit |

Country | United Kingdom |

Start | 01/2016 |

End | 12/2017 |

Description | LMS-CMI Research Schools |

Amount | £25,000 (GBP) |

Funding ID | RS-23 |

Organisation | London Mathematical Society |

Sector | Academic/University |

Country | United Kingdom |

Start | 08/2016 |

End | 12/2016 |

Title | Operator systems |

Description | We developed a mathematical methodology for the application of operator systems in questions concerning quantum correlations, and the applications of the latter to quantum chromatic numbers of graphs. In addition, we showed how C*-algebras with traces can be utilised to express quantum projective ranks of graphs. |

Type Of Material | Improvements to research infrastructure |

Year Produced | 2015 |

Provided To Others? | Yes |

Impact | There is no immediate impact outside academic research, but the methods developed will be useful in the further study of graphs in relation with their quantum parameters. |

Description | Collaboration |

Organisation | University of Houston |

Country | United States |

Sector | Academic/University |

PI Contribution | I was a PI on EPSRC funded project "Zero-error quantum information and operator theory - emerging links", in which Professor V. I. Paulsen from the University of Houston was a Visiting Researcher. |

Collaborator Contribution | Professor Paulsen provided world leading expertise in Operator Algebra Theory and Operator Space Theory needed for the successful implementation of the original research plan. |

Impact | "Quantum chromatic number via operator systems", a preprint; "Estimating quantum chromatic numbers", a preprint. |

Start Year | 2008 |

Description | Cross disciplinary collaboration |

Organisation | Autonomous University of Barcelona (UAB) |

Country | Spain |

Sector | Academic/University |

PI Contribution | I provided expertise in the area of Operator Algebras to feed in into the project entitled "Zero error quantum information and operator theory - emerging links", funded by the EPSRC. This project included Professor Winter as a Visiting Researcher. |

Collaborator Contribution | Professor A. Winter participated in collaboration within the project "Zero error quantum information and operator theory - emerging links", funded by the EPSRC. His world leading expertise in Quantum Information Theory was crucial for producing one of the outputs for this project. |

Impact | "Estimating quantum chromatic numbers", a preprint. |

Start Year | 2013 |

Description | Cross disciplinary collaboration |

Organisation | Carnegie Mellon University |

Country | United States |

Sector | Academic/University |

PI Contribution | I was the PI of the grant on which Dr Dan Stahlke (Carnegie Mellon University) was a Visiting Researcher. |

Collaborator Contribution | Dr Stahlke provided expertise in the area of quantum graph homomorphisms, which was essential for the completion of the second output of the grant. |

Impact | "Estimating quantum chromatic numbers", preprint |

Start Year | 2013 |

Description | Cross discipline collaboration |

Organisation | University College Hospital |

Department | University College London Hospitals Charity (UCLH) |

Country | United Kingdom |

Sector | Charity/Non Profit |

PI Contribution | An EPSRC grant, led by I. G. Todorov as a PI, had a co-PI a member of academic staff at University College London, namely S. Severini. |

Collaborator Contribution | Dr Severini provided important expertise in Graph Theory in the research project entitled "Zero error quantum information and operator theory - emerging links", funded by the EPSRC. |

Impact | "Estimating quantum chromatic numbers", a preprint that resulted from this partnership. This is a paper on the border between the areas of Operator Algebra Theory, Quantum Information Theory and Graph Theory. |

Start Year | 2013 |

Description | Instructional conference on Capacity of Communication |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Participants in your research and patient groups |

Results and Impact | The talks that took place at the Instructional Conference on Capacity of Communication were by experts in various ares: Operator Algebras, Graph Theory, Quantum Information Theory, Theoretical Physics. The conferences sparked lively discussions between its participants. After the conference, members of the Pure Mathematics group at Queen's University Belfast were motivated to learn more about the area of quantum information and its mathematical implications. |

Year(s) Of Engagement Activity | 2014 |