aa r X i v : . [ qu a n t - ph ] M a r Contextuality and the fundamental theorem of noncommutativealgebra
Arkady Bolotin ∗ Ben-Gurion University of the Negev, Beersheba (Israel)
March 21, 2018
Abstract
In the paper it is shown that the Kochen-Specker theorem follows from Burnside’s theorem onnoncommutative algebras. Accordingly, contextuality (as an impossibility of assigning binaryvalues to projection operators independently of their contexts) is merely an inference from Burn-side’s fundamental theorem of the algebra of linear transformations on a Hilbert space of finitedimension.
Keywords:
Quantum mechanics; Kochen-Specker theorem; Contextuality; Truth values; In-variant subspaces; Irreducible; Noncommutative algebra; Burnside’s Theorem
Consider the set of the propositions {⋄} where the symbol ⋄ stands for any proposition, compoundor simple.Let v C be a truth-value assignment function that denotes a truth valuation in a circumstance C ,that is, a mapping from the set {⋄} to the set of truth-values { v i } Ni =1 (where N ≥
2) relative to acircumstance of evaluation indicated by C : v C : {⋄} → { v i } Ni =1 . (1)Commonly (see, e.g., [1]), the image of ⋄ under v C is written using the double-bracket notation,namely, v C ( ⋄ ) = [[ ⋄ ]] C . (2)The truth-value assignment function v C expresses the notion of not-yet-verified truth values : Itspecifies in advance the truth-value obtained from the verification of a proposition. ∗ Email : arkadyv @ bgu.ac.il
1o relate the set {⋄} to the states of a physical system, one can employ a predicate – i.e., astatement whose truth value depends on the values of its variables. For example, in the case ofa system associated with the classical phase space Γ, the predicate P on Γ can be defined as afunction from the phase space to the set of truth-values: P : Γ → { v i } Ni =1 . (3)Let { P ⋄ } denote the set of the predicates uniquely (i.e., one-to-one) connected to the set of thepropositions {⋄} . Then, one can introduce the valuation equivalence v γ ( P ⋄ ) = [[ ⋄ ]] γ , (4)which signifies that the truth-value of the proposition ⋄ in the state γ ∈ Γ is equated with the valueof the corresponding predicate P ⋄ obtained in this state, i.e., v γ ( P ⋄ ) = P ⋄ ( γ ).Provided that in the case of a classical system a predicate is just an indicator function that onlytakes the values 0 or 1 (where 0 denotes the falsity and 1 denotes the truth), explicitly, v γ ( P ⋄ ) = (cid:26) , γ ∈ U ⋄ , γ ∈ V ⋄ , (5)in which U ⋄ and V ⋄ are some linear subspaces of Γ such that Γ = U ⋄ ⊕ V ⋄ , the relation between theset of the predicates and the set of truth-values is a bivaluation: v γ : { P ⋄ } → { , } . (6)Accordingly, the elements γ of the classical phase space Γ represent categorical properties that theclassical system possesses or does not. What is more, the bivaluation relation (6) is a total func-tion . This means that any proposition related to a classical system obeys the principle of bivalence (asserting that a proposition can be either true or false [2]).To define the truth-value assignment for a quantum system associated with a Hilbert space H , onecan assume the valuation equivalence analogous to (4) v | Ψ i ( ˆ P ⋄ ) = [[ ⋄ ]] | Ψ i , (7)where ˆ P ⋄ denotes a projection operator on H uniquely connected with a proposition ⋄ , while | Ψ i stands for a vector in H describing system’s state. In line with this equivalence, the truth value ofthe proposition ⋄ in the state | Ψ i is equated with the value of the corresponding projection operatorˆ P ⋄ obtained in this state.As it can be readily seen, the difference between the equivalence (7) and its classical counterpart(4) is not only one that in the former the argument of the value assignment function is an operator2n a Hilbert space H (instead of a predicate on the classical phase space Γ in the latter) but also(and more importantly) one that the relation between the set { ˆ P ⋄ } and the set { , } , namely, v | Ψ i : { ˆ P ⋄ } → { , } , (8)cannot be a total function in accordance with the Kochen-Specker theorem [3, 4]. This means thatat least one proposition related to a quantum system does not obey the principle of bivalence: Thesaid proposition may have a truth-value different from 0 and 1 (as it is argued in [5, 6]) or notruth-value at all (in line with the supervaluation approach suggested in [7]).Assuming that the quantum value assignment function v | Ψ i can be presented as an indicator func-tion similarly to the case of a classical system, that is, v | Ψ i ( ˆ P ⋄ ) = (cid:26) , | Ψ i ∈ U ⋄ , | Ψ i ∈ V ⋄ , (9)where U ⋄ and V ⋄ are some linear subspaces in H such that H = U ⋄ ⊕ V ⋄ , the question is, whatalgebraic properties of the linear subspaces in H cause the failure of the principle of bivalence?Correspondingly, can the Kochen-Specker theorem be derived from the algebra over H ?Let us answer these questions in the presented paper. Recall the following definitions.
The column space (a.k.a. range ), ran( ˆ P ), of the projection (i.e.,self-adjoint and idempotent) operator ˆ P is the subset of the vectors | Ψ i ∈ H that are in the imageof ˆ P , namely, ran( ˆ P ) = n | Ψ i ∈ H : ˆ P | Ψ i = | Ψ i o . (10)Likewise, the null space (a.k.a. kernel ), ker( ˆ P ), of the projection operator ˆ P is the subset of thevectors | Ψ i ∈ H that are mapped to zero by ˆ P , namely,ker( ˆ P ) = n | Ψ i ∈ H : ˆ P | Ψ i = 0 o . (11)Thus, any ˆ P is the identity operator ˆ1 on ran( ˆ P ) and the zero operator ˆ0 on ker( ˆ P ).The column and null spaces are complementary in the same way as ˆ P and ˆ1 − ˆ P , that is,ran( ˆ P ) = ker(ˆ1 − ˆ P ) , (12)3er( ˆ P ) = ran(ˆ1 − ˆ P ) . (13)Moreover, they produce the direct sumran( ˆ P ) ⊕ ran( ˆ P ) = ran(ˆ1) = H , (14)and they are orthogonal to each other:ran( ˆ P ) ∩ ker( ˆ P ) = ran(ˆ0) = { } , (15)where 0 denotes the zero vector in any vector space and { } stands for the zero subspace. Thus,ker( ˆ P ) is the orthogonal complement of ran( ˆ P ), and vice versa.Also recall that a subspace U ⊆ H is called an invariant subspace under ˆ P if | Ψ i ∈ U = ⇒ ˆ P | Ψ i ∈ U , (16)that is, ˆ P ( U ) is contained in U and so ˆ P : U → U . (17)Obviously, the space H itself as well as the zero subspace { } are trivially invariant subspaces forany projection operator ˆ P . Observation 1.
For each projection operator ˆ P there are two nontrivial invariant subspaces,namely, ran( ˆ P ) and ker( ˆ P ) .Proof. To see this, let | Ψ i ∈ ran( ˆ P ). Since ˆ P | Ψ i = | Ψ i one gets ˆ P | Ψ i ∈ ran( ˆ P ), and so ˆ P :ran( ˆ P ) → ran( ˆ P ). Similarly, let | Ψ i ∈ ker( ˆ P ). This means that ˆ P | Ψ i = 0. On the other hand,0 ∈ ker( ˆ P ), which implies ˆ P : ker( ˆ P ) → ker( ˆ P ).The presence of two invariant subspaces for each projection operator ˆ P motivates the definition ofthe valuation v | Ψ i as a bivalent function , that is, v | Ψ i ( ˆ P ) = (cid:26) , | Ψ i ∈ ran( ˆ P )0 , | Ψ i ∈ ker( ˆ P ) . (18)Take the identity operator ˆ1. Given ran(ˆ1) = H and ker(ˆ1) = { } , from the said definition it followsthat for any | Ψ i ∈ H and | Ψ i 6 = 0, i.e., for any admissible state | Ψ i of a system, v | Ψ i (ˆ1) = 1. Thisindicates that the identity operator ˆ1 relates to a tautology ⊤ (i.e., a proposition that is true in4ny admissible state of the system), namely, v | Ψ i (ˆ1) = [[ ⊤ ]] | Ψ i = 1.For zero operator ˆ0, one gets in accordance with (18) that in any admissible state of the system, v | Ψ i (ˆ0) = 0. This implies that the zero operator ˆ0 relates to a contradiction ⊥ (i.e., a propositionthat is false in any admissible state of the system): v | Ψ i (ˆ0) = [[ ⊥ ]] | Ψ i = 0. Let L ( H ) denote the algebra of linear transformations on H and let Σ represents the collection ofprojection operators on H . Consider a nonempty subset Σ ( q ) ⊂ Σ comprising projection operatorsˆ P ( q ) i that meet the conditions ˆ P ( q ) i ˆ P ( q ) j = ˆ P ( q ) j ˆ P ( q ) i = ˆ0 , (19)where i = j , and X ˆ P ( q ) i ∈ Σ ( q ) ˆ P ( q ) i = ˆ1 . (20)Such a subset Σ ( q ) is said to be a maximal (a.k.a. complete ) context .Let Lat( ˆ P ( q ) i ) be the family of subspaces invariant under the projection operator ˆ P ( q ) i , namely,Lat( ˆ P ( q ) i ) = n ran(ˆ0) , ran( ˆ P ( q ) i ) , ker( ˆ P ( q ) i ) , ran(ˆ1) o , (21)such that Lat( ˆ P ( q ) i ) forms a lattice : The operation meet ⊓ of this lattice corresponds to the in-terception Q ∩ W and the lattice operation join ⊔ corresponds to the smallest closed subspaceof Lat( ˆ P ( q ) i ) containing the union Q ∪ W , where
Q 6 = W and Q , W ∈
Lat( ˆ P ( q ) i ). This lattice isbounded, i.e., it has the greatest element ran(ˆ1) = H and the least element ran(ˆ0) = { } .Now, consider the invariant subspaces invariant under each projection operator ˆ P ( q ) i in the maximalcontext Σ ( q ) : Lat(Σ ( q ) ) = \ ˆ P ( q ) i ∈ Σ ( q ) Lat( ˆ P ( q ) i ) . (22)Given that for any maximal context the following interceptions holdran( ˆ P ( q ) i ) ∩ ran( ˆ P ( q ) j ) = ran( ˆ P ( q ) i ˆ P ( q ) j ) = { } , (23)ran( ˆ P ( q ) i ) ∩ ker( ˆ P ( q ) j ) = ran( ˆ P ( q ) i ) ∩ ran(ˆ1 − ˆ P ( q ) j ) = ran( ˆ P ( q ) i ) , (24)5er( ˆ P ( q ) i ) ∩ ran( ˆ P ( q ) j ) = ran(ˆ1 − ˆ P ( q ) i ) ∩ ker( ˆ P ( q ) j ) = ran( ˆ P ( q ) j ) , (25)ker( ˆ P ( q ) i ) ∩ ker( ˆ P ( q ) j ) = ran(ˆ1 − ˆ P ( q ) i − ˆ P ( q ) j ) = ran (cid:18) X k = i,j ˆ P ( q ) k (cid:19) , (26)the interception Lat(Σ ( q ) ) can be presented asLat(Σ ( q ) ) = (cid:26) ran(ˆ0) , n ran( ˆ P ( q ) i ) , ker( ˆ P ( q ) i ) o i =1 , R ( q ) , ran(ˆ1) (cid:27) , (27)where R ( q ) stands for R ( q ) = ( ran (cid:18) X k = i,j ˆ P ( q ) k (cid:19) , ran (cid:18) X j = i,k ˆ P ( q ) j (cid:19) , ran (cid:18) X i = j,k ˆ P ( q ) i (cid:19) , ran (cid:18) X l = i,j,k ˆ P ( q ) l (cid:19) , . . . ) . (28)Suppose that the system is prepared in a pure state | Ψ i ∈ ran( ˆ P ( q ) i ). Then, according to (18), v | Ψ i ( ˆ P ( q ) i ) = 1. As a result of (23) (expressing that all ran( ˆ P ( q ) i ) are orthogonal to each other), thevector | Ψ i also resides in the null space of any other projection operator in the maximal contextΣ ( q ) , i.e., | Ψ i ∈ ker( ˆ P ( q ) j ), which gives v | Ψ i ( ˆ P ( q ) j ) = 0. Hence, in the maximal context Σ ( q ) only oneprojection operator can be assigned the value 1, and so v | Ψ i (cid:18) X ˆ P ( q ) i ∈ Σ ( q ) ˆ P ( q ) i (cid:19) = X ˆ P ( q ) i ∈ Σ ( q ) v | Ψ i (cid:16) ˆ P ( q ) i (cid:17) = 1 . (29)Consider the invariant subspaces invariant under each maximal context Σ ( q ) ⊂ Σ, that is,Lat(Σ) = \ Σ ( q ) ⊂ Σ Lat(Σ ( q ) ) . (30) Observation 2.
Where
Lat(Σ) to contain some nontrivial invariant subspace(s), a logic definedas the relations between projection operators in Σ would have a bivalent semantics.Proof. Suppose that Lat(Σ) contains a nontrivial invariant subspace ran( ˆ P ( q ) i ). Since Lat(Σ) isthe intersection of all the lattices Lat(Σ ( q ) ), this would mean that ran( ˆ P ( q ) i ) is the member ofeach Lat(Σ ( q ) ) and thus orthogonal to all other column spaces in each Σ ( q ) . In the case where v | Ψ i ( ˆ P ( q ) i ) = 1, all other truth values of the projection operators in Σ would be zero, which wouldproduce P i v | Ψ i ( ˆ P ( q ) i ) = 1 for each Σ ( q ) . In this way, all the projection operators in Σ would obeythe principle of bivalence. 6ssume Σ = L ( H ), i.e., the collection Σ includes all the projection operators on system’s Hilbertspace H . If H is finite-dimensional (and dim( H ) is greater than 1), then, according to Burnside’sTheorem [8, 9, 10], Lat(Σ) is irreducible , i.e., has no nontrivial invariant subspace:Σ = L ( H ) = ⇒ Lat(Σ) = (cid:8) ran(ˆ0) , ran(ˆ1) (cid:9) . (31) Observation 3.
This means that for the given system the principle of bivalence fails.Proof.
Suppose that P i v | Ψ i ( ˆ P ( q ) i ) = 1 for the certain Σ ( q ) . Because it is irreducible, Lat(Σ)does not have any ran( ˆ P ( q ) i ) ∈ Lat(Σ ( q ) ). So, at least one nontrivial invariant subspace, say,ran( ˆ P ( w ) k ) ∈ Lat(Σ ( w ) ), where w = q , is not orthogonal to ran( ˆ P ( q ) i ) ∈ Lat(Σ ( q ) ). In consequence,the proposition associated with ˆ P ( w ) k cannot be bivalent alongside the propositions connected withˆ P ( q ) i ∈ Σ ( q ) , i.e., v | Ψ i ( ˆ P ( w ) k ) = { , } . Corollary 1.
There is a collection of the projection operators relating to a system with a finite-dimensional Hilbert space, namely, Σ ′ ⊂ L ( H ) , such that Lat(Σ ′ ) contains no nontrivial elements.Proof. This follows directly from the version of Burnside’s Theorem presented in (31). Truly, sincefor the said system Lat(Σ) is irreducible, there must exist maximal contexts Σ ( q ) and Σ ( w ) whosenontrivial column spaces ran( ˆ P ( q ) i ) ∈ Lat(Σ ( q ) ) and ran( ˆ P ( w ) k ) ∈ Lat(Σ ( w ) ) are not orthogonal toeach other. Correspondingly, the family of subspaces Lat(Σ ′ ) invariant under both Σ ( q ) and Σ ( w ) would have no nontrivial invariant subspace. C Let us demonstrate the application of Burnside’s Theorem to the set of the projection operatorson the two dimensional Hilbert space H = C . Comprised of the eigenvectors of the Pauli matrices σ z , σ x and σ y these projection operators areˆ P ( z )1 = (cid:20) (cid:21) , ˆ P ( z )2 = (cid:20) (cid:21) , (32)ˆ P ( x )1 = 12 (cid:20) (cid:21) , ˆ P ( x )2 = 12 (cid:20) − − (cid:21) , (33)ˆ P ( y )1 = 12 (cid:20) − ii (cid:21) , ˆ P ( y )2 = 12 (cid:20) i − i (cid:21) . (34)Since ˆ P ( q )1 ˆ P ( q )2 = ˆ P ( q )2 ˆ P ( q )1 = ˆ0 and ˆ P ( q )1 + ˆ P ( q )2 = ˆ1, they make up three maximal contexts Σ ( q ) ,namely, Σ ( q ) = n ˆ P ( q ) i o i =1 ⊂ L ( C ) , (35)7here L ( C ) denotes the collection of all linear transformations C → C (i.e., the algebra over C ).The invariant subspaces Lat(Σ ( q ) ) invariant under each ˆ P ( q ) i ∈ Σ ( q ) take the formLat(Σ ( z ) ) = (cid:26) { } , (cid:26)(cid:20) a (cid:21)(cid:27) , (cid:26)(cid:20) a (cid:21)(cid:27) , (cid:26)(cid:20) b (cid:21)(cid:27) , (cid:26)(cid:20) b (cid:21)(cid:27) , C (cid:27) , (36)Lat(Σ ( x ) ) = (cid:26) { } , (cid:26)(cid:20) aa (cid:21)(cid:27) , (cid:26)(cid:20) a − a (cid:21)(cid:27) , (cid:26)(cid:20) b − b (cid:21)(cid:27) , (cid:26)(cid:20) bb (cid:21)(cid:27) , C (cid:27) , (37)Lat(Σ ( z ) ) = (cid:26) { } , (cid:26)(cid:20) iaa (cid:21)(cid:27) , (cid:26)(cid:20) aia (cid:21)(cid:27) , (cid:26)(cid:20) bib (cid:21)(cid:27) , (cid:26)(cid:20) ibb (cid:21)(cid:27) , C (cid:27) , (38)where a, b ∈ R .Within each maximal context Lat(Σ ( q ) ) the corresponding projection operators ˆ P ( q ) i are bivalent.For example, suppose the system is prepared in the state | Ψ i = [1 , T , then | Ψ i = (cid:20) (cid:21) ∈ ran( ˆ P ( z )1 ) = (cid:26)(cid:20) a (cid:21)(cid:27) = ⇒ v | Ψ i ( ˆ P ( z )1 ) = 1ker( ˆ P ( z )2 ) = (cid:26)(cid:20) b (cid:21)(cid:27) = ⇒ v | Ψ i ( ˆ P ( z )2 ) = 0 . (39)Since the Pauli matrices σ q form an orthogonal basis for the space C , any matrix M × ∈ C canbe expressed as M × = w I × + X q =1 u q σ q , (40)where w and u q are complex numbers, and I × is the identity matrix on C .Consequently, the collection of the maximal contexts Σ = { Σ ( z ) , Σ ( x ) , Σ ( y ) } contains all the projec-tion operators on C . As L ( C ) is the span of all such operators, Σ = L ( C ).By Burnside’s Theorem it must be thenLat(Σ) = Lat(Σ ( z ) ) ∩ Lat(Σ ( x ) ) ∩ Lat(Σ ( y ) ) = (cid:8) { } , C (cid:9) , (41)which implies that the bivaluation v | Ψ i : { Σ ( q ) } → { , } cannot be a total function.8 Concluding remarks
As it has been just shown, the Kochen-Specker theorem is the consequence of Burnside’s theoremon the algebra of linear transformations on H .Indeed, according to the Kochen-Specker theorem, in a Hilbert space H of a finite dimension (greaterthan 3), it is impossible to assign to every projection operator in a set Σ ′ one of its eigenvalues, i.e.,1 or 0, in such a way that for any admissible state of the system | Ψ i the values v | Ψ i ( ˆ P ( w ) i ) assignedto members ˆ P ( w ) i of a maximal context Σ ( w ) ⊂ Σ ′ resolve to 1, that is, P i v | Ψ i ( ˆ P ( w ) i ) = 1.On the other hand, the bivaluation of a projection operator is associated with the existence ofits two nontrivial invariant subspaces. So, the inability to assign binary values, 1 or 0, to eachprojection operator in the set Σ ′ is the consequence of the fact that the family of subspaces Lat(Σ ′ )invariant under each maximal context in the set Σ ′ is irreducible, i.e., has no nontrivial invariantsubspace.In this way, contextuality (as an impossibility of assigning binary values to projection operatorsindependently of their maximal contexts) is merely an inference from the fundamental theorem ofnoncommutative algebra, i.e., Burnside’s Theorem.It is worth mentioning that this theorem fails for finite dimensional vector spaces over the reals[11], such as the classical phase space Γ. This can be regarded as the algebraic reason for bivalenceof classical mechanics. References [1] D. van Dalen.
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