Direct-sum sector code entropy model

Let $I$ be a finite nonempty index set. For each $\alpha \in I$, let $H_{\alpha }$ and $K_{\alpha }$ be nonzero finite-dimensional complex inner-product spaces, and give

$H_A:={⨁}_{\alpha \in I}(H_{\alpha }\otimes K_{\alpha })$

the orthogonal direct-sum inner product: distinct summands are orthogonal and the inner product within a summand is the Hilbert tensor-product inner product. On elementary tensors that inner product is $⟨x\otimes y,x^{\mathrm{\prime }}\otimes y^{\mathrm{\prime }}⟩=⟨x,x^{\mathrm{\prime }}⟩⟨y,y^{\mathrm{\prime }}⟩$. A map $T\otimes U$ acts by $(T\otimes U)(x\otimes y)=Tx\otimes Uy$, and a direct-sum map acts blockwise with zero off-diagonal blocks.

A complex-linear endomorphism is a linear map from a vector space to itself. A density matrix on a finite-dimensional complex inner-product space is a self-adjoint endomorphism with nonnegative eigenvalues and trace one; it is faithful when all eigenvalues are strictly positive. These are the conventions used for the finite density matrices in the earlier modular-balance result. The summand indexed by $\alpha$ is called the $\alpha$ sector. For each $\alpha$, fix a faithful density matrix ${\chi }_{\alpha }$ on $K_{\alpha }$. Let $\mathbf{p}=(p_{\alpha }{)}_{\alpha \in I}$ satisfy $p_{\alpha }>0$ and ${\sum }_{\alpha \in I}{p}_{\alpha }=1$, and let ${\rho }_{\alpha }$ be density matrices on $H_{\alpha }$. Define the encoded endomorphism

${\mathcal{E}}_{\chi }(\mathbf{p},\rho ):={⨁}_{\alpha \in I}{p}_{\alpha }{\rho }_{\alpha }\otimes {\chi }_{\alpha }\mathrm{.}$

When this encoded endomorphism is a density matrix, call it the encoded reduced state. The logical data are $\mathbf{p}$ and $\{{\rho }_{\alpha }\}$; the ancillary matrices $\{{\chi }_{\alpha }\}$ are held fixed. For a second strictly positive probability vector $\mathbf{q}$ and faithful density matrices ${\sigma }_{\alpha }$ on the same $H_{\alpha }$, write ${\sigma }_A:={\mathcal{E}}_{\chi }(\mathbf{q},\sigma )$ for the encoded reference endomorphism.

This is a finite-dimensional algebraic model. The definition makes no gravitational interpretation and does not identify a matrix entropy term with a geometric area.

Finite relative entropy equals modular energy minus entropy

Ordered tensor registers, their inner product, and wire permutations

Tensor products of finitely many linear maps

Linear map between vector spaces over the same field

The basis-independent trace of an endomorphism of a finite-dimensional vector space

Daniel Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Appendix A.7.2–A.7.3

John Watrous, The Theory of Quantum Information, §§5.2.1–5.2.2