Exact finite algebra entropy identities and holography boundaries

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

Let $I$ be a finite nonempty set. For each $\alpha \in I$, let $H_{\alpha },K_{\alpha }$ be nonzero finite-dimensional complex inner-product spaces. A density matrix is a self-adjoint endomorphism with nonnegative eigenvalues and trace one; it is faithful when all its eigenvalues are strictly positive. Let $p_{\alpha },q_{\alpha }>0$ with ${\sum }_{\alpha }{p}_{\alpha }={\sum }_{\alpha }{q}_{\alpha }=1$, let ${\rho }_{\alpha }$ be a density matrix on $H_{\alpha }$, let ${\sigma }_{\alpha }$ be a faithful density matrix on $H_{\alpha }$, and let ${\chi }_{\alpha }$ be a fixed faithful density matrix on $K_{\alpha }$. Form the finite orthogonal direct sums

${\rho }_A:={⨁}_{\alpha \in I}{p}_{\alpha }{\rho }_{\alpha }\otimes {\chi }_{\alpha },{\sigma }_A:={⨁}_{\alpha \in I}{q}_{\alpha }{\sigma }_{\alpha }\otimes {\chi }_{\alpha }\mathrm{.}$

For a density matrix $\tau$, set $s(\tau ):=-{\sum }_{{\lambda }_j(\tau )>0}{\lambda }_j(\tau )\log {\lambda }_j(\tau )$, with $0\log 0:=0$, so entropy is in nats. For density matrices $\tau ,\omega$ with faithful reference $\omega$, set $D(\tau \mathrm{\parallel }\omega ):=\mathrm{tr}(\tau \log \tau )-\mathrm{tr}(\tau \log \omega )$ using spectral logarithms on positive eigenvalues. Also define $H(\mathbf{p}):=-{\sum }_{\alpha }{p}_{\alpha }\log p_{\alpha }$ and $D_{\mathrm{c}\mathrm{l}}(\mathbf{p}\mathrm{\parallel }\mathbf{q}):={\sum }_{\alpha }{p}_{\alpha }\log (p_{\alpha }\mathrm{/}{q}_{\alpha })$. Then ${\rho }_A$ and ${\sigma }_A$ are density matrices, ${\sigma }_A$ is faithful, and

$s({\rho }_A)=H(\mathbf{p})+{\sum }_{\alpha \in I}{p}_{\alpha }(s({\rho }_{\alpha })+s({\chi }_{\alpha })),$

$D({\rho }_A\mathrm{\parallel }{\sigma }_A)=D_{\mathrm{c}\mathrm{l}}(\mathbf{p}\mathrm{\parallel }\mathbf{q})+{\sum }_{\alpha \in I}{p}_{\alpha }D({\rho }_{\alpha }\mathrm{\parallel }{\sigma }_{\alpha })\mathrm{.}$

These are exact finite-dimensional matrix identities; no literal black-hole area, universal Ryu–Takayanagi formula, continuum limit, or gravitational entropy claim follows.

Direct-sum sector code entropy model

Finite relative entropy equals modular energy minus entropy

Finite matrix spectral and state calculus

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

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm

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