The Bell states in noncommutative algebraic geometry

We introduce new mathematical aspects of the Bell states using matrix factorizations, nonnoetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial $p$ consists of two matrices $\phi_1,\phi_2$ such that $\phi_1\phi_2 = \phi_2\phi_1 = p \operatorname{id}$. Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on nonnoetherian singularities. Moreover, we find that the collapse of a Bell state is intimately related to the representation theory of the noncommutative blowup along its singular support. This presents an exchange in geometry: the nonlocal commutative spacetime of the entangled state emerges from an underlying local noncommutative spacetime.

Bell states. We introduce the notion that the spacetime nonlocality inherent in an entangled pair of particles (or more generally, qubits) emerges from an underlying local geometry which is noncommutative.
We briefly outline our results. In Section 2 we introduce a modification of quantum mechanics where the coefficient ring C of a complex Hilbert space is enlarged to the ring of matrices over C, We establish density matrices, inner products, normalization, and Born's rule in this setting.
In Section 3 we use this modification to factorize the Bell states using matrices. For example, the state Ψ = 1 √ 2 (↑ a ↓ b − ↓ a ↑ b ) ∈ H a ⊗ C H b ∼ = C 2 ⊗ C C 2 of two entangled particles a and b admits the matrix factorization Our first main result is the following.
Theorem A. (Theorem 3.1.) The emergent state ψ := Ψ1 2 is a separable product of two mixed states, each consisting of two pure states.
In Section 4.1 we introduce a new algebraic characterization of entanglement in an idealized setting where spacetime is an algebraic variety Max S with coordinate ring S. A commutative ring S is said to be noetherian if every ideal in S is finitely generated, and otherwise is nonnoetherian. As introduced in [B1, Section 2], a geometric space (variety or scheme) whose algebra of functions is nonnoetherian is often nonlocal, in the sense that it contains curves, surfaces, or other positive dimensional subvarieties that are single 'spread-out' points. 1 Using this property, the Einstein-Podolsky-Rosen nonlocality of Ψ [EPR] is captured by the nonnoetherian singularity where I ⊂ S is the ideal consisting of all polynomial functions on spacetime that vanish along the support of Ψ. The ring R 'sees' the support Z(I) as a single point since I is a maximal ideal of R (though I is a non-maximal ideal of S).
In Section 4.2 we present an exchange in geometry:

nonlocal commutative local noncommutative
The exchange comes about from the noncommutative blowup [L, Section R] of R at the point Z(I), 2 This endomorphism ring may be viewed as a coordinate ring of matrix-valued functions on spacetime Max S. Furthermore, it replaces the nonlocal point Z(I) of R with the set of distinct spacetime points in Z(I). Indeed, denote by d A ∈ Z ≥0 the maximal dimension of the simple (i.e., irreducible) representations of A. Then the following holds.
Proposition B. (Proposition 4.7.) The representation space To note, the space R(A) is important in the study of noncommutative resolutions of singularities, as it is often parameterized by a commutative resolution of the center of A.
We then introduce the following diagram to relate the matrix factorization of the Bell state Ψ to the noncommutative blowup A of R: Here ǫ n ∈ R(A) is the evaluation map at a point n in spacetime, c g specifies the summand ordering R ⊕ I or I ⊕ R in a matrix representation of A = End R (R ⊕ I), andφ is a morphism that encodes the matrix factorization. We will find that there does not exist a well-defined morphism M 2 (C) → M 2 (B) that would make the diagram commute, and this lack of commutativity corresponds to the lack of uniqueness of eigenstate that the Bell state Ψ may collapse onto.
Our second main result is the following, which shows how the representation theory of the noncommutative blowup A characterizes the collapse of the Bell states. In particular, the quantum randomness in the outcome of a measurement of Ψ arises from the fact that there is no preference of summand ordering, R ⊕ I or I ⊕ R, in A.
Theorem C. (Theorem 4.10.) The emergent collapsed Bell states ↑ a ↓ b 1 2 and ↓ a ↑ b 1 2 are obtained as 1-dimensional subspaces of the full Hilbert space H a ⊗ C H b , and generically only appear on the support Z(I) of Ψ: Furthermore, the constant identity function 1 ⊗ 1 ∈ A ⊗ A takes the values Notation. We will consider two entangled qubits a and b, such as two electrons with entangled spin (spin up and spin down), or two photons with entangled polarization (horizontal and vertical). 3 Denote by H a ∼ = H b ∼ = C 2 the respective Hilbert spaces of a and b. Set ↑:= ( 1 0 ) and ↓: We will denote a general Bell state by Ψ. The subscripts a and b allow us to symmetrize the tensor products. In particular, it will be useful to view the Bell states as elements of the symmetric tensor algebra of H a ⊕ H b over C, The Bell states possess maximal entanglement when θ equals 0 or π. Furthermore, each Bell state has density matrix (4) ρ = 1 2 1 1 1 1 , whence ρ 2 = ρ, and so each Bell state is pure. We will use the term 'local' in the physics sense (e.g., a wavefunction is nonlocal if it contains space-like separated points in its support), rather than in the algebraic sense (a ring is local if it contains a unique maximal ideal). Furthermore, by nonlocality we mean quantum nonlocality, and thus it is assumed that information cannot be transmitted faster than the speed of light.
Finally, denote by ε ij ∈ M n (C) the matrix with a 1 in the ijth slot and zeros elsewhere.
Matrix factorizations. Eisenbud introduced the following definition in commutative algebra to study a class of singularities [E].
We will use these factorizations to study the Bell states (1) and (2), by replacing the variables x, y, z, w with the spin states ↑ a , Remark 1.3. The Dirac equation is an example of a matrix factorization of the Klein-Gordon equation: In particular, its polynomial form with m = 0 is

Hilbert spaces over matrix rings
In this section we introduce a modification of quantum mechanics, where the ground field C is replaced by M n (C), the algebra of n × n matrices over C.
Definition 2.1. Fix a finite dimensional Hilbert space H with basis |1 , . . . , |m , an integer n ≥ 1, and setH : with coefficients c i in M n (C) and |i in H. We define the density matrix ρ ∈ M m (M n (C)) of ψ, with respect to the ordered basis |1 , . . . , |m , to have entries . We say ψ and ρ are normalized if the full trace of ρ is 1 ∈ C, and partially normalized if the partial trace of ρ is the identity matrix 1 n ∈ M n (C). We call ψ emergent if it is partially normalized and proportional to 1 n .
We introduce the following inner product onH.
Proof. Consider |ψ = i c i |i for some c i ∈ M n (C) and |i ∈ H. Let {|ℓ } ℓ be an orthonormal basis for H. Then for each i we may write |i = ℓ a iℓ |ℓ with a iℓ ∈ C. Furthermore, we may write with equality if and only if |ψ = 0. Furthermore, it is straightforward to check that if and only if ψ is normalized.
Proof. Let ρ be the density matrix of ψ. Then Remark 2.4. Generalized Born rule. If ψ = i c i |i is the normalized wavefunction for a particle written in terms of an eigenbasis {|i } i for H, then we postulate that the probability of finding a particle in the state |i is tr(c † i c i ). This of course reduces to the usual Born probability separable if it can be written as a product ψ = φ 1 ⊗ φ 2 with φ i ∈H i , and entangled otherwise.

From pure entangled to mixed separable via matrix factorizations
In this section we analyze the Bell states as emergent states with coefficients in M 2 (C).
Theorem 3.1. Let Ψ be a Bell state Ψ θ or Φ θ as in (1) and (2). Then the emergent state ψ := Ψ1 2 is a separable product of two mixed states, each consisting of two pure states.
Proof. Fix θ ∈ [0, 2π] and set ξ := e i(θ+π)/2 . It suffices to consider the Bell state Ψ : (5), Ψ admits the matrix factorization The emergent state Ψ1 2 is then proportional to the product of the normalized states which are elements ofH a ⊕H b ⊂ M 2 (B). Specifically, Remark 3.2. The commutation φ 1 φ 2 = φ 2 φ 1 generalizes the commutation The normalized density matrix of ψ with respect to the ordered basis and its partially normalized density matrix iŝ Thus Therefore ψ is mixed when normalized, and pure when partially normalized. In other words, ψ appears to be pure when viewed as an emergent state, but is really a mixture when its internal degrees of freedom-its matrix components-are taken into account.
Since ψ is a mixture, we would like to determine what pure states it is composed of. For i = 1, 2, set Then similar to (4), ψ i has density matrix Thus ρ 2 ψ i = ρ ψ i , and therefore ψ i is pure. It follows that Therefore ψ is a mixture of the two pure states ψ 1 and ψ 2 , and these states occur with equal probability.
We now analyze the states φ 1 and φ 2 . Using (6) and (7), their normalized density matrices with respect to the ordered basis Their partially normalized density matrices arê Therefore φ 1 and φ 2 are mixed when normalized, and pure when partially normalized.
Since φ 1 and φ 1 are mixed states, we would like to determine what pure states they are composed of, as before. Consider the states constructed from the columns of φ 1 , and the columns of φ 2 , It is straightforward to check that their normalized density matrices satisfy ρ 2 η ij = ρ η ij , and so each η ij is a pure state. For example, Furthermore, for i = 1, 2 we have Therefore φ i is an ensemble consisting of the two pure states η i1 and η i2 , and these states occur with equal probability, In future work it would be interesting to consider matrix factorizations of higher level and multiparticle entanglement.

An algebraic framework
Preliminaries. We begin by recalling some elementary algebraic geometry.
To any commutative algebra S containing C we may associate a geometric space Max S. The points of Max S are nonzero algebra homomorphisms ρ : S → C, or equivalently, their kernels ker ρ. These homomorphisms are the simple (i.e., irreducible) representations of S, and their kernels are the maximal ideals of S, the set of which is also denoted Max S. 6 S may be viewed as a ring of functions on Max S: for each f ∈ S and simple representation ρ of A with kernel n = ker ρ ∈ Max S, set or equivalently, f (n) := f + n ∈ S/n ∼ = C.
Associate to each set Y of Max S the ideal which is the set of functions in S that vanish identically on Y . Conversely, associate to each ideal J of S the subset which is the common zero locus of the functions in J. These subsets form the closed sets of a topology on Max S, called the Zariski topology. If S is a finitely generated C-algebra with no nonzero nilpotent elements, then by Hilbert's Nullstellensatz [GW2, Proposition 1.12, Corollary 1.47], for any closed set Y ⊂ Max S and ideal J ⊂ S satisfying we have Z (I(Y )) = Y and I (Z(J)) = J.
In this case S and Max S uniquely determine each other up to isomorphism, 7 Max S is called an algebraic variety, and S is called its coordinate ring.
6 In a commutative ring, the maximal and primitive ideal spectrums coincide [GW,Proposition 2.15]. Here we are focusing on primitive ideals, or closed points, rather than prime ideals, because in the next section we will be interested in the geometry that arises from the simple representations of a noncommutative algebra.
7 An algebra homomorphism h : S → S ′ determines a morphism Max S ′ → Max S by sending the

A new characterization of nonlocality: nonnoetherian singularities.
In this section we present a new characterization of quantum nonlocality as a nonnoetherian singularity birational to spacetime. We begin by introducing the following definitions.
Definition 4.1. The real support of a wavefunction Ψ is the locus of events in spacetime where it is possible in principle to measure Ψ, while its instrumental support is the locus of events where Ψ is actually measured.
Remark 4.2. Recall that an ontic state is a state of reality, while an epistemic state is a state of knowledge. The definition of real support fits into the framework of both ontic and epistemic realist quantum theories, and merely specifies the events where a measuring apparatus could be placed so that Ψ may be measured. In contrast, the definition of instrumental support fits into the framework of epistemic non-realist quantum theories.
As a mathematical toy model, we make the following assumptions. 8 Assumptions 4.3.
• The complexification of spacetime is a smooth algebraic variety X = Max S with coordinate ring S. • Emergent pure states are supported on (Zariski) closed subsets of X.
For example, we may take spacetime to be flat, in which case X = C 4 and S = C[x, y, z, t].
Definition 4.4. We define the supporting coordinate ring of a pure state Ψ with support Y ⊂ X to be the subalgebra Recall that a commutative ring S is noetherian if every ideal in S is finitely generated, and otherwise S is nonnoetherian. Furthermore, S is connected if S admits a grading S = i≥0 S i such that S 0 = k.
Remark 4.5. By [B2,Lemma 4.8], if S is a connected domain and Z(I) has positive dimension, then the supporting coordinate ring R = C + I of Ψ is nonnoetherian.
The following lemma shows that Definition 4.4 captures what Einstein called 'spooky action at a distance', which is a fundamental property of any pure state supported on more than one point of a spatial slice of spacetime. Recall that two varieties are birational if they are isomorphic on nonempty open subsets.
Lemma 4.6. (The spooky lemma.) Let Ψ be a pure state with support Z(I) ⊂ X, and let R = C + I ⊂ S be its supporting coordinate ring. Then Max R coincides with Max S except that the locus Z(I) ⊂ Max S is identified as one single 'spread-out' point in Max R. In particular, the locus Consequently, the possibly nonnoetherian singularity Max R is birational to the algebraic variety Max S.
Proof. I is clearly a maximal ideal of R, and so I is a closed point of Spec R. The claim that U = Max S \ Z(I) follows from [B1,Proposition 2.7], and birationality follows from [B1,Theorem 2.5.2].
We note that the birational morphism Max S → Max R, n → n ∩ R, is in general not proper. Now let Ψ denote an entangled Bell state, and let us assume in our toy model that Alice's particle a and Bob's particle b are point-like. Consider flat complexified spacetime X = C 4 with coordinate ring S = C[x, y, z, t]. Suppose that the entangled particles are traveling at a constant speed v in the z direction away from each other, relative to their center-of-mass frame. The real support Y re of Ψ is then the zero locus Y re = {x = y = z − vt = 0} ∪ {x = y = z + vt = 0} ⊂ X. Y re has radical ideal (9), 9 I re := I(Y re ) = (x, y, z − vt) (x, y, z + vt) = xS + yS + (z − vt)(z + vt)S.
Further suppose Alice and Bob each measure their respective particles at the spacetime events p a := (x a , y a , z a , t a ) and p b : are the maximal ideals of S consisting of all functions that vanish at the respective events p a and p b .
By Remark 4.5, the real supporting coordinate ring R re = C + I re of Ψ is nonnoetherian since its real support Y re is 1 (complex) dimensional and S is connected. In contrast, it is not known whether the instrumental supporting coordinate ring R in = C + I in of Ψ is nonnoetherian since its instrumental support Y in is 0 dimensional.

Collapse from the representation theory of a noncommutative blowup.
Without loss of generality we will consider the Bell state Ψ = ↑ a ↓ b − ↓ a ↑ b . Fix a type of support, real or instrumental, and denote by R = C + I ⊂ S the supporting coordinate ring for Ψ as in Section 4.1.
Throughout, given a left R-module M = M 1 ⊕ · · · ⊕ M ℓ with each M i indecomposable, we will denote by end R (M 1 ⊕ · · · ⊕ M ℓ ) the matrix ring whose ij-th entry is Hom R (M j , M i ); this matrix ring is isomorphic to the endomorphism ring End R (M) by fixing a particular basis. 10 The noncommutative blowup of R along the support Z(I) of Ψ is the endomorphism ring The algebra A := end R (R ⊕ I) is a modification of R in the sense that R ∼ = End R (R), and its center is isomorphic to R, (In the introduction we took A to be End R (R ⊕ I) rather than end R (R ⊕ I), for ease of exposition.) Furthermore, A may be viewed as a noncommutative coordinate ring on the spacetime variety Max S: the evaluation of a function f ∈ A at a point n ∈ Max S is the image of f under the representation (10) ǫ n : A −→ R/ (n ∩ R) S/n that is, f (n) := ǫ n (f ) ∈ M 2 (C).
Note that this is analogous to the commutative case (8). However, in the following proposition we show that a representation ρ : A → M 2 (C) is simple if and only if ρ is isomorphic to ǫ n for some n ∈ Z(I), and this occurs precisely when n is not in the support of Ψ. Consequently, we will find that the Bell states only collapse on representations which are not simple. Denote by d A ∈ Z ≥0 the maximal C-dimension of the simple representations of A. Further, given a representation ρ : A → M n (C), denote by [ρ] its representation isoclass. Consider the representation space Proof. We first claim that d A = 2. Indeed, since the corner rings ε 11 Aε 11 ∼ = R and ε 22 Aε 22 ∼ = S are commutative algebras over the algebraically closed field C, any simple representation ρ of A over C will be at most two dimensional and satisfy dim ρ (ε 11 Aε 11 ) = dim ρ (ε 22 Aε 22 ) = 1. Now suppose ρ : A → M 2 (C) is a representation whose isoclass is in R(A). The conditions dim ρ (ε 11 Aε 11 ) = 1 and dim ρ (Aε 11 ) = 2 imply that dim ρ (ε 22 Aε 22 ) = 1 and ρ (ε 12 ) = 0. Therefore, since R and S are commutative C-algebras, their kernels are maximal ideals m ∈ Max R and n ∈ Max S: Furthermore, since ρ (ε 12 ) = 0 and ρ is an algebra homomorphism, m = n ∩ R. Therefore ρ is isomorphic to ǫ n .
Finally, ǫ n is simple if and only if I/(n ∩ I) = 0, if and only if n does not contain I.
We introduce the following notion to capture a representation-theoretic perspective of wavefunction collapse.
There are two summand orderings of End R (R ⊕ I), namely given by the respective isomorphisms of A, g R⊕I := 1 2 and g I⊕R := 0 1 1 0 .
11 Specifically, if an endomorphism ring of the form A = End R (R ⊕ M ) is a noncommutative resolution of its singular center Z(A) ∼ = R ∼ = ε 11 Aε 11 , then R(A) is often parameterized by a commutative resolution of R.
In the following we will show that the choice of summand ordering of the R-module R ⊕ I determines what eigenstate the entangled Bell state Ψ collapses onto.
Consider the evaluation representation of end R (I ⊕ R) at a point n ∈ Max S as in (10), The conjugation map c g for g ∈ {g R⊕I , g I⊕R }, defined by c g (a) := gag −1 , commutes with ǫ n , 12 and therefore g may also be viewed as a particular choice of basis for the representation ǫ n : A → M 2 (C).
Recall the symmetric tensor algebra B of H a ⊕ H b over C, given in (3). For each spacetime point n ∈ Max S and summand ordering g ∈ {g R⊕I , g I⊕R } of A, consider the diagram, Note that the tensor products are over the centers of the respective algebras. The morphisms are defined as follows: • Each vertical morphism is the multiplication map, µ(a 1 ⊗ a 2 ) := a 1 a 2 .
• The representation ǫ n c g : A → M 2 (C) is extended to an R, R-bimodule homomorphism on A ⊗ R A by ǫ n c g (a 1 ⊗ a 2 ) := gǫ n (a 1 ) ⊗ ǫ n (a 2 )g −1 .
This further extends to a representation of the full tensor algebra ǫ n c g : T R (A) → T C (M 2 (C)), although we will not use this representation here.
• The morphismφ is the C, C-bimodule homomorphism defined on the basis {ε jk } of M 2 (C) by with (φ 1 , φ 2 ) the matrix factorization of Ψ given in (6) and (7).φ is then extended C-linearly to M 2 (C) ⊗ C M 2 (C). In particular, We call the composition φ := µφ the state morphism of Ψ.
Remark 4.9. We note that the left square in (11) commutes, whereas there is no morphism M 2 (C) → M 2 (B) that would make the right square commute, since any such morphism would necessarily not be well-defined (see (14) in the following theorem). We propose that this ambiguity is what gives rise to the randomness in the outcome of a measurement of Ψ.
Theorem 4.10. Consider the morphism φ in (12) corresponding to the Bell state Ψ. The emergent collapsed eigenstates ↑ a ↓ b 1 2 and ↓ a ↑ b 1 2 are obtained as 1-dimensional subspaces of the full Hilbert space H a ⊗ C H b , and generically only appear on the support Z(I) of Ψ: Furthermore, the constant identity function 1 ⊗ 1 ∈ A ⊗ A takes the values Proof. First suppose n ∈ Z(I), i.e., n ⊇ I. Then I/(n ∩ I) = 0. If g = g R⊕I = 1 2 , then φǫ n c g (A ⊗ A) = φ · R/(n ∩ R) S/n I/(n ∩ I) S/n ⊗ R/(n ∩ R) S/n I/(n ∩ I) S/n Similarly, if g = g I⊕R = ( 0 1 1 0 ) = g −1 , then Now suppose n is not in Z(I). Then I/(n ∩ I) ∼ = C, and the first claim (13) follows.
The second claim (14) is straightforward to verify.
Remark 4.11. The roles of the orderings R ⊕ I and I ⊕ R in Theorem 4.10 can be exchanged by considering the bimodule homomorphism φ ′ defined as in (12) with the matrix factorization in place of φ. This matrix factorization is isomorphic to (φ 1 , φ 2 ) since the following diagram commutes, The following are notable observations that follow from Theorem 4.10.
• The randomness inherent in the outcome of a measurement of the Bell state Ψ arises from the choice of summand ordering R ⊕ I or I ⊕ R, noting that there is no mathematical preference of one summand ordering over the other. • The center Z(A) = R1 2 of A determines the possible emergent observed states, by (14). • Ψ only collapses within its support Z(I), by (13). For n ∈ Z(I), the dimension of the subspace φǫ n c g (A ⊗ A) | 1 2 is greater than one, and thus no state (eigenstate or superposition) is specified. This corresponds to the fact that the particles a and b cannot be observed outside the support of Ψ. Equivalently, Ψ does not collapse at representations of A (points in spacetime) that are simple, by Proposition 4.7. In regards to the definition of real support and Remark 4.2, we conclude with a final observation. This observation explains the physical sense in which A is the coordinate ring for a local noncommutative geometry.
Remark 4.12. Suppose the particles a and b are entangled at the spacetime event n 0 , and the polarization of a is measured at the event n 1 . According to standard quantum mechanics, at all points n along the worldline of a strictly between n 0 and n 1 , we should find the superposition subspace φǫ n c g (A ⊗ A) | 1 2 = (↑ a ↓ b − ↓ a ↑ b )C, while at the event n 1 we should find the eigenspaces φǫ n 1 c g (A ⊗ A) | 1 2 = ↑ a ↓ b C and ↓ a ↑ b C, depending on g. However, according to Theorem 4.10, Ψ collapses along its entire support Z(I). Thus, if we take the support to be real, then Ψ collapses along the entire worldline of a between n 0 and n 1 .
In contrast, the morphism φ does not depend on n ∈ Max S, and thus exists independently of spacetime. Therefore the information of the non-collapsed state Ψ continues to exist as the particles fly apart. We are thus led to a perspective that is analogous to the de Broglie-Bohm waveguide interpretation, where the morphism φ plays the role of the waveguide and the representation ǫ n c g of A plays the role of the particles. In particular, if Ψ interacts reversibly (or unitarily) with its environment, such as when a photon passes through a polarizer, then presumably the interaction occurs with the state morphism φ, while if Ψ interacts irreversibly, such as in a measurement of Ψ, then the interaction occurs with the non-faithful morphism ǫ n c g . We leave these speculations for future work.