While having been the uncontested workhorse in the field for more than eighty years, the Dirac-von Neumann formalism of quantum mechanics is neither terribly elegant, nor universally practical.

For one, the use of Hilbert spaces results in structural overkill: Every physical system \(\Psi\) corresponds to an infinite number of unit vector representations \(e^{i\theta}|\Psi\rangle,\ \theta\in\mathbb{R}\). For another, the true clumsiness of the Dirac-von Neumann approach lies in its failure to adequately capture the compositionality of the theory.

To illustrate our claims, consider the following example which is taken from a quantum secret sharing protocol [1, p.99]:

It needs to be shown that this expression is proportional to \(|0\rangle+e^{i\alpha}|1\rangle\). This can be done brute-force, using the matrix representations of the individual gates. The above circuit involves the manipulation of four distinct physical \(2^n\) qubits, which are the smallest informational unit in the realm of quantum mechanics. Matrix size scales as \(2^n\), meaning that \(4\) qubits result in $16\times 16$ matrices, with roughly $n^3=4096$ scalar multiplications performed per matrix multiplication.

Performing these calculations densely on a classical computer won’t work beyond roughly a dozen qubits. Exploiting the problem’s underlying sparsity is usually difficult due to its non-trivial structure. Many recent important quantum computation models, such as measurement-based quantum computation [2], depend on thousands or millions of entangled qubits.

But even if tractable, brute-force calculations yield little physical insight. If one wishes to understand the physical operation of the described system better, one may resort to circuit representation:

How does one arrive at this circuit diagram? Each qubit corresponds to a distinct line. The tensor product $\otimes$ is represented by juxtaposition. Unitary evolution, corresponding to $\circ$, is represented by sequential composition. Two-qubit gates, such as CNot, are depicted by connecting the lines of the respective qubits, which intuitively indicates that there is some information flow between control and gate. The power of this pictorial representation lies in abstracting away the well-known interchange identity \((g\\circ f)\\otimes(g’\\circ f’) = (g\\otimes g’)\\circ(f\\otimes f’)\) with $f,f’,g,g’$ being linear maps, which allows to trace individual qubits through the system. Provided that only standard gates are being employed and the network isn’t too dense, this method can be used to avoid the complexity of matrix manipulations.

However, questions remain: Firstly, can the map \(\\text{Dirac-von Neumann term}\\mapsto \\text{circuit diagram}\) be expressed in a mathematically rigorous fashion? Secondly, there are some obvious rewrites that can be directly performed on the circuit diagram, for example, $\text{CNot}_{12} \circ \text{CNot}_{12} := \text{Id}_{12}$ corresponding to:

Are there further opportunities for graphical manipulation? Apart from being intrinsically suited to human abilities, this could open new avenues for efficient automatized manipulation beyond physically-unaware sparse-matrix methods.


In their seminal paper [3], Abramsky and Coecke developed a rigorous mathematical framework for translating Dirac-von Neumann terms into circuit diagrams based on
category theory. The study of interactions between complementary observables lead to a simple, yet powerful, set of graphical rewrite rules, dubbed the ZX-calculus [4].

Returning to the above example, the ZX-calculus allows to convert the circuit diagram to the following form:

A short series of rewrites transforms this diagram to the expected form:

Categorical quantum mechanics doesn’t stop there. The mathematical framework is sufficiently abstract to accommodate a wide range of graphical calculi. Instead of interactions between complementary qubits, one can study the interactions of three-partite entangled states. This results in the graphical GHZ/W-calculus [5]. The ZX-calculus diagram from above may now be represented as follows:

A different approach may be taken to handle so-called mixed states, which correspond to classical superpositions of quantum states: Selinger’s CPM construction allows the graphical representation of density matrices and completely positive maps using the ZX-calculus [6].

For example, suppose one would like to apply a conditional $X$-gate to the above example state with probability $\frac{1}{2}$, leaving the state intact otherwise. In von Neumann-formalism, the resulting density matrix is obtained by treating both scenarios individually. In Selinger’s CPM construction, this corresponds to the following graphical manipulation:

, where the resultant diagram corresponds to the maximally mixed state $|0\rangle\langle 0| + |1\rangle\langle 1|$. This may not immediately seem like a clear advantage. Measurement correlations of and conditional gates on $n$-qubit entangled states, however, in the worst case result in $2^n$ distinct scenarios to be considered. The CPM construction comes with clear and simple high-level rules that help to prune unnecessary complexity, whereas such strategies may be hard to come by using the Dirac-von Neumann notation.

Categorical Quantum Mechanics gives rise to more than just graphical calculi. It is a thoroughly principled framework that sheds light on the underlying compositionality of the physical world.

Yet again, Categorical quantum mechanics offers much more than just a rigorous framework for graphical calculi. It provides an abstraction away from Hilbert spaces based on information-theoretic axioms. Many characteristic features of quantum information processing, such as the no-cloning theorem [7], quantum teleportation [8], entanglement swapping [9], quantum steering [10] and gate teleportation [11], are trivially encompassed [12]. The formalism also seems ideal for the study of multi-partite entanglement[13].

One way to better understand the oddness of quantum mechanics is to compare and contrast quantum phenomena within a large space of generalised theories. Categorical quantum mechanics helped to clarify issues around one particularly popular such theory, namely Spekken’s toy model [14]. This in turn may have deeper interpretational consequences [15].

Can categorical quantum mechanics help coming up with a theory of quantum gravity information processing?

But why is it so important to thoroughly understand the information-theoretic properties of quantum mechanics? Information is always physical, and conversely, every physical theory gives rise to an associated information processing theory. It is clear that a successful theory of quantum gravity will need to have both quantum mechanics and general relativity as limiting cases. It follows that a theory of quantum gravity information processing needs to, at least in the limit, respect both a general relativistic information processing theory and quantum information processing. This latter issue has received a lot of attention recently: Tentative theories of quantum gravity information processing have been suggested [16], and the interaction between motion under spatial curvature and quantum entanglement has been studied experimentally [17].

At the same time, there is an ongoing debate questioning the role of string theory in physics [18][19][20]. Undoubtedly, more than 30 years of research in string theory have yielded plenty of exciting mathematics [21], however, whether or not string theory will ever result in a useful formulation of physical reality seems as of yet undecidable. Even if string theory one day turns out to offer a concise description of physical reality, alternative routes may deepen our understanding of it.

We need more research into Categorical Quantum Mechanics!

Despite the unfortunate case that infinite dimensional Hilbert spaces have so far resisted full categorical treatment[31], the framework of Categorical Quantum Mechanics does hold some promise to yield insight into more fundamental questions. A particularly interesting feature is that its finite dimensional formalism seems to be well-suited to relativistic extensions [15].

References

[1] Hillebrand, A. Quantum protocols involving multiparticle entanglement and their representations in the ZX-calculus. MSc Thesis, University of Oxford (Sept. 2011)

[2] Roy, S. A compositional characterization of multipartite quantum
states.
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[3] Abramsky, S., and Coecke, B. A categorical semantics of quantum protocols. arXiv e-print quant-ph/0402130, Feb. 2004. Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004).

[4] Coecke, B., and Duncan, R. Interacting quantum observables: Categorical algebra and diagrammatics. arXiv e-print 0906.4725, June 2009. New J. Phys. 13 (2011) 043016.

[5] (GHZ/W calculus)

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[10] Schroedinger, E. Proc. Cam. Phil. Soc. 31, 555 (1935)

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[12] Coecke, B., and Duncan, R. Interacting quantum observables: Categorical algebra and diagrammatics. arXiv e-print 0906.4725, June 2009. New J. Phys. 13 (2011) 043016.

[13] Coecke, B., and Kissinger, A. The compositional structure of multipartite quantum entanglement. arXiv e-print 1002.2540, Feb. 2010.

[14] Coecke, B., and Edwards, B. Spekkens’s toy theory as a category of processes. arXiv e-print 1108.1978, Aug. 2011.

[15] Spekkens, R. W. In defense of the epistemic view of quantum states: a toy theory. arXiv e-print quant-ph/0401052, Jan. 2004. Phys. Rev. A 75, 032110 (2007).

[16] Hardy, L. Quantum gravity computers: On the theory of computation with indefinite causal structure. arXiv e-print quantph/0701019, Jan. 2007.

[17] Adesso, G., Fuentes-Schuller, I., and Ericsson, M. Continuous variable entanglement sharing in noninertial frames. Physical Review A 76, 6 (Dec. 2007), 062112.

[18] Duff, M. J. String and m-theory: answering the critics. arXiv eprint 1112.0788, Dec. 2011

[19] Smolin, L. The case for background independence. arXiv e-print hep-th/0507235, July 2005.

[20] Johansson, L.-G., and Matsubara, K. String theory and general methodology; a reciprocal evaluation. arXiv e-print 0912.3160, Dec. 2009. Stud.Hist.Phil.Mod.Phys.42:199-210,2011.

[21] J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/9711200.