Fermionic tensor products, partial traces, and partial transpose

Fermionic operators anticommute, so the tensor product, partial trace and partial transpose of fermionic systems differ from kron, the naive partial trace and the ordinary partial transpose. This page explains why, and which conventions the package uses. The tensor product and partial trace follow Szalay et al.^[1] and the partial transpose follows Shapourian et al.^[2]

The CAR algebra

Fermionic creation and annihilation operators $c_i^\dagger, c_i$ satisfy the canonical anticommutation relations (CAR)

\[\{c_i, c_j^\dagger\} = \delta_{ij}, \quad \{c_i, c_j\} = 0, \quad \{c_i^\dagger, c_j^\dagger\} = 0.\]

The Fock space is spanned by occupation number states $\ket{n_1,\ldots,n_N}$ built by acting with $c_i^\dagger$ on the vacuum, and is represented in this package as bitstrings of an integer.

The fermionic tensor product

Suppose we split the modes into two groups $X$ and $\bar X$, with Hilbert spaces $\mathcal{H}_X$ and $\mathcal{H}_{\bar X}$, and we want to embed local operators $A$ on $\mathcal{H}_X$ and $B$ on $\mathcal{H}_{\bar X}$ into the joint space $\mathcal{H} = \mathcal{H}_X \otimes \mathcal{H}_{\bar X}$. The naive choice $A \otimes \mathbb 1$ and $\mathbb 1 \otimes B$ always commutes, $(A \otimes \mathbb 1)(\mathbb 1 \otimes B) = (\mathbb 1 \otimes B)(A \otimes \mathbb 1)$, regardless of what $A$ and $B$ are. This is inconsistent with the CAR relations whenever $A$ and $B$ are fermionic.

To embed a fermion into a larger space, it must be dressed with a parity operator counting the fermions occupying the modes it is moved past, known as a Jordan-Wigner (JW) string. We refer to the tensor product that includes this JW string as a fermionic tensor product and denote it by $\tilde{\otimes}$. We define the embedding of $A$ into the joint space as $E_{\mathcal{H}_X \rightarrow \mathcal{H}}[A] = A \tilde{\otimes} \mathbb 1$ and the code for it is embed(A, HX => H).

There are two ways to combine $A$ and $B$ into an operator on $\mathcal{H}$:

1. tensor_product((A, B), (HX, HXbar) => H) is the ordered product of embeddings $A \hat{\otimes} B = (A \tilde{\otimes} \mathbb 1)(\mathbb 1 \tilde{\otimes} B)$, the algebraic product of eq. (26) in ^[1]. The order of the arguments matters: swapping two odd operators flips the sign.
2. generalized_kron((A, B), (HX, HXbar) => H) is the basis-wise fermionic tensor product $A \tilde{\otimes} B$ of eqs. (14b) and (15) in ^[1]. It equals $A \hat{\otimes} B$ when each subsystem consists of consecutive modes of $\mathcal{H}$ passed in that order, and otherwise can differ, for example by a sign; see tensor products of arrays.

Use tensor_product for product states, for products of local observables, and for anything that should equal a product of embedded operators, such as representation of a product of symbolic operators.

The fermionic partial trace

The partial trace is most often used to calculate reduced density matrices. The fundamental property is that for an observable $A$ on the subsystem $X$, the expectation value of $A$ in a state $\rho$ on the full system is equal to the expectation value of $A$ in the reduced state $\rho_X = \text{Tr}_{\bar X}[\rho]$. A more mathematical description is that the partial trace is the adjoint of the embedding: With $(A|B) = \text{Tr}[A^\dagger B]$ the Hilbert-Schmidt inner product, we have

\[(E_{\mathcal{H}_X \rightarrow \mathcal{H}}[A] | \rho) = (A | Tr_{\mathcal{H} \rightarrow \mathcal{H}_X}[\rho]).\]

Since the embedding acquires fermionic phases depending on the mode ordering, so does the partial trace. This is handled automatically by the package when using partial_trace(A, H => HX).

The fermionic partial transpose

For bosonic and spin systems, partial_transpose is the ordinary matrix transpose on the selected subsystem. For fermions, the ordinary partial transpose does not capture the fermionic structure of the operator algebra. Following ^[2], the package instead implements the fermionic partial transpose, also called partial time reversal.

In the fermionic operator basis, this operation swaps the bra and ket labels on the selected subsystem and multiplies by a phase determined by the fermionic operator content there. Thus it is not, in general, the ordinary matrix transpose in the occupation-number basis. The convention is independent of the ordering of the fermionic modes. Set phase_factors=false in partial_transpose to disable these phases and obtain the ordinary subsystem transpose.

Unlike the ordinary partial transpose, the fermionic result is not generally Hermitian. If $p_A$ is the fermion-parity operator on the transposed subsystem, it obeys $(\rho^{R_A})^\dagger = p_A \rho^{R_A} p_A$.