Reasonable fermionic quantum information theories require relativity

We show that any quantum information theory based on anticommuting operators must be supplemented by a superselection rule deeply rooted in relativity to establish a reasonable notion of entanglement. While quantum information may be encoded in the fermionic Fock space, the unrestricted theory has a peculiar feature: The marginals of bipartite pure states need not have identical entropies, which leads to an ambiguous definition of entanglement. We solve this problem, by proving that it is removed by relativity, i.e., by the parity superselection rule that arises from Lorentz invariance via the spin-statistics connection. Our results hence unveil a fundamental conceptual inseparability of quantum information and the causal structure of relativistic field theory.

* nicolai.friis@uibk.ac.at attempt to construct an abstract fermionic quantum information theory, see, e.g., Refs. [2][3][4]. However, as we shall show here, the physically unrestricted theory suffers from a disconcerting malady: The marginals ρ A and ρ B of bipartite pure states may not have matching spectra, leaving the typical notion of entropy of entanglement in a state of ambiguity, since S(ρ A ) = S(ρ B ). Depending on the choice of subsystem, different amounts of entanglement would be attributed to the system. This problem does not occur in theories with a natural tensor product structure, like bosonic modes or qubits, where the Schmidt decomposition guarantees symmetric marginal entropies for pure states. For fermions, on the other hand, mappings to a tensor product space, i.e., to qubits, do not generally preserve the structure of the subsystems [5], and the issue persists.
We demonstrate here that this problem can be overcome by imposing a superselection rule (SSR) that forbids coherent superpositions of even and odd numbers of fermions. Although the problem of asymmetric pure state marginals is thus removed, it seems rather artificial to enforce such a restriction within the abstract theory. Only once the model is embedded in a physical context, in this case relativistic field theory, does the SSR arise naturally from Lorentz invariance and the constraints of signalling. With this, we provide an alternative view on the connection between abstract quantum information theory and relativistic quantum field theory, arguing that the latter is indeed necessary for the reasonable construction of the former. This work hence adds a new facette to the discussion of informational constructions of quantum theory (see, e.g., Refs. [6][7][8]), by introducing an information-theoretic aspect of SSRs -a fascinating topic in its own right (see, e.g., Refs. [9][10][11][12][13][14][15][16][17] for a selection of literature).
In the following, we will first outline the construction of the fermionic Fock space, as well as of the pure and mixed states in such a Hilbert space. To understand the origin of the problem described above, we will then discuss the subtleties involved in forming subsystems of fermionic modes, and give an example for a pure state that features marginals with different entropies.
Finally, the role and the origin of SSRs are discussed, and we show how the problem can be disposed of. To highlight the intrinsically different character of fermionic modes and qubits, we supplement our discussion with an Appendix, where an example for a pure state that satisfies the SSR, but still cannot be consistently mapped to a multi-qubit state, is presented.
Fermionic Fock space. Let us consider a system of n fermionic modes with mode operators b k and b † k for (k = 1, . . . , n), which satisfy the anticommutation relations for all i, j. The vacuum state is annihilated by all b k , i.e., b k ||0 = 0 ∀ k, and the purpose of the double-lined notation for the state vectors will become apparent shortly. The creation operators b † k populate the vacuum with single fermions, that is, b † k ||0 = ||1 k . When two, or more, fermions are created, the corresponding tensor product of single-particle states needs to be antisymmetrized due to the indistinguishability of the particles. We use the convention where we use the double-lined notation to imply the antisymmetrized wedge product "∧" between single-mode state vectors with particle content (as opposed to the standard notation | · | · = | · ⊗ | · ). With this definition at hand, and postponing possible physical restrictions, one may write arbitrary pure states on the Fock space as where the complex coefficients γ 0 , γ i , γ jk , . . . are chosen such that the state is normalized. Similarly, mixed state density operators can be written as convex sums of projectors on such pure states. For more details on this notation and the fermionic Fock space, see, e.g., Refs. [5,18].
Partitioning the Hilbert space. To construct a quantum information theory, it is then necessary to establish a meaningful notion of subsystems. Since the particle number in the Fock space need not be fixed, we will consider entanglement between different modes. However, due to the antisymmetrization, the Fock space is not naturally equipped with a tensor product structure with respect to the individual mode subspaces. These subspaces may nonetheless be defined by invoking consistency conditions [5] that ensure that the expectation values of all local observables O A (i.e., as in, operators pertaining only to the modes of the subspace A) yield the same result for the global state ρ AB , and for the corresponding local reduced states ρ A = Tr B (ρ AB ), i.e., The global n-mode fermionic state ρAB may be mapped to an isomorphic n-qubit stateρAB. The marginals of ρAB, e.g., ρA = TrB(ρAB), are well-defined by Eq. (4), and may also be mapped to isomorphic qubit states (e.g., ρA ↔ρ A ). However, as shown in Ref. [5], it is in general impossible to match all the marginals of the n-mode state to all marginals of the nqubit state,ρ A =ρA = TrB(ρAB). An example for a state featuring this problem is given in the Appendix.
This procedure uniquely defines the mode subspace marginals of any global state, i.e., the partial trace operation, via That is, operators corresponding to modes that are being traced out are anticommuted towards the vacuum projector, before being removed. At this point, it is helpful to understand the differences between fermionic modes and qubits. For any fixed number n, the fermionic n-mode Fock space is isomorphic to an n-qubit space. A widely known example for such an isomorphism is the Jordan-Wigner transformation, (see, e.g., [2]). Such mappings generally do not commute with the procedure of partial tracing [5], since local mode operators are mapped to global qubit operations. In other words, it is generally not possible to establish isomorphisms between a fermionic n-mode state and an n-qubit state in such a way that also all of the respective fermionic marginals are isomorphic to their qubit counterparts. An illustration of this problem is shown in Fig. 1. Consequently, the (quantum) correlations between n fermionic modes may generally not be identified with those of the isomorphic n-qubit states.
In spite of this inequivalence, the partial trace, and hence the subsystems and their entropies remain well defined for fermionic modes. Moreover, the construction of the density operators and its marginals is based solely on the algebraic structure of Eqs. (1), together with the requirement that the expectation values of subsystem observables yield the same result when evaluated using either the global states or the corresponding marginals. No other assumptions are required for a consistent definition of the subsystems, and their total correlations. For instance, the mutual information I AB , a measure of the overall correlation between subsystems A and B, is in this context already well defined by the expression where ρ A(B) = Tr B(A) (ρ AB ), and S(ρ) = −Tr ρ ln(ρ) is the von Neumann entropy of the density operator ρ. But, as we shall elaborate on shortly, the same cannot be said for genuine quantum correlations, i.e., entanglement. Consider, for instance, the non-superselected two-mode pure state given by According to the prescription of Eq. (5), the single-mode reduced states can be quickly checked to be where the symmetry between the subsystems is broken by the different relative signs within the offdiagonal elements. The eigenvalues of the two reduced states do not match in general. For example, when γ 0 = γ k = γ k = γ kk = 1/2, the mode k appears to be in a pure state (with eigenvalues 0 and 1), whereas the state of the mode k is maximally mixed (both eigenvalues are 1/2). Normally, the entropy of the subsystem of a pure state would be considered as an entanglement measure. Here, depending on the choice of subsystem, one would either conclude that the overall state is maximally entangled, or not entangled at all. This problem is not limited to pure states. It persists for mixed states, where the entropy of entanglement is of central importance for the entanglement of formation.
Such an ambiguity in the definition of entanglement is of course highly undesirable. One possibility to resolve the issue, would be to change the definition of entanglement, and work with at a non-symmetric quantity. On the other hand, such a drastic step may not be required, if one is willing to embed the abstract fermionic quantum information theory in a physical framework. As we shall show in the following, a reasonable definition of entanglement between fermionic modes is obtained when invoking an additional physical principle-the spinstatistics connection, which itself arises from (special) relativity.
Invoking relativity-superselection rules. No SSRs have been introduced up to this point. Note that the term SSR may refer to different restrictions. For instance, they may arise from fundamental symmetries of the system, such as parity [9], or charge conservation [10][11][12]. Alternatively, effective (or generalized) SSRs originate from practical limitations, such as particle number conservation due to energy constraints, see, e.g., Refs. [13-15, 19, 20]. Both type of restrictions may be formalized as constraints on the observables, see, e.g., Ref. [3,4].
Here, we will formulate such constraints in a different, but equivalent way, as restrictions on the components γ 0 , γ i , γ jk , . . . [see Eq. (3)], of pure state decompositions with respect to the Fock basis. In particular, we will consider any coherent superpositions of even and odd numbers of fermionic excitations as unphysical. The argument that we will advocate to defend this position is based on the well-known spin-statistics connection, relating the anticommutation relation algebra to the transformation properties associated to objects of half-integer spin. Recall that, a priori, we have made no assumptions on the physical realization of the anticommutation relations of Eq. (1a). Nonetheless, as an inevitable consequence of this anticommutation relation algebra, the excitations of the mode operators must satisfy Fermi-Dirac statistics. At this stage, we may surrender some level of abstraction and provide a physical context. When we place the fermionic quantum information theory in the context of relativistic field theory, the spin-statistics connection follows from Lorentz invariance (see, e.g., Refs. [21] and [22, pp. 52]). Thus, we must interpret the fermionic excitations of our theory as particles of half-integer spin. The corresponding states switch their sign when subjected to spatial rotations by 2π. If a superposition of even and odd numbers of fermions was permitted in such a physical theory, it would entail that rotations of our reference frame by 2π would change the relative sign of these contributions within the superposition. Put bluntly, an observer spinning around once, could switch a remote quantum system between two orthogonal states at will. Since such instantaneous signals are forbidden in relativity, one arrives at the aforementioned SSR. Crucially, this line of reasoning is intimately tied to relativity. On the other hand, as we will show next, the thus imposed constraint provides the essential ingredient to guarantee a meaningful quantum (information) theory based on anticommuting operators.
Superselection rules & symmetric pure state marginals. We shall now provide the main technical statement of this work, and its proof. Proof. To show this, let us consider a pure state ||ψ N even in an n-mode fermionic Fock space, where N = {1, 2, . . . , n} denotes the set of modes, and without loss of generality we have chosen ||ψ N even to be a superposition of states with even numbers of fermions. The set N is then partitioned into the subsets M = {µ i | µ i,j ∈ N : µ i = µ j if i = j; i, j = 1, 2, . . . , m < n} and M C = N \M , such that N = M ∪ M C and M ∩ M C = ∅. With respect to this bipartition, we may write the state ||ψ N even in the pure state decomposition where γ µi , . . . , γ µ1...µm ∈ C, ψ N even ||ψ N even = 1, and without loss of generality we have here selected m to be odd. The states ||ψ M C µi...µ k ,even(odd) contain only even (odd) numbers of excitations, and only in modes from the set M C . Any sign changes that may occur when rewriting a given state in such a decomposition can be absorbed into the γ-coefficients. Adhering to the "outside-in" tracing rule of Eq. (5), we note that the state has been brought to a form, where the partial trace over M C is achieved by simply removing all projectors ||ψ M C µi...µ k ψ M C µi...µ k || pertaining to M C from the projector on ||ψ N even , without incurring any additional sign flips. On the other hand, if we trace over the modes in the set M instead, anticommuting the operators corresponding to modes in M towards the vacuum projector in the process, we may generate sign changes. However, for the superselected state, all the nonzero contributions to the partial trace over M are generated from elements such as There, the parity of the number of anticommutations towards ||0 from the left, is the same as the parity of the number of anticommutations towards 0|| from the right. In other words, once the state has been brought to the form of Eq. (9), the partial trace can be carried out as if operating on a tensor product of Hilbert spaces, i.e., as if H N = H M ⊗ H M C . In particular, this implies that the two reduced states on H M and H M C , which are isomorphic to the corresponding m-mode and (n − m)-mode reduced fermionic states, respectively, have the same spectrum. For any bipartition of the Fock space, a decomposition with this property can be found, although, in general, no decomposition exists that simultaneously accomplishes the required task for all bipartitions at once. An example is presented in the Appendix. We hence conclude that the SSR forbidding superpositions of even and odd numbers of fermions, guarantees that the spectra of the marginals for any bipartition are pairwise identical. An analogous argument applies if the initial state is a superposition of only odd numbers of fermions, or if m is even, and the proof therefore applies without restriction.
Discussion. Within quantum information theory and quantum computation, discussing problems in an abstract context has proven to be very useful. However, when attempting a similar approach to a quantum information theory based on a fermionic Fock space one encounters difficulties. As we have shown, an unrestricted fermionic model features pure states with non-symmetric bipartitions. That is, pairs of reduced states across bipartitions need not have the same spectra, which is very problematic for the definition of entanglement. As we have shown, this problem is removed, when superpositions of even and odd numbers of fermions are forbidden. Nonetheless, the removal of an inconvenience appears to be a rather weak justification for the introduction of a restriction of generality within the abstract model. On the other hand, when placing the fermionic system within the physical framework of a relativistic quantum field theory, the SSR follows naturally from the requirements of Lorentz invariance and the constraints of signalling.
Let us now turn to the bipartitions into pairs of modes, where the reduced state density operators in the even and odd subspaces are given by p even 1,2 ρ even 1,2 = |α 0 | 2 + |α 34 | 2 ||0 0|| (A.5a) for the subspace of modes 1 and 2. Similarly, for 3 and 4 we obtain For the superselected state the even and odd subspaces decouple. We may therefore compare the characteristic polynomials for the even and odd subspaces separately. A simple computation reveals that both p even 1,2 ρ even 1,2 and p even 3,4 ρ even 3,4 yield the characteristic polynomial det p even 1,2 ρ even 1,2 − λ1 = det p even 3,4 ρ even 3,4 − λ1 Similarly, for the odd subspace we find We hence find that the eigenvalues of the marginals ρ 1,2 and ρ 3,4 coincide. Some straightforward computation along the same lines confirm that this is also the case for the bipartitions (1, 3|2, 4) and (1, 4|2, 3). Now, the interesting aspect of this insight pertains to the fact that the 4-mode system may not be consistently mapped to a 4-qubit state. For the latter, the matching subsystem spectra would be guaranteed by the Schmidt decomposition. Recall that the consistency conditions (see Ref. [5]) for partial tracing demand that the numerical value of the expectation values of any "local" operator (in the sense of mode-subspaces) is independent of being evaluated for the overall state, or for the corresponding local reduced state. These consistency conditions then fix the relative signs of different contributions from matrix elements of the total state to the matrix elements of the reduced states. For the example state at hand, the resulting off-diagonal matrix elements of all 2-mode vs. 2-mode bipartitions are collected in Table I. Now, one wonders, whether the pure state ||ψ (4) even and its marginals can be faithfully represented as a 4-qubit state | ψ (4) even ∈ (C 2 ) ⊗4 . Here, we will call such a mapping faithful, if all the diagonal matrix elements of | ψ (4) even ψ (4) even | and its marginals with respect to the computational basis match the diagonal elements of ||ψ (4) even ψ (4) even || with respect to the Fock basis. For the off-diagonal elements we impose a slightly weaker condition, i.e., that the absolute values of the off-diagonals (with respect to the respective bases) match. This corresponds to demanding that measurements in the Fock basis are reproduced, and that the marginals have the same spectra. These conditions imply that a faithful mapping from the Fock basis of four fermionic modes to the computational basis of four qubits must be of the form ||0 → e iφ0 | 0000 , (A.9a) Since this must hold independently of the values of α 0 , α 12 , α 34 , and α 1234 , we arrive at where n 1 ∈ Z. Similarly, the other off-diagonals from Table I provide  which cannot be satisfied, since the left-hand side is an even integer for all n 1 , n 3 ∈ Z, while the right-hand side is an odd integer for all n 5 ∈ Z. We hence conclude that the state of Eq. (A.1) cannot be consistently mapped to a 4qubit state, even though it satisfies the SSR, and despite the fact that for any of its bipartitions the respective marginals have the same spectra.