Fermionic Subsystems, Parity, and Graded Tensor Products

Fermionic subsystems are graded: odd operators in disjoint regions anticommute, while physical even observable algebras commute. Tensor-product formulas therefore depend on an ordering convention unless they are translated back to the graded algebra. Entanglement and partial-transpose constructions must respect parity and distinguish physical nonlocality from a Jordan–Wigner string artifact.

Required background. Symmetry-constrained operations fixes parity-preserving operations.

Helpful background. Factorization failure and local algebras supplies the algebraic subsystem viewpoint.

Graded locality

For homogeneous fermionic operators $X_A$ and $Y_B$ in disjoint regions,

$$X_A Y_B =(-1)^{|X_A||Y_B|}Y_BX_A,$$

where $|X|=0$ for even and $|X|=1$ for odd parity. The even subalgebras commute at spacelike separation. Physical density operators obey the parity superselection rule in the usual setting, so expectation values of isolated odd operators vanish unless a parity reference or enlarged algebra is explicitly introduced.

The CAR-algebra and quasifree-state framework used for this restriction is set out in Araki 1970, §§ 2–3.

A mode ordering maps the canonical anticommutation relations to qubits through Jordan–Wigner strings. Changing the ordering moves those strings. Results for even local observables must agree after the corresponding algebra map; an apparent long string is not automatically physical nonlocality.

Fermion parity is a graded superselection structure. The accessible algebra is the even algebra, while parity sectors and fermionic partial-transpose data require conventions compatible with the grading. Schematic and not to scale.

Reduced states and ordering

For a Gaussian state, a spatial subsystem is specified by the subalgebra generated by its Majorana operators. The restricted two-point function determines the even reduced state. In a qubit representation, a noncontiguous fermion region can acquire Jordan–Wigner strings across the complement. One must either use the graded tensor product directly or transform observables and states consistently.

Two safe checks are:

1. even correlators computed before and after reordering agree; and
2. the subsystem spectrum obtained from the restricted covariance matrix agrees with the consistently ordered spin representation.

If only the spin sites are traced without mapping the algebra, one has changed the subsystem question.

Fermionic partial transpose

The bosonic partial transpose is basis dependent but well defined on an ordinary tensor product. For fermions, applying it after an arbitrary Jordan–Wigner map can violate the graded structure. Fermionic partial transpose or partial time reversal introduces parity-dependent phases so that the construction is intrinsic to the fermionic algebra.

For a bipartite state $\rho_{AB}$, a fermionic logarithmic negativity has the schematic form

$$\mathcal E_f =\log\left\lVert\rho_{AB}^{R_A}\right\rVert_1,$$

where $R_A$ denotes the specified fermionic partial time-reversal or transpose convention. Different conventions in the literature must not be mixed term by term; compare their stated monotonicity and normalization properties.

The partial-time-reversal construction is Shapourian, Shiozaki, and Ryu 2017, §§ II–III; its parity-preserving LOCC monotonicity and few-mode tests are Shapourian and Ryu 2019, §§ II–IV.

Two-partition benchmark

Take four fermionic modes in a parity-even Gaussian state and compare partitions $(1,2)|(3,4)$ and $(1,3)|(2,4)$. For the contiguous partition, a standard Jordan–Wigner order makes the qubit and fermion cuts coincide. For the interleaved partition, strings cross the cut.

Compute the restricted covariance spectra and all even two-point functions in both orderings. They agree after the algebra is mapped. A naive qubit partial transpose on the interleaved sites can disagree with the fermionic negativity; the discrepancy diagnoses a changed tensor structure, not a violation of locality.

Continuum boundary

Continuum CAR algebras avoid a literal spatial Hilbert factorization. Entropy calculations therefore use a lattice, split property, or other regulator and must demonstrate stability of even-algebra observables. Parity remains meaningful, but the finite-dimensional density-matrix representation is a model of the local algebra rather than the fundamental definition.

Validity map for graded subsystems. The regional even algebra, parity-preserving operations, mode ordering, and regulator must be fixed. A qubit cut that omits the Jordan–Wigner algebra map fails the first gate. Schematic and not to scale.

Common pitfalls

Letting odd operators commute across regions. Graded locality gives anticommutation for two odd operators. Physical commuting local algebras are the even subalgebras.

Reading Jordan–Wigner strings as physical signals. The strings encode a representation and ordering. Compare invariant even observables.

Using bosonic partial transpose without a convention check. Fermionic negativity requires a graded construction such as partial time reversal or fermionic partial transpose.

References

• Araki, Huzihiro. “On Quasifree States of CAR and Bogoliubov Automorphisms.” Publications of the Research Institute for Mathematical Sciences 6 (1970): 385–442. DOI.
• Shapourian, Hassan, Ken Shiozaki, and Shinsei Ryu. “Partial Time-Reversal Transformation and Entanglement Negativity in Fermionic Systems.” Physical Review B 95 (2017): 165101. DOI.
• Shapourian, Hassan, and Shinsei Ryu. “Entanglement Negativity of Fermions: Monotonicity, Separability Criterion, and Classification of Few-Mode States.” Physical Review A 99 (2019): 022310. DOI.