Fermionic Subsystems, Parity, and Graded Tensor Products

Fermionic subsystems are graded: odd operators in disjoint regions anticommute, whereas the physical even observable algebras commute. An ordinary qubit tensor product can represent the canonical anticommutation relations only after a mode ordering has been chosen. Entanglement, reduction, and partial-transpose calculations must therefore carry the algebra map and parity restriction along with the state; otherwise a Jordan–Wigner string can be mistaken for physical nonlocality.

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

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

Chapter map. See the chapter-wide path from a task specification to an information resource, claim-validity table, and three gates before an operational claim.

Graded locality and parity

Fermionic creation and annihilation operators obey

$$\{c_i,c_j^\dagger\}=\delta_{ij}, \qquad \{c_i,c_j\}=0.$$

The parity operator is $P=(-1)^N$. A homogeneous operator $X$ has degree $|X|=0$ if $PXP=X$ and degree $|X|=1$ if $PXP=-X$. For homogeneous operators $X_A$ and $Y_B$ supported on disjoint mode sets,

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

Thus two odd operators anticommute, while an even operator commutes with every disjoint homogeneous operator. Relativistic observables are taken from the even local algebras, which are compatible with ordinary spacelike commutativity.

In the standard physical setting, density operators commute with global parity. Every odd expectation then vanishes:

$$\operatorname{Tr}(\rho X_{\rm odd}) =\operatorname{Tr}(P\rho P X_{\rm odd}) =-\operatorname{Tr}(\rho X_{\rm odd})=0.$$

This global fermion-parity or univalence rule should not be described as if a generic external “parity reference” made even–odd superpositions observable. Shared fermionic modes can assist tasks restricted by local parity while preserving total parity, but they do not authorize globally parity-violating observables. The relativistic motivation for this distinction is developed in Friis 2016, §§ 2–3.

A Jordan–Wigner representation chooses an order and maps

$$c_j =\left(\prod_{k<j}Z_k\right)\sigma_j^-.$$

Changing the mode order moves the strings and changes occupation-basis phases. The state and every observable must be transformed together. The embedding of fermionic subalgebras, fermionic partial trace, and parity restriction are given explicitly in Szalay et al. 2021, §§ 2–4 and 6–7.

Reduced Gaussian states

A finite fermionic subsystem is specified by the subalgebra generated by its modes, or equivalently by its Majorana operators

$$\gamma_{2j-1}=c_j+c_j^\dagger, \qquad \gamma_{2j}=i(c_j-c_j^\dagger),$$

with $\{\gamma_r,\gamma_s\}=2\delta_{rs}$. A Gaussian state is determined by its two-point data. Restricting the Majorana covariance matrix to the chosen modes determines the reduced Gaussian state, including pairing. For a number-conserving Gaussian state, the smaller correlation matrix

$$C_{ij}=\langle c_i^\dagger c_j\rangle$$

is sufficient. If its restricted eigenvalues are $\nu_k$, then

$$S_A =-\sum_k\left[ \nu_k\ln\nu_k +(1-\nu_k)\ln(1-\nu_k) \right].$$

These density-matrix and covariance constructions are finite regulator statements. The quasifree CAR framework underlying them is developed in Araki 1970, §§ 2–3.

Two checks catch most ordering mistakes:

1. all even correlators agree before and after a consistent reordering; and
2. the spectrum from the restricted covariance matrix agrees with the spin representation after the state and the embedded subsystem algebra have been mapped.

Tracing a convenient set of qubit sites without transforming the fermionic algebra changes the subsystem question.

One fermionic partial-time-reversal convention

There are inequivalent operations called a fermionic partial transpose in the literature. This page uses the partial-time-reversal convention of Shapourian, Shiozaki, and Ryu and does not combine its termwise rules with another convention.

Split a finite subsystem into $A_1\cup A_2$ and expand an even density operator in ordered Majorana monomials:

$$\rho_A =\sum_{\substack{p+q\\ \mathrm{even}}} w_{\boldsymbol\mu,\boldsymbol\nu} \gamma_{\mu_1}\cdots\gamma_{\mu_p} \eta_{\nu_1}\cdots\eta_{\nu_q},$$

where the $\gamma$ Majoranas lie in $A_1$ and the $\eta$ Majoranas lie in $A_2$. Partial time reversal on $A_1$ is the linear map

$$\rho_A^{R_1} =\sum_{\substack{p+q\\ \mathrm{even}}} i^p w_{\boldsymbol\mu,\boldsymbol\nu} \gamma_{\mu_1}\cdots\gamma_{\mu_p} \eta_{\nu_1}\cdots\eta_{\nu_q}.$$

This is Shapourian, Shiozaki, and Ryu 2017, § II.B.2, Eq. (15). The transformed operator need not be Hermitian, so its trace norm uses singular values:

$$\mathcal E_f(\rho_A) =\ln\left\lVert\rho_A^{R_1}\right\rVert_1 =\ln\operatorname{Tr} \sqrt{\rho_A^{R_1}(\rho_A^{R_1})^\dagger}.$$

Natural logarithms are used. Within the parity-preserving LOCC class specified in Shapourian and Ryu 2019, § III.D, this logarithmic negativity has the stated monotonicity properties. A bosonic partial transpose applied after an arbitrary Jordan–Wigner map is a different operation and must not be substituted silently.

Exact four-mode, two-partition benchmark

Take four modes in canonical order $1<2<3<4$ and occupy the two orthogonal orbitals

$$d_1^\dagger =\frac{c_1^\dagger+c_3^\dagger}{\sqrt2}, \qquad d_2^\dagger =\frac{c_2^\dagger+c_4^\dagger}{\sqrt2}.$$

The parity-even, number-conserving Gaussian state is

$$\begin{aligned} |\Psi\rangle &=d_1^\dagger d_2^\dagger|0\rangle\\ &=\frac12\bigl( |1100\rangle+|1001\rangle -|0110\rangle+|0011\rangle \bigr). \end{aligned}$$

The minus sign is forced by moving $c_3^\dagger$ past $c_2^\dagger$ into canonical order.

For the cut $A=\{1,2\}$ and $B=\{3,4\}$,

$$C_A=\frac12 \begin{pmatrix}1&0\\0&1\end{pmatrix},$$

whose eigenvalues are $(1/2,1/2)$. Therefore

$$S_A=2\ln2.$$

This is $2$ bits.

For the interleaved cut $A'=\{1,3\}$ and $B'=\{2,4\}$,

$$C_{A'}=\frac12 \begin{pmatrix}1&1\\1&1\end{pmatrix},$$

whose eigenvalues are $(1,0)$. Hence $S_{A'}=0$: in the graded tensor product,

$$|\Psi\rangle =d_1^\dagger|0\rangle_{A'} \mathbin{\widehat{\otimes}} d_2^\dagger|0\rangle_{B'}.$$

Now perform the adversarial calculation: retain the original qubit order $1,2,3,4$, call qubit sites $1$ and $3$ the subsystem, and trace sites $2$ and $4$ without mapping the algebra. The result is

$$\rho_{13}^{\rm naive} =\frac12|10\rangle\langle10| +\frac12|01\rangle\langle01|,$$

with entropy $\ln2$. This disagrees with the graded entropy because the fermionic even operator $c_1^\dagger c_3$ contains a Jordan–Wigner parity string through site $2$; it is not represented by an operator on qubits $1$ and $3$ alone. The naive trace has changed the algebra, not discovered physical entanglement.

Thus the adjacent graded cut, interleaved graded cut, and naive interleaved qubit cut give $2$, $0$, and $1$ bits, respectively.

The same lesson applies to a mode permutation. Reordering modes while leaving occupation amplitudes unchanged is not a physical relabeling: fermionic exchange signs must transform the state and algebra together. As an out-of-domain operation, the formal unitary

$$U_{\rm odd}=e^{-i\pi\gamma_1/4}$$

coherently mixes global parity sectors because $[U_{\rm odd},P]\ne0$. Its action on the vacuum produces an even–odd superposition and is excluded by the declared operation class. This does not mean that every odd Kraus operator is forbidden: an odd Kraus operator can occur as one charge-transfer branch of a parity-preserving complete channel, just as on the preceding page.

Continuum boundary

For one-particle spaces $\mathcal H_A$ and $\mathcal H_B$, Fock space admits a graded factorization

$$\mathcal F(\mathcal H_A\oplus\mathcal H_B) \simeq \mathcal F(\mathcal H_A) \mathbin{\widehat{\otimes}} \mathcal F(\mathcal H_B).$$

Representing this as an ordinary tensor product requires an ordering convention. A separate obstruction appears for bounded regions in relativistic QFT: their von Neumann algebras are generally type III rather than type-I tensor factors, so a regional density matrix and its entropy require a lattice cutoff, split inclusion, or another controlled regulator. Parity and even-algebra correlators remain intrinsic; the finite matrix representation is the calculational model, not the definition of the continuum subsystem.

Common pitfalls

Making odd operators commute across regions. The graded sign is physical algebraic structure. Ordinary commutativity applies to the disjoint even observable algebras.

Calling a Jordan–Wigner string a signal. The string records an ordering-dependent representation. Compare consistently mapped even observables.

Mixing partial-transpose conventions. Fix the transformation on Majorana monomials first, and use the trace norm and monotonicity theorem belonging to that same convention.

Exercises

1. Odd expectations and graded signs

Prove that an even state has zero expectation for every odd operator. Then evaluate the sign acquired when an odd operator in $A$ passes an odd operator in $B$.

Solution

For $[\rho,P]=0$ and $PXP=-X$,

$$\operatorname{Tr}(\rho X) =\operatorname{Tr}(P\rho P X) =\operatorname{Tr}(\rho PXP) =-\operatorname{Tr}(\rho X),$$

so the expectation is zero. Each elementary odd factor from $A$ anticommutes with each elementary odd factor from disjoint $B$. For two homogeneous odd operators the total degree product is one, hence

$$X_AY_B=-Y_BX_A.$$

If either operator is even, the degree product vanishes and the sign is positive.

2. Diagnose the interleaved-cut error

Compute the two restricted correlation matrices of the benchmark and explain the naive qubit entropy $\ln2$.

Solution

The occupied single-particle orbitals are $(1,0,1,0)/\sqrt2$ and $(0,1,0,1)/\sqrt2$. Restricting their projector to modes $1,2$ gives $C_A=I_2/2$, with eigenvalues $(1/2,1/2)$ and entropy $2\ln2$. Restricting to modes $1,3$ gives the rank-one projector

$$C_{A'}=\frac12 \begin{pmatrix}1&1\\1&1\end{pmatrix},$$

with eigenvalues $(1,0)$ and zero entropy.

In the occupation-qubit vector, configurations with subsystem states $|10\rangle_{13}$ and $|01\rangle_{13}$ are correlated with orthogonal states of qubits $2,4$. A bare qubit trace therefore produces an equal mixture and entropy $\ln2$. It omits the parity string required to represent $c_1^\dagger c_3$, so it is not the fermionic reduction for $A'$.

3. Two-mode fermionic logarithmic negativity

Using the convention above, find $\mathcal E_f$ for the product vacuum and for

$$|\Phi\rangle =\frac{|00\rangle+|11\rangle}{\sqrt2}.$$

Solution

For the vacuum, the Majorana rule leaves a rank-one projector (up to an irrelevant local phase convention). Therefore $\lVert\rho_{\rm vac}^{R_1}\rVert_1=1$ and $\mathcal E_f=0$.

For the Bell state, in the ordered basis $(|00\rangle,|01\rangle,|10\rangle,|11\rangle)$, the rule gives

$$\rho_\Phi^{R_1} =\frac12 \begin{pmatrix} 0&0&0&i\\ 0&1&0&0\\ 0&0&1&0\\ i&0&0&0 \end{pmatrix}.$$

The four singular values of this matrix are all $1/2$. Hence

$$\left\lVert\rho_\Phi^{R_1}\right\rVert_1=2, \qquad \mathcal E_f=\ln2.$$

The answer uses the partial-time-reversal trace norm defined on this page; substituting a different fermionic transpose requires redoing the calculation with that convention.

References

• Araki, Huzihiro. “On Quasifree States of CAR and Bogoliubov Automorphisms.” Publications of the Research Institute for Mathematical Sciences 6 (1970): 385–442. DOI.
• Friis, Nicolai. “Reasonable Fermionic Quantum Information Theories Require Relativity.” New Journal of Physics 18 (2016): 033014. 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.
• Shapourian, Hassan, Ken Shiozaki, and Shinsei Ryu. “Partial Time-Reversal Transformation and Entanglement Negativity in Fermionic Systems.” Physical Review B 95 (2017): 165101. DOI.
• Szalay, Szilárd, Zoltán Zimborás, Mihály Máté, Gergely Barcza, Christian Schilling, and Örs Legeza. “Fermionic Systems for Quantum Information People.” Journal of Physics A: Mathematical and Theoretical 54 (2021): 393001. DOI.