We propose a local model for general fermionic systems, which we express in the Heisenberg picture. To this end, we shall use a recently proposed formalism, the so-called “Raymond-Robichaud” construction, which allows one to construct an explicitly local model for any dynamical theory that satisfies no-signaling, in terms of equivalence classes of transformations that can be attached to each individual subsystem. By following the rigorous use of the parity superselection rule for fermions, we show how this construction removes the usual difficulties that fermionic systems display in regard to the definition of local states and local transformations.

Since the well-known works of Bell,1,2 quantum theory has been labeled as “non-local.” This terminology arises from the fact that quantum systems have been proven, both theoretically and experimentally, to violate Bell's inequalities. As a consequence of this fact, Bell's theorem implies that quantum theory cannot admit a local hidden variable model in terms of a real-valued stochastic theory—hence the term non-local.

However, quantum theory has long been known to be local in a deeper, dynamical sense. First, it satisfies the so-called “no-signaling theorem”; moreover, it satisfies the principle of no-action at a distance, as formulated by Einstein.3 

The fact that the principle of no-action at a distance is satisfied is most explicit in the Heisenberg picture formulation. That is the case in both non-relativistic and relativistic quantum theory.4,5 Specifically, for a system of N qubits, it is possible to define elements of reality attached to each qubit, which fully specify the dynamics of the composite system of N qubits jointly. We refer to these elements of reality as “descriptors.”4 Such descriptors are not real-valued elements of a stochastic hidden-variable theory, precisely as Bell's theorem requires. They are q-numbers,6 i.e., elements of a non-commutative algebra, which generate the algebra of local observables of each qubit. One can recover the dynamics of the system of all qubits by simply considering the set of all local descriptors of each qubit.

This relation between no-signaling and no-action at a distance, which emerges in quantum theory, is not accidental. A recent work7 proved formally and in a general way that every no-signaling theory with reversible dynamics admits a so-called local-realistic model, which Raymond-Robichaud7 defines as a model where (i) each system has a mathematical object that describes its real factual situation; (ii) given the objects (say, a and b) fully describing one of two disjoint subsystems A and B, the objects describing the global system AB are entirely determined by the ordered pair (a, b); (iii) a dynamical transformation that affects system A only can only change the object a describing A, but cannot change the object b describing B. The objects a and b are called local elements of reality, or the ontic states of the subsystems A and B. Furthermore, the authors give a constructive method to define such local elements of reality of any theory that is no-signaling and dynamically reversible—which we shall refer to as the equivalence class formalism. Raymond-Robichaud8 uses this formalism to find the local elements of reality in qubit networks; such local elements of reality can be represented in terms of the standard local descriptors of qubits, i.e., the generators of the algebra of observables.9 

In this paper, we shall apply these recent results to fermionic quantum field theory. We will show how it admits local elements of reality as q-numbers by explicitly constructing them following the equivalence class formalism. As one would naturally expect, we will demonstrate that the familiar generators of the fermionic algebra can be used to represent the local elements of reality. Our work provides the essential ground to understand better the foundations of fermionic phenomena, such as the fermionic Aharonov–Bohm effect.10 Moreover, it helps to ease further the conceptual difficulties with the locality of fermionic quantum field theory and seek a unified description of classical and quantum theories.

The structure of this paper is as follows. We first define fermionic systems (Sec. II) and explain the formal notion of local realism in fermionic field theory (Sec. III). Then, we use the structure of fermionic theory to derive the fermionic descriptors using the equivalence class formalism (Sec. IV). Finally, we discuss the results, commenting on the subtleties of the formalism and suggesting generalizations of our results (Sec. V).

Fermions can be regarded as a type of quantum system. In this work, we shall adopt a general information-theoretic approach to represent them, following previous works.11,12 This perspective adheres to the position of studying locality in fermionic systems in terms of modes,13 not particles.14,15 Within this approach, a fermionic system consists of a set of modes I and for each mode iI there are fermionic creation and annihilation operators f̂i,f̂i. The ground state |Ω is defined by the requirement that f̂i|Ω=0. The characteristic fermionic operator algebra is given by the creation and annihilation operators of all modes iI obeying the following commutation relations:

{f̂i,f̂j}=0̂,{f̂i,f̂j}=δijÎ.
(1)

We focus throughout the paper on systems with a finite set of modes for simplicity. In the Schrödinger picture, the allowed physical states are spanned by the states obtained by acting with the creation operators on the ground state |Ω, allowing linear combinations of states with the same parity only. With parity, we mean the parity of the number of fermions in the state. This restriction to allow only superpositions of states with the same parity is the parity superselection rule. Remarkably, this rule must be introduced for fermions to satisfy the no-signaling principle.

As an example, the density operator ρ=|Ω1|+|1Ω| is not allowed to physically exist, otherwise we could have faster than light communication between two parties. Formally, one can state the parity superselection rule as requiring that any physically allowed density operator ρ has to commute with the parity operator . has as eigenstates of eigenvalue 1(–1) the states with an even (odd) number of fermions. This parity superselection rule is long known16 to restrict the set of physically allowed density operators in a fermionic system. However, it also restricts the set of unitaries and observables that one can apply to the fermionic system. It has been shown recently11 that physical unitaries and observables also need to commute with the parity operator in order not to be able to signal.

More concretely, in a two-mode system, unitaries and observables with the form f̂1+f̂1 are not physically allowed. It follows from the parity superselection rule that if ÛI is a parity superselection rule unitary that involves modes in the set I, and we consider a mode jI, then the action of ÛI on f̂j leaves it invariant: ÛIf̂jÛI=f̂j. This result is a direct implication of ÛI commuting with the parity operator. This fact is crucial for when we analyze in the Heisenberg picture the local mode structure. We will see that a local unitary in A does not affect the elements of reality of a disjoint region B. However, before going into detail about descriptors and the Heisenberg picture, let us first set the basis of what we mean by locality. We adhere to the position that locality is a structural requirement for a theory. We believe that the definition provided by Raymond-Robichaud7 formalizes perfectly this notion of locality at a structural level.

After providing a general definition of local-realistic physical theories, Raymond-Robichaud7 proves that every physical theory satisfying the no-signaling principle and having reversible dynamics is a local-realistic theory. In this section, we apply this theorem to fermions. We first introduce the notion of a realistic theory, then clarify the notion of locality that we use. Finally, we show how fermionic theory fits into the construction of the equivalence class formalism.

To define a realistic theory, we use a triad of sets (P,R,T) on which we impose the following structures.

The set P is the set of the elements of the physical theory that describe the phenomenal properties of the theory—i.e., those that are empirically accessible. In other words, an element ρP details all the properties that can be observed of a specific configuration of a physical system. For fermionic theories, this is the set of density operators ρ that satisfy the parity superselection rule. We call the set P the phenomenal state space and its elements phenomenal states since they give us the observational properties of the system.

The set T is the set of operations that describe the allowed physical transformations. In the fermionic case, this corresponds to the set of parity superselected unitary operators. The set of transformations for fermions forms a group under composition °. This fact gives reversible dynamics in the theory. The transformations group T acts on the phenomenal state space P, defining a group action. The group action in the fermionic case is given by ρ=Û·ρ·Û, where ÛT and ρ,ρP.

The set R is the set of states of a physical system describing all properties that entirely specify the configuration of that physical system, rather than merely describing the observational properties of it. We call the set R the ontic state space. Since the term ontic appears more often in the literature17 than the terms real or noumenal, we will be using it throughout this work. The group of operations T also acts on R, but not necessarily with the same action as in P. We denote the action of an element UT to an element rR as r=Ur. We will see in Sec. IV how the descriptors are a representation of these ontic states.

Let us consider a familiar example to exemplify the meaning of the ontic states. Consider a qubit composite system with the singlet state |ψ=12(|01|10), and we apply a local rotation in one of the qubits. We know that the Schrödinger state |ψ will remain unchanged. One can imagine in this case that the phenomenal state is |ψ since it gives the observational properties of the system. However, we can consider that the fact that we have applied a rotation is relevant for the constitutive properties of the system, even though such rotation does not produce any observational effect. In that case, we could consider that the realized rotation is part of our ontic state.

The phenomenal state must be completely characterized by the ontic state of the system: there must be an epimorphism φ from R to P. In other words, there has to be a mapping φ:RP where for every ρP exists a rR such that φ(r)=ρ. Moreover, φ has to be faithful with respect to the two actions of the group of operations T onto the sets R and P. In the fermionic case, for all ÛT,rR, then φ(Ûr)=Û·φ(r)·Û.

Thus, if we want to see that the fermionic theory is a realistic theory, we have to find a set R, an appropriate group action and an epimorphism φ that satisfy the conditions above. One could choose R as being P, where the action of T is the same as in P and the epimorphism is the trivial identification φ(ρ)=ρ. However, this trivial identification may not follow the conditions to be considered a local-realistic theory. In order to show that fermionic quantum field theory is a local-realistic theory, we need to find a suitable ontic state space for a fermionic theory that satisfies the locality properties described below.

The notion of locality presented7 does not refer to position in any way. It refers to the structure of subsystems.18 So, when we talk about locality, we refer to being localized in a subsystem of the global system in consideration.

As we comment in Sec. II, we take the fermionic mode as the primary subsystem of a fermionic system. More concretely, if we have a physical system of N modes N={1,,N}, we consider that the systems where the set of modes is a subset of N are the physical subsystems of the theory.

The notion of locality refers to the relation between different fermionic modes of the set N. We need to specify which systems our states refer to and how our transformations act in terms of mathematical structure. We will call an object local to a subsystem A if it only refers to or acts on that subsystem A. We are also interested in restricting objects that live or act in the global system to the particular subsystems.

In the phenomenal state space P, this so-called projection map πP is given by the partial trace. If we have a phenomenal state of the whole system of N modes ρ, we say that the object that gives all the phenomenal properties of the system within the subsystem of modes M={r1,,rM}N is given by the fermionic partial trace of ρ over the set of modes N\M denoted by ρr1,,rM=trN\M(ρ). Indeed ρr1,,rM is a valid phenomenal state, since it is a parity superselected density operator. Moreover, it can be considered to be local to the subset of modes M={r1,,rM} since ρr1,,rM can be given by combining creation and annihilation operators of the M modes alone.

For operations, it is also necessary to have a notion of an operation being local to a subset of modes M. For fermions, an operation that acts on the global system of modes N,ÛT, is local to the subsystem given by the subset of modes M iff Û can be given by a combination of creation and annihilation operators of the modes in M alone.

We now analyze what we are required to obtain in order to find a local-realistic theory for fermions. First, we need a realistic theory; thus, we need an ontic state space with an action of the transformation space and the epimorphism φ to the fermionic phenomenal state space. In order for a realistic theory to be local for a lattice of systems, we need to be able to define two operations in the ontic state space R. First, we need a projection map πSR:RR. The projection map sends global ontic states to local ontic states. We require that the ontic projections and the phenomenal projections act in parallel. It is required that for any ρ being the phenomenal state of the whole system of fermionic modes N, then for the ontic state(s) RR such that φ(R)=ρ, φ(πSR(R))=ρS=πSP(ρ), where S is any subset of the modes of N. So, we want a projection to subsystems that is faithful under the φ epimorphism. These properties give us a notion of ontic states belonging to a subsystem.

Last and crucially, we want a structure where knowing the ontic states of the parts, we know the ontic state of the whole. More specifically, we want to be able to define a map :RA×RBRAB. Consider a phenomenal state ρP of the whole system of modes, such that it has an underlying ontic state RR given by φ(R)=ρ. Also, consider that then we split the global system of modes in two disjoint pieces by choosing two disjoint subsets of modes A and B, such that AB=N and AB=. Then, we can define ontic states RA=πAR(R),RB=πBR(R) by the previously defined projection operation. Now, we require that we can assign RARB=RAB=R uniquely. Observe that this is the condition of separability in the Einstein locality principle.19 Moreover, notice that with the usual understanding that density operators are the phenomenal states and the ontic states of the system, such join products cannot be defined. Given an entangled state ρ for the disjoint systems A and B, one can obtain πAP(ρ)=trB(ρ)=ρA and πBP(ρ)=trA(ρ)=ρB, where the projection map is the partial trace. However, from the un-entangled product state ρ=ρAρB, using the same map we also get to πAP(ρ)=trB(ρ)=ρA=ρA and πBP(ρ)=trA(ρ)=ρB=ρB. Thus, we cannot define a join product ρAρB=ρ and ρAρB=ρ since ρρ. Thus, the usual realistic view of fermionic theory is not local-realistic. However, this does not stop us from discussing if it is possible to find a different ontic state space where the local realistic structure manifests. We see in Sec. IV how using the constructive proof of the equivalence class formalism theorem, we find such a structure.

In the equivalence class formalism, an operational no-signaling theory needs a phenomenal state space P, a set of transformations T with an action · on P. Moreover, it needs a lattice of systems with associated faithful projectors πSP and that the no-signaling condition is respected. We can understand the condition of T having a group structure and giving a group action on P as that the dynamics of the theory are reversible. This reversibility is the case in fermionic theory since the set of transformations T corresponds to the set of fermionic parity superselected unitaries for a given complete set of modes. Such unitaries set, of course, conforms a group with the usual product. The phenomenal state space P for fermionic theory is, as mentioned before, the set of parity superselected density operators for the given total set of modes. Given ÛT,ρP the group action is defined as Û·ρ·Û. Furthermore, the partial trace over modes in the complete set of modes gives the projections of the system lattice in the fermionic case. If we have a total set of modes N, we can define any subsystem by any subset MN and we obtain πMP()=trN\M().

Finally, we express the no-signaling principle presented in the equivalence class formalism as: given a system with a bipartition of disjoint subsystems A and B, any operation local in AUAT, any operation local in BVBT, and any phenomenal state of the global system ABρP, the following equation is satisfied:

πAP((UAVB)·ρ)=UA·(πAP(ρ)),
(2)

which in fermionic theory corresponds to the usual no-signaling principle equation given by

trB(ÛA·V̂B·ρ·V̂B·ÛA)=ÛA·trB(ρ)·ÛAtrB(V̂B·ρ·V̂B)=trB(ρ).
(3)

Therefore, since the above no-signaling condition is satisfied for the fermionic system that considers the parity superselection rule for both states and unitaries,11 all the conditions of the equivalence class formalism are satisfied. Therefore, we can find with their construction the ontic states that dote fermionic theory of the structure of local realism.

The proof that any operational no-signaling theory is a local-realistic theory is constructive.7 The authors construct for a general operational theory with a group of transformations T, a lattice of systems, a phenomenal state space P, with the associated group action · and projection operators πAP the ontic state space R, the ontic group action , the epimorphism φ, the ontic projection operators πAR, and the join product . They do so by defining an equivalence relation A that states that two transformations U, V are equivalent in a subsystem AUAV if and only if there exists a transformation WA¯ local on the subsystem A¯, complementary to A, such that U=WA¯V. The equivalence classes [U]A under this equivalence are the ontic states of the subsystem A, forming an ontic state space R with all the desired properties. Some of them are

V[U]A=[VU]A;πAR([U]AB)=[U]A,[U]A[U]B=[U]AB;WA[V]B=[V]B,
(4)

where A and B are systems, with AB= and U,V,WA are transformations with WA being local in A.

In order to find these equivalence classes in the fermionic setting, we utilize the equivalence between the descriptors and the equivalence class formalism in qubit networks.20 The paper aims to generalize it to the fermionic case and use the excellent properties and physical intuition that the descriptor picture provides to unveil the set of ontic states that give the fermionic theory a local realistic structure.

As we have explored in Sec. III, in order to provide a fermionic theory with a local realistic structure for an orthonormal set of modes, we need to find a set of ontic states R with proper operations πAR,φ and . In order to construct these objects presented in the equivalence class formalism, we utilize the descriptor formalism presented by Deutsch and Hayden4 and then we use the result by Bédard20 that establishes the equivalence between the two formalisms.

Their work focuses on qubit networks, but we will see that one can apply their ideas to fermionic systems. Here, we will be applying their procedure directly to the fermionic case. Using the Heisenberg picture, the descriptor formalism consists of re-interpreting the real elements of quantum mechanics. The formalism considers pure states with unitary evolutions, assuming that one can always increase the dimension of the Hilbert space and use the Stinespring dilation theorem21 to treat any quantum channel as a unitary evolution. Thus, from here on, we consider the Heisenberg state being pure unless stated otherwise.

The derivation of descriptors starts with the fact that in quantum theory, the expectation value of an observable Ô of a system in a particular ρ is given by tr(Ô·ρ). Now, if we apply a unitary transformation Û to ρ, the updated data will be given by tr(Ô·Û·ρ·Û). We use the Heisenberg picture. We consider that the state ρ remains invariant under the transformation Û, but the observables Ô evolve under the transformation given by the unitary operator Û as Ô=Û·Ô·Û. Since the observables of the system have evolved and ρ has not, it is reasonable to assume that we could find a framework where the observables are the ones that determine the state of the system and not ρ.

In order to do so, it is helpful to try to find which operators we can track to determine any Û·Ô·Û. We start by defining that the space that we work with consists of a fermionic space of N modes under the parity superselection rule. Therefore, the observables Ô are Hermitian operators that commute with the parity operator. Using the anticommutation relations of the creation and annihilation operators, we can write any observable Ô as

Ô=jαjÔ1(j),,·ÔN(j),
(5)

where Ôi(j) are monomials of any degree of the creation and annihilation operators f̂i,f̂i. Notice that for each j in the sum, the global operator Ô1(j)··ÔN(j) is a monomial of even degree of (f̂i,f̂i)i=1N. Now, we can see that for any Ô, we have this decomposition and, then,

Û·Ô·Û=jαjÛ·(Ô1(j)··ÔN(j))·Û=jαj(Û·Ô1(j)·Û)··(Û·ÔN(j)·Û).
(6)

Therefore, if we know how all the possible local operators Ôi(j) transform under Û, then we know how any observable Ô evolves under Û. However, we can reduce this further. Due to the fermionic algebra, it is easy to see that there are only four linearly independent monomials of fi,fi. Thus, the transformation under Û of Ôi(j) will be given by

Û·Ôi(j)·Û=Û·f̂i·ÛorÛ·f̂i·ÛorÛ·(f̂i·f̂i)·ÛorÛ·(f̂i·f̂i)·Û.
(7)

If we draw our attention to the second line, we can observe that Û·(f̂i·f̂i)·Û=(Û·f̂i·Û)·(Û·f̂i·Û) and that Û·(f̂i·f̂i)·Û=(Û·f̂i·Û)·(Û·f̂i·Û). Therefore, we can conclude that if we know the transformation under Û of the 2N operators f̂i and f̂i for i=1,,N, then we know the evolution under Û of any observable Ô. However, we can even reduce this further, since it can be observed that (Û·f̂i·Û)=Û·f̂i·Û.

This fact implies that we only need to know the evolution under Û of the N operators f̂i in order to know the evolution under Û of any observable of the N-mode fermionic system. We call the set (f̂1,,f̂N) the set of fermionic descriptors of an N-mode fermionic system.

By knowing the initial state of the system |ψψ| and knowing the expression of (Û·f̂1·Û,,Û·f̂N·Û), one can recover all the expectation values of any observable in the current state of the system. In the Schrodinger picture, such a phenomenal state would be given by Û|ψ. A few things to note is that each Û·f̂i·Û can be a global N-mode operator, so it can have contributions in its expression from other modes ji. However, at the same time, the “updated” descriptors satisfy the fermionic algebra. In the sense that if we name f̂¯i=Û·f̂i·Û, then {f̂¯i,f̂¯j}=0 and {f̂¯i,f̂¯j}=δijÎ.

The curious reader may be wondering how these objects relate to the notion of local realism that we have introduced in Sec. III. The answer is inspired by the work of Bédard.20 The set of descriptors Û·(f̂1,,f̂N)·Û is a compact way to represent the equivalence classes [Û]i that according to the equivalence class formalism can be considered as the ontic states of the system. This process endows fermionic theory with a local-realistic structure.

We develop the specifics of this claim in the following lines, where we present the results that justify the connection between the fermionic descriptors that we have proposed and the equivalence class formalism of local-realistic theories.

Theorem 1.Given a fermionic theory with a set of modesI={i1,,iN}. Given the equivalence relation on the group of parity superselected unitaries of the theory for each modeijIgiven byÛijV̂iffÛ=ŴI\{ij}·V̂, then

ÛijV̂Û·f̂ij·Û=V̂·f̂ij·V̂.
(8)

Thus, [Û]ij={V̂T|Û·f̂ij·Û=V̂·f̂ij·V̂}.

With Theorem 1, we have obtained a direct connection between the fermionic descriptors and the ontic states of the equivalence class formalism. The proof follows easily from the algebraic properties of unitaries, and it is in the  Appendix. We can define the equivalence classes that are the local elements of reality in terms of properties satisfied by the descriptors. Moreover, we see that we can define the equivalence class that gives the local ontic state in a subsystem in terms of only the descriptors associated with that subsystem.

1. Ontic action

Using the result from Theorem 1, we can represent the ontic fermionic states of the equivalence class formalism using the fermionic descriptors. The action of the group of transformations T on the ontic state space R,W[U]S that we have defined in Sec. III in the representation by descriptors is given by

W((Ûf̂1Û,,Ûf̂NÛ),|ψ0ψ0|)=((ŴÛf̂1ŴÛ,,ŴÛf̂NŴÛ),|ψ0ψ0|).
(9)

In what follows, we express the join product , the ontic projectors πAR, and the epimorphism φ in respect of fermionic descriptors. Consider an N-mode fermionic system with set of modes I={i1,,iN}. We have an initial phenomenal Heisenberg state ρ0=|ψ0ψ0|. The system undergoes a specific transformation Û given as a polynomial of the operators f̂ij,f̂ik. After such a transformation is applied, the pure state |ψ=Û|ψ0 is the phenomenal state of the system. In regard to the descriptor picture, we have that in respect of the lattice of systems given by the canonical mode set I, the set of descriptors (f̂i1,,f̂iN) and the initial Heisenberg state |ψ0 represent the initial ontic state of the system. After the transformation Û is applied, the ontic state of the system can be represented by (Ûf̂i1Û,,Ûf̂iNÛ) and the Heisenberg state |ψ0.

2. Descriptor epimorphism

A result that is important to understand the completeness of the descriptor picture and its relationship with Theorem 1, the group of transformations T, and its equivalence classes is the following Theorem 2. The following theorem helps to understand the definition of the epimorphism φ for a complete set of descriptors.

Theorem 2.Using the complete set of descriptors(f¯̂1,,f¯̂N), it is possible to find uniquely theÛthat acts on the N-mode system such that(f¯̂1,,f¯̂N)=(Ûf̂1Û,,Ûf̂NÛ).

The proof of this result is entirely algebraic, but the idea is to use a decomposition of the unitary operators on a basis of the operators that act as descriptors. The complete proof is in the  Appendix.

The construction of the epimorphism φ seems straightforward from this theorem. We choose the annihilation operators to describe the ontic state of our initial configuration, with a given Heisenberg state |ψ0ψ0|. We say that for such a configuration, the associated phenomenal state is the Heisenberg state. Then, if we have a different set of descriptors that is unitarily conjugated to the canonical one, one can obtain the unitary Û that connects them, and associate to the new configuration the phenomenal state φ((Ûf̂i1Û,,Ûf̂iNÛ),|ψ0ψ0|)=Û|ψ0ψ0|Û.

This definition is consistent with what one might expect since we start with a phenomenal state |ψ0ψ0| that we use as a Heisenberg state. The mentioned Heisenberg state and a corresponding set of canonical annihilation operators give the initial ontic state. Then, a unitary evolution makes the states evolve. The action of the unitary updates the descriptors. Moreover, since we can deduce the unitary, the new phenomenal state is the unitary acting on the initial phenomenal state. So, the definition of the epimorphism is consistent with our physical intuition on how transformations should act on the ontic and phenomenal states.

However, this only defines the epimorphism for a total set of descriptors, so for the global states of the system. We now present how from the ontic local state ((Ûf̂i1Û,Ûf̂iMÛ),|ψ0ψ0|), we associate the phenomenal state ρM, which is a parity superselected density operator for the system of M modes i1,,iM. The idea is the same as for the proof of Theorem 2. It is not difficult to see that ρM can be decomposed in an operator basis ÔM(j) given by monomials of creation and annihilation operators f̂i1,f̂i1,,f̂iM,f̂iM, then manipulating the expression we get

ρM=jtr(ÔM(j)ρM)ÔM(j)=jtr(ÔM(j)ρ)ÔM(j)=jtr(ÛÔM(j)Û|ψ0ψ0|)ÔM(j).
(10)

Since ÛÔM(j)Û can be calculated from the descriptors ((Ûf̂i1Û,Ûf̂iMÛ),|ψ0ψ0|) and |ψ0ψ0| is part of the ontic state, we can assign from our ontic local states the phenomenal local states through the epimorphism φ given by

φ((Ûf̂i1Û,,Ûf̂iMÛ),|ψ0ψ0|)=jtr(ÛÔM(j)Û|ψ0ψ0|)ÔM(j).
(11)

The complete details of the decomposition we use are in the proof of Theorem 2 in the  Appendix.

Once we have a representation of the epimorphism φ in terms of fermionic descriptors, we can discuss separability. Separability is the condition that gives the notion of locality to the ontic state-space construction. In the fermionic descriptor setting, we now discuss how we represent the ontic local states, how the join product operates, and how they relate to the local phenomenal states (the reduced density operators).

1. Ontic projections

We know that given a subset IA of the mode set IAI: {a1,,aM}{i1,,iN} induces an M mode subsystem A of the N mode system S, where A is the fermionic system of M modes generated by the mode set IA={i1,,iM}. At the ontic state space in the equivalence class formalism to define the ontic projection mappings of the equivalence classes, one just defines πAR([U]AB)=[U]A. It is straightforward to see that the following simple operation defines the ontic projection mappings in the descriptor representation of fermionic systems by using Theorem 1.

Definition 3.If we have an ontic global state of the system S represented by((Ûf̂i1Û,,Ûf̂iNÛ),|ψ0ψ0|). Then, we define

πAR((Ûf̂i1Û,,Ûf̂iNÛ),|ψ0ψ0|)=((Ûf̂a1Û,,Ûf̂aMÛ),|ψ0ψ0|).
(12)

2. Joint ontic product

To define the ontic join product, we need to see that for any fermionic subsystem A defined in terms of modes subsets, we can consider a bipartition of the global system S. We can consider the fermionic system B of NM modes spanned by the set of modes IB=I/IA. It is straightforward to see that A and B are disjoint systems and that the global system S can be considered the join space of both subsystems.

Now, we are in a position to define ontic states of systems A and B to be compatible. Consider an ontic state of system A represented with descriptors ((Ûf̂a1Û,,Ûf̂aMÛ),|ψψ|) and an ontic state of system B represented with descriptors ((V̂f̂b1V̂,,V̂f̂bNMV̂),|ηη|). Then, the state is said to be compatible if there exists a global state of the system S=AB, represented by descriptors as ((Ŵf̂i1Ŵ,,Ŵf̂iNŴ),|ϕϕ|) such that when projected to subsystems A and B equals the beforementioned states.

Note that this means that for two states to be compatible, they have to have the same Heisenberg state and that there must exist a unitary transformation W such that [W]A=[U]A and [W]B=[V]B. Then, it is straightforward for compatible states to see how the join product can be defined and operates in the descriptor representation.

Definition 4.Consider two compatible local ontic states of disjoint subsystems A and B,((Ûf̂a1Û,,Ûf̂aMÛ),|ψ0ψ0|)and((V̂f̂b1V̂,,V̂f̂bNMV̂),|ψ0ψ0|). Then, there exists a unique global state((Ŵf̂i1Ŵ,,Ŵf̂iNŴ),|ψψ|)such that the definition of the join product of the compatible states can be defined as

(Û(f̂a1,,f̂aM)Û,|ψ0ψ0|)(V̂(f̂b1,,f̂bNM)V̂,|ψ0ψ0|)=((Ŵf̂a1Ŵ,,Ŵf̂bNŴ),|ψ0ψ0|).
(13)

Thus, due to the uniqueness property of the definition, knowing the ontic states of the subsystems is the same as knowing the ontic state of the global system. The projection and join structures are the same for the fermionic case to the qubit case.4,7 However, the critical result for the fermionic case is that for a global system where we know the unitaries that are applied, the projection and joining operations consist in separate and merge the descriptors of the relevant set of modes that conform to our subsystems of interest.

Of course, we can repeat the bipartition process until we reach the point of individual modes forming subsystems. We see how the local and global states are on the same footing, which gives us strong notions to believe in their physicality given a particular partition of the global system in terms of subsystems. A particular excellent result that one can obtain from the representation of ontic states with descriptors is that splitting into subsystems is a faithful operation because the diagram in Fig. 1 commutes. Theorem 5 condenses the information in Fig. 1.

Fig. 1.

Commuting diagram that represents the actions of taking the projection into subsystems and the ontic-phenomenal epimorphism. Diagram (a) represents the spaces and diagram (b) represents the action of the mappings in the descriptor picture.

Fig. 1.

Commuting diagram that represents the actions of taking the projection into subsystems and the ontic-phenomenal epimorphism. Diagram (a) represents the spaces and diagram (b) represents the action of the mappings in the descriptor picture.

Close modal

Theorem 5.Given a fermionic system with the mode setI={1,,N}. For any set of descriptors(f¯̂1,,f¯̂N)together with any Heisenberg state|ψ0ψ0|and any non-empty subset of modesJI. The associated diagram of Fig. 1 commutes. In other words,

(πJP°φ)((f¯̂1,,f¯̂N),|ψ0ψ0|)=(φ°πJR)((f¯̂1,,f¯̂N),|ψ0ψ0|).
(14)

The proof of the theorem is algebraic. We apply the form of the epimorphism φ for descriptors that we have on Eq. (11). We also use the definition of ontic projection πAR that consists in forgetting the descriptors for the modes that are not in A. Furthermore, finally, we use the properties of πAP being the fermionic partial trace. Moreover, we use some algebraic properties of fermionic systems under the parity superselection rule derived in previous works.11 The complete proof is in  Appendix.

The definitions and theorems allow us to claim that fermions also have as qubits a local-realistic structure. We have found explicitly a neat representation of the ontic states using the widely known Heisenberg picture of quantum mechanics. Furthermore, we have seen that tracking the creation or annihilation operators is enough to have a complete description of the ontic states of the system. In Sec. V, we discuss the implications of our findings and the possible uses that this way of representing fermionic systems may have.

Following the equivalence class formalism, this work shows how one can formally prove that fermionic systems are local-realistic. We have seen how a compact representation of the local structure can be given by extending the descriptor picture of qubit networks to fermionic modes. We have also seen that fermionic annihilation operators form a valid set of fermionic descriptors.

Readers familiar with the Heisenberg picture in quantum field theory may wonder how the analysis we have performed differs from it. We are indeed just using the usual Heisenberg picture. We have seen that the most efficient way to use the Heisenberg picture is to track the fermionic annihilation operators since, from their evolution, we can retrieve the evolution of any observable of the theory. However, the interpretation of these objects by the equivalence class formalism differentiates the usual Heisenberg picture from our treatment. First, we have stressed the relevance and necessity of having local objects associated with each fermionic mode, which are the subsystems in our theory and second, we propose to interpret the dynamical evolution of the annihilation operators as a representation of the equivalence classes of parity superselected unitaries that correspond to the ontic states in our system. This consistent interpretation is a novel element that introduces a possible new angle on the meaning of states and dynamics in fermionic systems.

Marletto et al.22 have analyzed how a phase in a fermionic Mach–Zender interferometer is acquired locally by using the Heisenberg picture and tracking some relevant observables. This paper is a generalization of that work, using the new construction of the equivalence class formalism. We specify the notion of a local element of reality in terms of local ontic states, and we can then describe any fermionic physical process with a local description.

We emphasize, as also pointed out by Chiribella,18 that the lattice of systems of a theory is an extra layer of the structure that the fermionic algebra alone does not determine. More concretely, in the fermionic case, the lattice of systems is determined by which notion of modes one chooses. Moreover, under that partition, we can assign the ontic local states in terms of subsystems of the theory. However, the choice of modes is not unique. We could perfectly define a different set of modes by unitary conjugation to the chosen one that would define a different partitioning of the global system in terms of subsystems. Furthermore, in this other partition, we would be able to identify the ontic local states. Therefore, even though we can uniquely identify the local elements of reality given a notion of locality, the locality is not unique for a group of physical transformations.

For a set of local operators to be a suitable set of descriptors, we need that the evolution of such operators suffices to determine the evolution of any observable. One can wonder if there is a straightforward way to understand why the annihilation operators are suitable descriptors. There is. We first observe that fermionic observables are fermionic operators, which form an algebra. We now notice that in this algebra, in particular, the 2N creation and annihilation operators are generators of the algebra of fermionic operators and since Û(αÂ+B̂)Û=αU(A)U+U(B)U and Û(ÂB̂)Û=Û(Â)ÛÛ(B̂)Û, tracking the generators of the algebra is enough to track any element of it. Using the fact that (f̂)=f̂, we can claim that tracking f̂ is enough to track f̂, so it is enough to track the annihilation operators.

This observation is useful in understanding why in the N-qubit network case, the descriptors can be the σ̂jx,σ̂jz operators for each qubit j. These σjx,σjz are the generators of the N-qubit network operator algebra. Following the same spirit as in this article, we can now also use the operation properties. It is straightforward to see that if one defines q̂j=12(σ̂jx+iσ̂jy), then q̂j,q̂j are generators of the N-qubit algebra (see in  Appendix). We can see that since (q̂j)=q̂j, it is enough to track all the q̂j to track any N-qubit observable. Therefore, we only need to track one operator for each qubit, with the caveat of that operator not being an observable since it is a non-Hermitian operator.

These operators q̂j behave like annihilation operators. They satisfy the condition that q̂j|0|0=0,q̂j2=0,{q̂j,q̂j}=Î and [q̂j,q̂i]=0. At the individual qubit/mode level, the anticommutation relations are the same as for fermionic modes, but operators of different modes/qubits commute. This realization is not related to the Jordan–Wigner transformation; it is just a property that arises from the defining commutation/anticommutation relations of generators of the qubit network algebra.

We can now postulate what the descriptors will be for any quantum system of particles. For the bosonic case, we can see that the direct candidate for bosonic descriptors in a system of N bosonic modes is the set of bosonic annihilation operators b̂j. We know that the set of b̂j,b̂j is the generators of the bosonic operator algebra. We can simplify further, by tracking only the set of b̂j—this is enough to track any bosonic observable. In our future research, we would like to use the notion of algebra generators to study the locality and descriptors of anyonic systems,23 where the identifications are not so clear due to the lack of field theory creation and annihilation operators. Similarly, we would like to expand on these concepts focusing on well-known quantum algebras such as the Virasoro algebra used in string theory to find a compact local description of such systems.

A confusing question can arise in our main fermionic case. Due to the parity superselection rule, f̂i are not fermionic observables, nor fermionic unitaries, nor a linear combination of them; how can the collection of them together with |ψ0ψ0| be the ontic state of the parity super selected fermionic system? We answer this by regarding the set of fermionic annihilation operators and |ψ0ψ0| not as the ontic state of the fermionic system, but as a representation of the ontic state. We use here representation as a broad term, not the precise meaning used in group theory. The ontic states are the equivalence classes of the parity superselected unitaries that we have described with the equivalence relation. Note that only the parity superselected unitaries (real, physically allowed) are required to define such equivalence classes as the ontic states. We use the descriptors and the Heisenberg picture to have a compact and convenient way of representing the ontic states of the parity superselected unitaries' equivalence classes and their structure.

Finally, in the fermionic and the general equivalence class formalism case, only the set of physically allowed transformations is necessary to define the ontic states of a physical system. Thus, we think these results suggest that we should study the transformation structures by themselves to see what range of structures and phenomena can offer. Identifying which transformations are physically possible seems crucial in understanding the constitutive properties and the locality features of a physical system. Further future work could involve extending the equivalence class formalism to nonreversible dynamics so that theories such as the objective collapse model24 could be studied in detail.

The authors thank David Deutsch, Paul Raymond-Robichaud, and Charles Alexandre Bédard for fruitful discussions. N.T.V. acknowledges financial support from “la Caixa” Foundation (ID 100010434, LCF/BQ/EU18/11650048). C.M. and V.V. thank the John Templeton Foundation and the Eutopia Foundation. V.V.'s research is supported by the National Research Foundation and the Ministry of Education in Singapore and administered by the Centre for Quantum Technologies, National University of Singapore. This publication was made possible through the support of the ID 61466 grant from the John Templeton Foundation, as part of the Quantum Information Structure of Spacetime (QISS) Project (qiss.fr). The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

The authors have no conflicts to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

In this section, we include mathematical details of the paper and laborious proofs of statements made in the main article.

(Theorem 1). Given a fermionic theory with a set of modesI={i1,,iN}. Given the equivalence relation on the group of parity superselected unitaries of the theory for each modeijIgiven byÛijV̂iffÛ=ŴI\{ij}·V̂, then

ÛijV̂U·f̂ij·Û=V̂·f̂ij·V̂.

Thus,[Û]ij={V̂T|Û·f̂ij·Û=V̂·f̂ij·V̂}.

Proof. The last statement follows directly from the definition of an equivalence class, so the equation that needs to be proven is Eq. (8):

”: ÛIjV̂ implies Û=ŴI\{ij}·V̂ for some ŴI\{ij} being a parity superselected unitary local on the set of modes that excludes the mode ij. Thus, since ŴI\{ij} is an even operator that does not contain any terms involving f̂ij,f̂ij, it is straightforward to check that [ŴI\{ij},f̂ij]=0. Thus, Û·f̂ij·Û=V̂·ŴI\{ij}·f̂ij·ŴI\{ij}·V̂=V̂·f̂ij·ŴI\{ij}·ŴI\{ij}·V̂=V̂·f̂ij·V̂.

”: We have that Û·f̂ij·Û=V̂·f̂ij·V̂, and to see that ÛijV̂ we need to see that Û=ŴI\{ij}·V̂. Or, equivalently, since we have a group structure, where transformations are unitaries, proving that Û·V̂=ŴI\{ij} is enough to prove that ÛijV̂. From Û·f̂ij·Û=V̂·f̂ij·V̂, it is straightforward to deduce that f̂ij·(Û·V̂)=(Û·V̂)·f̂ij. Naming Û·V̂=Ŵ, noticing Ŵ is a parity superselected unitary and taking the dagger of the previous equation, we have that the two following equalities hold:

Ŵ·f̂ij=f̂ij·Ŵ,Ŵ·f̂ij=f̂ij·Ŵ.
(A1)

Moreover, now since W is a priori a general parity superselected unitary, is not difficult to see that we can decompose it as follows: Ŵ=Ô0+f̂ijÔ1+f̂ijÔ2+f̂ijf̂ijÔ3, where Ô0,Ô3,Ô1,Ô2 are local operators on the set of modes I\{ij}, where O0, O3 are even operators and Ô1,Ô2 are odd operators (since the operator Ŵ is an even operator). Using this decomposition of Ŵ in the first condition of Eq. (A1) and commuting/anticommuting the f̂ij,f̂ij terms with the Ôk operators, we obtain that

f̂ij(Ô0+Ô3)+f̂ijf̂ijÔ2Ô2=f̂ijÔ0+f̂ijf̂ijÔ2.

Is not difficult to see that this implies that Ô2=0̂ and Ô3=0̂. Then, using that Ŵ=Ô0+f̂ijÔ1 and replacing in the second condition of Eq. (A1), we obtain

f̂ijÔ0+f̂ijf̂ijÔ1=f̂ijÔ0+Ô1f̂ijf̂ijÔ1.

It is also easy to see that it implies that Ô1=0̂. Thus, we have seen that the conditions imply that Ŵ=Ô0, thus being a local parity superselected unitary on the set of modes I\{ij}, thus allowing us to name Ŵ=ŴI\{ij}, and therefore we have proven that ÛijV̂. ◻

(Theorem 2). Using the complete set of descriptors(f¯̂1,,f¯̂N), it is possible to find uniquely theÛthat acts on the N-mode system such that(f¯̂1,,f¯̂N)=(Ûf̂1Û,,Ûf̂NÛ).

Proof. First, note that the unitary conjugation action is up to a global phase. Keeping this in mind, considering the unitary Û as an operator, we can regard it as a vector on the vector space of operators. We can consider in the operator algebra an orthonormal basis, where the scalar product between two operators Â,B̂ is given by tr(ÂB̂). It is straightforward to see that for the fermionic operators, if one labels the orthonormal Fock basis as {|k}k=12N, then |kl| is an orthonormal basis of the operator vector space with scalar product given by tr{ÂB̂}.

Considering this, now Û can be written as Û=k,l=12Ntr(Û|kl|)|kl|. So, if we know tr(Û|kl|) we know the unitary. |kl| is a product of creation and annihilation operators, since |k=f̂i1f̂in|Ω and |ΩΩ|=f̂Nf̂1f̂1f̂N. Using this fact, having at our disposal (f̂1,,f̂N) and (f¯̂1,,f¯̂N), we can construct |kl| and |k¯l¯|, where to construct the second we have replaced the f̂i,f̂i in the decomposition of |kl| by f¯̂i,f¯̂i. We can see easily that |k¯l¯|=Û|kl|Û.

We take the scalar product of any two of these objects. In other words, consider tr(|k¯l¯||mn|)=tr(Û|kl|Û|mn|). Now, if we insert the decomposition of Û found above and we use the linearity properties of the trace, we obtain that

tr(|k¯l¯||mn|)=o,p,q,r=12Ntr(Û|op|)tr(Û|qr|)tr(|po|·|kl|·|rq|·|mn|)=o,p,q,r=12Ntr(Û|op|)tr(Û|qr|)δokδlrδqmδnp=tr(Û|kn|)tr(Û|ml|),

where we have used the orthonormality of the Fock basis and the properties of the Kronecker delta. Using the cyclic properties of the trace and complex conjugation, we obtain that tr(Û|kn|)tr(Û|ml|)=tr(Û|nk|)*tr(Û|ml|). The question that now arises is that if knowing all the values of tr(|k¯l¯||mn|) (that we can obtain since we only use (f1,,fN) and (f¯1,,f¯N)), we can retrieve tr(Û|ml|). We can. We need to notice that tr(Û|ml|) is a complex number, so knowing its polar form is enough. We see that tr(|l¯l¯||mm|)=|tr(Û|ml|)|2. Thus, we obtain the modulus of the complex number. We can see now that tr(Û|ml|)=tr(|l¯l¯||mm|)eiϕm,l. So, only the phases are up to determine. Here is where the issue of the overall phase freedom intervenes. Since Û is unitary, we know there must exist m0, l0 such that |tr(Û|m0l0|)|2=tr(|l0¯l0¯||m0m0|)>0. We have the freedom to fix the phase ϕm0,l0=0 due to the overall phase redundancy. In other words, we could always choose a global phase to cancel the phase ϕm0,l0 so it is set to 0. Now, we can see that if we consider tr(|l0¯l¯||mm0|)=tr(Û|m0l0|)*tr(Û|ml|)=tr(|l0¯l0¯||m0m0|)tr(|l¯l¯||mm|)eiϕml. Thus, we obtain that

tr(Û|ml|)=tr(|l0¯l¯||mm0|)tr(|l0¯l0¯||m0m0|).

Therefore, indeed we can retrieve the unitaries that conjugate (f¯̂1,,f¯̂N) to (f̂1,,f̂N). □

(Theorem 5). Given a fermionic system with the mode setI={i1,,iN}. For any set of descriptors(f¯̂1,,f¯̂N)together with any Heisenberg state|ψ0ψ0|and any non-empty subset of modesJI. The associated diagram of Fig. 1 commutes. In other words,

(πJPφ)((f¯̂1,,f¯̂N),|ψ0ψ0|)=(φπJR)((f¯̂1,,f¯̂N),|ψ0ψ0|).

Proof. We begin by expanding the right-hand side of the equation, by applying the definition of the ontic projection operator, obtaining

(φπJR)((f¯̂1,,f¯̂N),|ψ0ψ0|)=φ((f¯̂j1,,f¯̂jn),|ψ0ψ0|).

Applying now the definition of φ of Eq. (11), we obtain

φ((f¯̂j1,,f¯̂jn),|ψ0ψ0|)=p,ltr(Û(f̂l1f̂lm|ΩΩ|f̂prf̂p1)Û|ψ0ψ0|)f̂l1f̂lm|ΩΩ|f̂prf̂p1,
(A2)

where we have chosen as orthonormal basis for the operator space of the fermionic system with modes j1,,jn the basis given by {f̂l1f̂lm|ΩΩ|f̂prf̂p1}. Where, l1<<lm,p1<<pr with l1,,lm,p1,,prJ, where |Ω is the vacuum for the set of modes I, and thus also J.

We now turn to expand the left-hand side of the initial equation by applying the definition of the epimorphism and choosing the same basis as before, but for the global set of modes {1,…, N}, we obtain

(πJP°φ)((f¯̂1,,f¯̂N),|ψ0ψ0|)=πJP(p,ltr(Û(f̂l1f̂lm|ΩΩ|f̂prf̂p1)Û|ψ0ψ0|)f̂l1f̂lm|ΩΩ|f̂prf̂p1).

The main difference with the expanded version of the right-hand side is the values of the vectors l,p; in the first expression, they can have length up to n with the values being elements of J, while in the second case they can have lengths up to N with the values being elements of I.

Now, let us take the fermionic partial trace. In order to ease up on notation, let us reorder our set of modes such that J corresponds to the first n elements of our set of modes I. First, let us trivially observe that if J = I the equality holds, so, now we consider J a strict subset of I. Using the reordering, we can see how the expression of the left-hand side becomes

πJP(p,l,u,vtr(Û(f̂l1f̂lmf̂u1f̂us|ΩΩ|f̂vtf̂v1f̂prf̂p1)Û|ψ0ψ0|)f̂l1f̂lmf̂u1f̂us|ΩΩ|f̂vtf̂v1f̂prf̂p1),

where p,l are vectors of elements of J up to length n and the vectors u,v are vectors of elements of I\J up to length Nn. Once we have the expression of this form, we first notice that ρ will be a parity superselected state, since ρ0 is and the unitary is also, and the trace of an odd operator is always zero. With this notion, we can apply the properties of the fermionic partial trace11 where we need u=v for the component not to vanish. Thus, we obtain that the left-hand side becomes

p,l,utr(Û(f̂l1f̂lmf̂u1f̂us|ΩΩ|f̂usf̂u1f̂prf̂p1)Û|ψ0ψ0|)f̂l1f̂lm|ΩΩ|f̂prf̂p1.

Moreover, we can focus now only on the coefficients to see that we can pull the sum over the vector u inside. By using the cyclic property of the trace and then using its linearity, we obtain that the coefficients of the left-hand side are

utr((f̂l1f̂lmf̂u1f̂us|ΩΩ|f̂usf̂u1f̂prf̂p1)Û|ψ0ψ0|Û)=tr((f̂l1f̂lm(uf̂u1f̂us|ΩΩ|f̂usf̂u1)f̂prf̂p1)Û|ψ0ψ0|Û).

Now, we can use the fact that f̂l1f̂lmf̂u1f̂us|Ω can be written as |l|u, thus, obtaining that the coefficients of the left-hand side can be written as

tr((f̂l1f̂lm(uf̂u1f̂us|ΩΩ|f̂usf̂u1)f̂prf̂p1)Û|ψ0ψ0|Û)=tr((|lp|(u|uu|))Û|ψ0ψ0|Û),

we have that u|uu|=I and ÔI=Ô, thus we obtain indeed that the left-hand side is p,ltr(Û(f̂l1f̂lm|ΩΩ|f̂prf̂p1)Û|ψ0ψ0|)f̂l1f̂lm|ΩΩ|f̂prf̂p1. Precisely as the right-hand side, proving the theorem. ◻

Proposition 6.If one definesq̂j=12(σ̂jx+iσ̂jy), thenq̂j,q̂jare generators of the N-qubit algebra.

Proof. It is well established that Ij,σ̂jx,σ̂jy,σ̂jz are generators of the N-qubit algebra. Moreover, since (σ̂jx)2=Ij and iσ̂jxσ̂jy=σ̂jz we can say that σ̂jx,σ̂jy are generators of the N-qubit algebra. We will see how q̂j,q̂j are enough to generate such operators that generate the algebra, thus, becoming generators. We first note that q̂j+q̂j=σ̂jx and finally, i(q̂jq̂j)=σ̂jy. ◻

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