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Strong Complementarity and Non-locality in Categorical Quantum Mechanics

Bob Coecke, Ross Duncan, Aleks Kissinger, Quanlong Wang

arXiv:1203.4988v2quant-phcs.LOmath.CT

TL;DR

The paper asks how non-locality and complementarity relate, and uses categorical quantum mechanics to generalize Mermin-style arguments. It introduces strong complementarity and shows that it is necessary for these arguments, while classifying strongly complementary observables in quantum theory.

  • Problem

    The paper investigates the relationship between non-locality and complementarity and how Mermin-style non-locality arguments extend across parties, measurement angles, and system dimensions.

    Source: §V · p. 7, §VIII · p. 9

  • Method

    The paper formulates Mermin’s argument using categorical quantum mechanics and algebraic structures defined within monoidal categories.

    Source: §VIII · p. 9, A. Some categorical background · p. 10

  • Results

    Strong complementarity is necessary for Mermin arguments, and the paper generalizes Mermin’s argument to multiple parties, arbitrary angles, and arbitrary-dimensional systems.

    Source: §I · p. 2, §VIII · p. 9

  • Takeaways & Limitations

    Strong complementarity has an operational interpretation in Mermin arguments, and strongly complementary observables are classified for quantum theory.

    Source: §VIII · p. 9

Abstract

from arXiv · show

Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.

I. INTRODUCTION

The paper uses categorical quantum mechanics to study how non-locality and complementarity are related, extending Mermin’s argument and introducing strong complementarity. It develops this approach in dagger-compact categories, whose compositional processes can be represented and manipulated diagrammatically.

I. INTRODUCTION

  • The paper focuses on compositional structure of processes, rather than only the convex probabilistic structure retained by generalized probabilistic theories.It frames this as an investigation of generalized process theories.Source: §I · p. 1
  • The paper casts Mermin’s argument in categorical quantum mechanics, generalizes it to multiple parties, arbitrary angles, and arbitrary-dimensional systems, and establishes strong complementarity as necessary for Mermin arguments.The necessity result gives strong complementarity an operational interpretation.Source: §I · p. 2
  • Strong complementarity connects the paper’s treatment of non-locality with its classification of strongly complementary observables in quantum theory.The paper also relates this stronger notion to ordinary complementarity.Source: §I · p. 2

A. Models of Physical Theories

  • Symmetric monoidal categories represent processes as morphisms, with sequential and parallel combinations represented by composition and tensor product.Finite-dimensional Hilbert spaces and linear maps provide one example; finite sets and relations provide another.Source: A. Models of Physical Theories · p. 2
  • Dagger-compact categories provide a framework for representing quantum concepts including Bell states and effects, unitarity, the Born rule, and complete positivity.The paper uses this framework to express its results diagrammatically.Source: A. Models of Physical Theories · p. 2

B. †-Compact Categories and Diagrammatic Calculus

  • In the diagrammatic calculus, wires denote systems and boxes denote processes; composition is plugging outputs into inputs, while tensor product is juxtaposition.The notation represents spatial separation horizontally and temporal separation vertically.Source: B. †-Compact Categories and… · pp. 2–3

C. Generalised Observables

The section represents observables as algebraic structures that correspond to orthonormal bases in finite-dimensional Hilbert spaces. Their classical points behave like eigenstates, and connected diagrams built from the observable’s operations reduce to a single spider.

C. Generalised Observables

  • In FHilb, orthonormal bases correspond one-to-one with †-special commutative Frobenius algebras, also called observable structures.A non-degenerate quantum observable is represented by its orthonormal basis.Source: C. Generalised Observables · p. 3
  • Any connected diagram built from one observable structure’s multiplication, unit, comultiplication, and counit is equal to a single spider.This spider rule captures how the observable’s operations compose.Source: C. Generalised Observables · p. 3
  • FRel has distinct observable structures, and the two specified structures on a two-element set are strongly complementary.This gives a non-Hilbert-space example of observable structures.Source: C. Generalised Observables · p. 3
  • An observable’s classical points are states copied by its comultiplication and deleted by its counit, abstractly paralleling eigenvectors.In FHilb, an observable structure copies basis vectors and erases them.Source: C. Generalised Observables · p. 3

D. Phase Group for an Observable Structure

The section defines phase groups from observable structures and uses their elements to decorate spiders with phases. It also abstracts measurement and probability distributions in dagger symmetric monoidal categories, linking classical and quantum data through measurement and preparation maps.

D. Phase Group for an Observable Structure

  • For an observable structure, its phase group is the Abelian group of points satisfying the stated multiplication condition; in the qubit case it is the circle group.For a qubit observable, the group consists of points unbiased to its eigenstates.Source: D. Phase Group for an Observable… · p. 4
  • A phase-group element defines a one-input, one-output phased spider that is unitary.Phased spiders compose by adding their phase labels.Source: D. Phase Group for an Observable… · p. 4

E. Generalised Classical-Quantum Interaction

  • The paper defines abstract measurements and Born vectors for an observable structure, extending Born vectors to tensor products of observables.The adjoint measurement map prepares a probabilistic mixture of outcome states with probabilities given by the Born vector.Source: E. Generalised Classical-Quantum… · pp. 4–5
  • The measurement map and its adjoint connect classical and quantum data, represented respectively by single and double wires.This diagrammatic paradigm applies to arbitrary observables in any †-SMC.Source: E. Generalised Classical-Quantum… · p. 5
  • In FHilb, phases relative to one observable can implement measurements associated with another; Pauli spin-Z phases yield X and Y measurements.Both measurements produce Born vectors of outcome probabilities with respect to the basis.Source: E. Generalised Classical-Quantum… · p. 5

III. STRONG COMPLEMENTARITY

The paper defines strong complementarity as a strengthening of ordinary complementarity and relates it to algebraic structure. It also characterizes the phase-group conditions associated with strongly complementary observables.

III. STRONG COMPLEMENTARITY

  • In FHilb, complementary observables can be represented by coherent observable structures whose classical points correspond to the eigenbases of the observables.This construction uses the freedom to choose global phases for the eigenbasis vectors.Source: §III · p. 5
  • Strong complementarity implies ordinary complementarity, but coherent complementary observables need not be strongly complementary.Strongly complementary observables also form a scaled Hopf algebra; the result relies on both structures being Frobenius algebras.Source: §III · pp. 5–6

A. Strong Complementarity and Phase Groups

  • Strong complementarity makes the classical points of one observable, after scaling, a subgroup of the other observable’s phase group.The converse holds when the observable has enough classical points.Source: A. Strong Complementarity and Phase… · p. 6
  • For strongly complementary observables, every element of that subgroup acts as a homomorphism for the other observable; with enough classical points, this condition characterizes strong complementarity.The homomorphism condition is stated as a converse when the relevant observable has enough classical points.Source: A. Strong Complementarity and Phase… · p. 6

B. Classification of Strong Complementarity in FHilb

The paper classifies strongly complementary observables in finite-dimensional Hilbert spaces and computes GHZ measurement distributions diagrammatically. The distributions are symmetric for arbitrary-angle measurements, and the computation covers arbitrary systems and phases.

B. Classification of Strong Complementarity in FHilb

  • Every strongly complementary observable pair in FHilb has the stated form indexed by a finite Abelian group, and every such pair is strongly complementary.This gives a classification of strongly complementary pairs in quantum theory.Source: B. Classification of Strong… · p. 6
  • For D ≥ 2, the largest set of pairwise strongly complementary observables has size two, and such a pair exists in every such dimension.The upper bound is stated in Theorem 3.11, and existence follows from Corollary 3.10.Source: B. Classification of Strong… · p. 6

IV. DIAGRAMMATIC COMPUTATION OF GHZ MEASUREMENT OUTCOME DISTRIBUTIONS

  • The diagrammatic computation treats n measurements at arbitrary phases on a generalized GHZ state for systems of any type.The paper introduces the computation to support a graphical Mermin/GHZ-style argument.Source: §IV · pp. 6–7
  • For arbitrary-angle measurements on a GHZ state, the resulting classical outcome distribution is symmetric.The theorem states symmetry for measurements of each system against an arbitrary angle.Source: §IV · p. 7
  • For three qubits, the three settings XYY, YXY, and YYX yield the same outcome distribution.The paper computes the XXX distribution separately from these three equivalent cases.Source: §IV · p. 7

V. MERMIN’S NON-LOCALITY ARGUMENT IN CQM

The section formulates the three-party Mermin scenario in CQM using hidden states that assign definite outcomes to measurements. A parity constraint on the combined measurement data conflicts with the locality assumption.

V. MERMIN’S NON-LOCALITY ARGUMENT IN CQM

  • For XXX, XYY, YXY, and YYX, locality models outcomes with hidden states assigning definite results to measurements on each subsystem.The hidden states are distributed probabilistically to simulate the theory's outcome probabilities.Source: §V · p. 7
  • The combined experimental data must not rule out any hidden state allowed by the locality model, even though the four settings cannot be measured simultaneously.The argument concerns possible hidden states, not only probabilities.Source: §V · pp. 7–8
  • A function that computes each experiment's parity and then the overall parity of the three variations is constant on the product distribution of the settings.The construction uses the support relation between the hidden-state distribution and the product of the setting distributions.Source: §V · p. 8
  • Substituting the locality condition into the consistency condition yields a contradiction, using the Z2 group structure to simplify the diagrammatic expression.The simplification deletes pairs of parallel edges connecting dots of different colours.Source: §V · p. 8

VI. GENERALISED NON-LOCALITY ARGUMENTS

The generalized argument extends the symmetry-based contradiction to broader lists of measurement settings. It applies under stated multiplicity and distinctness conditions, which also identify when the argument does not hold.

VI. GENERALISED NON-LOCALITY ARGUMENTS

  • The generalized construction frames the conflict as experimental data excluding symmetry imposed on hidden states by locality.Source: §VI · p. 8
  • The variation settings must produce a symmetric expression after applying Corollary 3.8.The multiplicity requirement ensures the needed symmetry.Source: §VI · p. 8
  • The generalized Mermin argument applies when each same-system measurement occurs in the variations a multiple of exp(K) times and the control and variation points are distinct classical points.Both conditions are required by Theorem 6.1.Source: §VI · p. 8
  • The theorem yields many generalized Mermin-type arguments and characterizes situations in which such an argument fails.Source: §VI · pp. 8–9

VII. NECESSITY OF STRONG COMPLEMENTARITY

The paper derives strong complementarity as a necessary condition for Mermin arguments under specified measurement assumptions, then relates that result to broader generalizations and diagrammatic insights. Its conclusion highlights which forms of complementarity matter and identifies possible extensions.

VII. NECESSITY OF STRONG COMPLEMENTARITY

  • For a coherent pair of observables, sharpness of the third outcome necessitates strong complementarity.The setup assumes phase-related measurements, coherence, and that after two measurements the remaining system is in an eigenstate of the third.Source: §VII · p. 9

VIII. CONCLUSION AND OUTLOOK

  • The paper generalizes Mermin's argument to multiple parties, arbitrary angles, and systems of arbitrary dimension.Source: §VIII · p. 9
  • Categorical quantum mechanics simplifies the computations using contraction of labeled nodes and commutation of differently colored multiplications and comultiplications.Source: §VIII · p. 9
  • The diagrammatic account depicts non-locality through the interaction of measurement phases across the GHZ state.Source: §VIII · p. 9
  • The paper gives strong complementarity an operational interpretation and classifies strongly complementary observables for quantum theory.Source: §VIII · p. 9
  • In the generalized argument, strong complementarity is essential, while complementarity between the X and Y observables is not essential.The relevant pair is the observable characterizing the GHZ state and the one whose classical points form a subgroup of the corresponding phase group.Source: §VIII · p. 9
  • The authors propose studying general process theories to investigate how key quantum concepts persist beyond quantum theory.Source: §VIII · pp. 9–10
  • The authors suggest that their results could support generalizations of quantum secret-sharing schemes based on X or Y measurements on a qubit GHZ state.They also point to possible treatment of other quantum-informatic protocols.Source: §VIII · p. 10

APPENDIX A ADDITIONAL BACKGROUND

The appendix supplies categorical background and algebraic tools used in the paper. It introduces monoids and comonoids, observable structures, and bialgebra normal forms that support diagrammatic derivations.

APPENDIX A ADDITIONAL BACKGROUND

  • The paper's categorical setting uses dagger symmetric monoidal categories, with strict monoidal structure assumed throughout.A dagger is an involutive functor that acts identically on objects.Source: A. Some categorical background · p. 10
  • The paper uses internal algebraic structures in monoidal categories, including monoids, comonoids, and their interacting algebraic forms.In finite-dimensional Hilbert spaces, a basis supplies copying and erasing maps for a comonoid.Source: A. Some categorical background · pp. 10–11
  • Connected diagrams built from a special commutative Frobenius algebra reduce to a single spider, the basis of the paper's spider notation.Source: A. Some categorical background · p. 11
  • A dagger special commutative Frobenius algebra defines an observable structure; in FHilb these structures arise from copying an orthonormal basis.Source: A. Some categorical background · p. 11
  • Bialgebra maps admit a factorization into comonoid structure, a permutation, and monoid structure, with paths preserved from every input to every output in the discussed normal form.The appendix uses this result together with the spider theorem in a proof of Corollary 4.1.Source: A. Some categorical background · pp. 11–12

B. Proof of Proposition 7.1

Proposition 7.1 establishes that Equation (16) implies strong complementarity for coherent observables. The proof concludes that any coherent pair satisfying the equation is strongly complementary.

B. Proof of Proposition 7.1

C. Some QM background

This background section introduces finite-dimensional quantum systems, measurement, and entanglement. It uses these concepts to motivate complementarity and non-local correlations.

C. Some QM background

  • A quantum system is represented by a finite-dimensional complex Hilbert space, and composite systems use tensor products that admit entangled states.Entangled states do not decompose into separate states of the subsystems.Source: C. Some QM background · pp. 12–13
  • Measurement outcomes are probabilistic in general, and measurement changes the state to the eigenvector associated with the observed outcome.Repeated measurement of the same observable then produces consistent results.Source: C. Some QM background · p. 13
  • Complementary observables are maximally incompatible; for Pauli X and Z, knowing one observable leaves the other's outcomes equiprobable.Measuring one can erase the previous value of the other.Source: C. Some QM background · p. 13
  • Entanglement combined with local measurements gives rise to non-local correlations, illustrated by perfect correlations between two parties measuring a Bell state.After one party's Z-basis measurement, the other party is guaranteed the same outcome.Source: C. Some QM background · p. 13
  • Quantum systems undisturbed by measurement evolve unitarily, and sequential or parallel quantum logic gates form the quantum circuit model.Source: C. Some QM background · p. 13

D. More Physical Models

The paper situates categorical quantum mechanics among other dagger-compact models and subcategories. Examples include classical process models and the Stab and Spek theories, whose phase groups distinguish them.

D. More Physical Models

  • Dagger-compact categories include models beyond quantum theory, including nCob and subcategories representing classical, reversible, or stochastic processes.FSet models classical processes, Perm reversible ones, and Stoch stochastic ones.Source: D. More Physical Models · pp. 13–14
  • Stab and Spek are quantum-like subcategories, representing the stabilizer restriction and Spekkens' toy theory, respectively.Stab restricts qubits to the six X, Y, and Z eigenstates; Spek is described as local despite quantum-like features.Source: D. More Physical Models · p. 14
  • Stab and Spek are generated by one observable and S(4) unitaries, while their phase groups differ: Z4 for Stab and Z2 × Z2 for Spek.The phase-group distinction captures essential differences between the theories, including their non-locality.Source: D. More Physical Models · p. 14

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