nLab quantum information theory via dagger-compact categories

under construction

Idea

Central aspects of quantum mechanics with finite-dimensional spaces of quantum states — such as tensor products of qbit states of relevance in quantum information theory and quantum computation — follow from the formal properties of the dagger-compact category of finite-dimensional Hilbert spaces.

Conversely, much of quantum information theory and quantum computation can thus be formulated internal to any dagger-compact category, and general reasoning about †-compact categories themselves yields results about quantum mechanics and quantum computation.

A transparent string diagram calculus in †-compact categories provides an intuitive and powerful tool for reasoning in $\dagger$-compact categories.

Quantum mechanical concepts in $\dagger$-compact categories

Let $(C,\otimes,I, \dagger)$ be a †-compact category.

We list various concepts in quantum mechanics and their corresponding incarnation in terms of structures in $C$.

Classical measurement outcomes

An observable in quantum mechanics formulated on a Hilbert space is modeled by a self-adjoint operator, and the classical measurement outcomes of this operator provide, at least under some assumptions, an orthogonal basis on the Hilbert space.

That, more abstractly, the notion of orthogonal basis of an object can be phrased intrinsically inside any suitable $\dagger$-compact category is the point made in (CoeckePavlovicVicary).

Complex phases

The underlying “algebra of quantum amplitudes” of the corresponding quantum mechanical system is the endomorphism monoid of the tensor unit

$$\mathbb{C}_C = End_C(I) \,.$$

In (Vicary) it is shown that in $\dagger$-compact categories with all finite limits over certain “tree-like” diagrams compatible with the $\dagger$-structure, this $\mathbb{C}_C$ has the properties that

• it is a field of characteristic 0 with involution $\dagger$;

• the subfield $\mathbb{R}_C$ fixed under $\dagger$ is orderable.

If furthermore every bounded sequence of measurements in $C$ with values in $\mathbb{R}_C$ has a least upper bound, then it follows that this field coincides with the complex numbers

$$\mathbb{C}_C = \mathbb{C}$$

and moreover

$$\mathbb{R}_C = \mathbb{R} \,.$$

Completely positive maps

The behaviour of quantum channels and completely positive maps has an elegant categorical description in terms of $\dagger$-compact categories. See (Selinger and Coecke).

Quantum logic

Symmetric monoidal categories such as †-compact categories have as internal logic a fragment of linear logic and as type theory a flavor of linear type theory. In this fashion everything that can be formally said about quantum mechanics in terms of †-compact categories has an equivalent expression in formal logic/type theory. It has been argued (Abramsky-Duncan 05, Duncan 06) that this linear logic/linear type theory of quantum mechanics is the correct formalization of “quantum logic”. An exposition of this point of view is in (Baez-Stay 09).

Related concepts

• quantum circuit

• quantum teleportation

• order-theoretic structure in quantum mechanics

• linear logic, linear type theory, quantum logic, quantum computing

• JBW-algebraic quantum mechanics

• tensor network, string diagram

References

Quantum information theory via String diagrams

General

The observation that a natural language for quantum information theory and quantum computation, specifically for quantum circuit diagrams, is that of string diagrams in †-compact categories (see quantum information theory via dagger-compact categories):

• Samson Abramsky, Bob Coecke, A categorical semantics of quantum protocols, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004) $[$arXiv:quant-ph/0402130, doi:10.1109/LICS.2004.1319636$]$

• Samson Abramsky, Bob Coecke, Categorical quantum mechanics, in Handbook of Quantum Logic and Quantum Structures, Elsevier (2008) $[$arXiv:0808.1023, ISBN:9780080931661, doi:10.1109/LICS.2004.1319636$]$

On the relation to quantum logic/linear logic:

• Samson Abramsky, Ross Duncan, A Categorical Quantum Logic, Mathematical Structures in Computer Science 16 3 (2006) $[$arXiv:quant-ph/0512114, doi:10.1017/S0960129506005275$]$

• Ross Duncan, Types for quantum mechanics, 2006 $[$pdf, slides$]$

Early exposition with introduction to monoidal category theory:

• Bob Coecke, Kindergarten quantum mechanics $[$arXiv:quant-ph/0510032$]$

• Bob Coecke, Introducing categories to the practicing physicist $[$arXiv:0808.1032$]$

• John Baez, Mike Stay, Physics, topology, logic and computation: a rosetta stone in: New Structures for Physics, Bob Coecke (ed.), Lecture Notes in Physics 813, Springer (2011) 95-174 $[$arxiv/0903.0340$]$

• Bob Coecke, Eric Oliver Paquette, Categories for the practising physicist, in: New Structures for Physics, Lecture Notes in Physics 813, Springer (2010) $[$arXiv:0905.3010, doi:10.1007/978-3-642-12821-9_3$]$

• Bob Coecke, Quantum Picturalism, Contemporary Physics 51 1 (2010) $[$arXiv:0908.1787, doi:10.1080/00107510903257624$]$

Review in contrast to quantum logic:

• Samson Abramsky, Bob Coecke, Physics from Computer Science: a Position Statement, International Journal of Unconventional Computing 3 3 (2007) $[$pdf, ijuc-3-3-p-179-197$]$

and with emphasis on quantum computation:

• Jamie Vicary, The Topology of Quantum Algorithms, (LICS 2013) Proceedings of 28th Annual ACM/IEEE Symposium on Logic in Computer Science (2013) 93-102 $[$arXiv:1209.3917, doi:10.1109/LICS.2013.14$]$

Generalization to quantum operations on mixed states (completely positive maps of density matrices):

• Peter Selinger, Dagger compact closed categories and completely positive maps, Electronic Notes in Theoretical Computer Science 170 (2007) 139-163 $[$doi:10.1016/j.entcs.2006.12.018, web, pdf$]$

• Bob Coecke, Chris Heunen, Pictures of complete positivity in arbitrary dimension, Information and Computation 250 50-58 (2016) $[$arXiv:1110.3055, doi:10.1016/j.ic.2016.02.007$]$

• Bob Coecke, Chris Heunen, Aleks Kissinger,

  Categories of Quantum and Classical Channels, EPTCS 158 (2014) 1-14 $[$arXiv:1408.0049, doi:10.4204/EPTCS.158.1$]$

Textbook accounts (with background on relevant monoidal category theory):

• Bob Coecke, Aleks Kissinger, Picturing Quantum Processes – A First Course in Quantum Theory and Diagrammatic Reasoning, Cambridge University Press (2017) $[$ISBN:9781107104228$]$

• Chris Heunen, Jamie Vicary, Categories for Quantum Theory, Oxford University Press 2019 $[$ISBN:9780198739616$]$

  based on:

  Chris Heunen, Jamie Vicary, Lectures on categorical quantum mechanics (2012) $[$pdf, pdf$]$

• Bob Coecke, Stefano Gogioso, Quantum in Pictures, Quantinuum Publications (2023) $[$ISBN 978-1739214715, Quantinuum blog$]$

  (focus on ZX-calculus)

Measurement & Classical structures

Formalization of quantum measurement via Frobenius algebra-structures (“classical structures”):

• Bob Coecke, Duško Pavlović, Quantum measurements without sums, in Louis Kauffman, Samuel Lomonaco (eds.), Mathematics of Quantum Computation and Quantum Technology, Taylor & Francis (2008) 559-596 $[$arXiv:quant-ph/0608035, doi:10.1201/9781584889007$]$

• Bob Coecke, Eric Oliver Paquette, POVMs and Naimark’s theorem without sums, Electronic Notes in Theoretical Computer Science 210 (2008) 15-31 $[$arXiv:quant-ph/0608072, doi:10.1016/j.entcs.2008.04.015$]$

• Bob Coecke, Eric Oliver Paquette, Duško Pavlović, Classical and quantum structuralism, in: Semantic Techniques in Quantum Computation, Cambridge University Press (2009) 29-69 $[$arXiv:0904.1997, doi:10.1017/CBO9781139193313.003$]$

• Bob Coecke, Duško Pavlović, Jamie Vicary, A new description of orthogonal bases, Mathematical Structures in Computer Science 23 3 (2012) 555- 567 $[$arXiv:0810.0812, doi:10.1017/S0960129512000047$]$

and the evolution of the “classical structures”-monad into the “spider”-diagrams (terminology for special Frobenius normal form, originating in Coecke & Paquette 2008, p. 6, Coecke & Duncan 2008, Thm. 1) of the ZX-calculus:

• Bob Coecke, Ross Duncan, §3 in: Interacting Quantum Observables, in Automata, Languages and Programming. ICALP 2008, Lecture Notes in Computer Science 5126, Springer (2008) $[$doi:10.1007/978-3-540-70583-3_25$]$

• Aleks Kissinger, §§2 in: Graph Rewrite Systems for Classical Structures in $\dagger$-Symmetric Monoidal Categories, MSc thesis, Oxford (2008) $[$pdf, pdf$]$

• Aleks Kissinger, §4 in: Exploring a Quantum Theory with Graph Rewriting and Computer Algebra, in: Intelligent Computer Mathematics. CICM 2009, Lecture Notes in Computer Science 5625 (2009) 90-105 $[$doi:10.1007/978-3-642-02614-0_12$]$

• Bob Coecke, Ross Duncan, Def. 6.4 in: Interacting Quantum Observables: Categorical Algebra and Diagrammatics, New J. Phys. 13 (2011) 043016 $[$arXiv:0906.4725, doi:10.1088/1367-2630/13/4/043016$]$

ZX-Calculus

Evolution of the “classical structures”-Frobenius algebra (above) into the “spider”-ingredient of the ZX-calculus for specific control of quantum circuit-diagrams:

• Bob Coecke, Ross Duncan, §3 in: Interacting Quantum Observables, in Automata, Languages and Programming. ICALP 2008, Lecture Notes in Computer Science 5126, Springer (2008) $[$doi:10.1007/978-3-540-70583-3_25$]$

• Aleks Kissinger, Graph Rewrite Systems for Classical Structures in $\dagger$-Symmetric Monoidal Categories, MSc thesis, Oxford (2008) $[$pdf, pdf$]$

• Aleks Kissinger, Exploring a Quantum Theory with Graph Rewriting and Computer Algebra, in: Intelligent Computer Mathematics. CICM 2009, Lecture Notes in Computer Science 5625 (2009) 90-105 $[$doi:10.1007/978-3-642-02614-0_12$]$

• Bob Coecke, Ross Duncan, Interacting Quantum Observables: Categorical Algebra and Diagrammatics, New J. Phys. 13 (2011) 043016 $[$arXiv:0906.4725, doi:10.1088/1367-2630/13/4/043016$]$

Relating the ZX-calculus to braided fusion categories for anyon braiding:

• Fatimah Rita Ahmadi, Aleks Kissinger, Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics $[$arXiv:2211.03855$]$

The role of complex numbers in general $\dagger$-compact categories:

• Jamie Vicary, Completeness of $\dagger$-categories and the complex numbers (arXiv:0807.2927)