Minimal Areas from Entangled Matrices

Our starting point is the quantum mechanics of $d$ $N\times N$ Hermitian matrices $X^{1}\dots X^{d}$, with a Lagrangian $L=\Tr\mathsf{L}(X^{a},\dot{X}^{a})$. The matrices can often be associated to directions in an ambient space $\mathds{R}^{d}$, which is distinct from the emergent noncommutative space we will be interested in. The theories we will consider have a $U(N)$ gauge symmetry³³3 More properly, the gauge group is the projective unitary group $PU(N)$, which is $U(N)$ quotiented by the overall phase, $\mathsf{U}\sim e^{i\phi}\mathsf{U}$. Our considerations will mostly be at large $N$, where the distinction between $PU(N)$ and $U(N)$ is subleading. In §4 we will need to remember that the correct group is $PU(N)$. under which

$$X^{a}\to\mathsf{U}X^{a}\mathsf{U}^{\dagger},\qquad\mathsf{U}\in U(N).$$

(2.1)

We define three Hilbert spaces: the extended, the invariant and the physical. The extended Hilbert space is just a copy of $L^{2}(\mathds{R})$ for every real degree of freedom,

	$\displaystyle\mathcal{H}_{\mathrm{ext}}$	$\displaystyle\equiv\mathrm{span}\,\left\{\ket{X^{a}}\Big{|}\left(X^{a}\right)^{\dagger}=X^{a},a=1\ldots d\right\}$
		$\displaystyle=\bigotimes_{a=1}^{d}\left[\bigotimes_{1\leq i<j\leq N}\mathrm{span}\,\left\{\ket{X^{a}_{ij}}\Big{|}X^{a}_{ij}\in\mathds{C}\right\}\otimes\bigotimes_{i=1}^{N}\mathrm{span}\,\left\{\ket{X^{a}_{ii}}\Big{|}X_{ii}^{a}\in\mathds{R}\right\}\right].$		(2.2)

The cumbersome second line has the advantage of clarifying automatically the inner product in this Hilbert space: it is just a $\delta$-function in each of these factors. The invariant Hilbert space is the quotient of the extended Hilbert space by all gauge transformations (2.1),

$$\mathcal{H}_{\mathrm{inv}}\equiv\mathcal{H}_{\mathrm{ext}}\Big{/}\left(\ket{X^{a}}\sim\ket*{\mathsf{U}X^{a}\mathsf{U}^{\dagger}}\right).$$

(2.3)

An equivalent definition of the invariant Hilbert space is as follows. Let the matrices $\hat{\Pi}^{a}$ be, component-wise, the momenta canonically conjugate to $\hat{X}^{a}$. The generators of the $U(N)$ transformation (2.1) are the matrix of operators $\hat{G}=2i\sum_{a}{:}[\hat{X}^{a},\hat{\Pi}^{a}]{:}$, where the normal ordering symbol ${:}\quad{:}$ means that all $\hat{X}_{ij}$ are to the left of $\hat{\Pi}_{kl}$ in terms of operator ordering. Explicitly,

$$\hat{G}_{ij}=2i\sum_{a=1}^{d}(\hat{X}^{a}_{ik}\hat{\Pi}^{a}_{kj}-\hat{X}^{a}_{kj}\hat{\Pi}^{a}_{ik})\,.$$

(2.4)

Then, $\mathcal{H}_{\mathrm{inv}}$ is the subspace of $\mathcal{H}_{\mathrm{ext}}$ annihilated by $\hat{G}$:

$$\mathcal{H}_{\mathrm{inv}}\equiv\left\{\ket{\Psi}\in\mathcal{H}_{\mathrm{ext}}\Big{|}\hat{G}_{ij}\ket{\Psi}=0\text{ for }1\leq i,j\leq N\right\}\,.$$

(2.5)

A point of notation: for the rest of §2 (and this section only), we denote Hilbert space operators with hats.

The physical Hilbert space $\mathcal{H}_{\mathrm{phys}}$ of the theory is isomorphic to the invariant Hilbert space. It will be crucial for us, however, that (2.5) is just one of many ways of realising the physical Hilbert space within the extended Hilbert space. This fact is illustrated in Fig. 3 and elaborated on below. The essential point is that

gauge-fixed states are an alternative way to realise $\mathcal{H}_{\mathrm{phys}}$. We will see how this works in detail in the following §2.2. The first step is to decompose $\mathcal{H}_{\mathrm{ext}}$ into the gauge orbits of a particular gauge-fixed slice. We will use the parametrisation

$$X^{d}=\mathsf{U}\lambda\mathsf{U}^{\dagger},\quad\mathsf{U}\in U(N),\,\quad\lambda\text{ diagonal and ordered.}$$

(2.6)

Here $X^{d}$ is the last of the $d$ matrices. The remaining $d-1$ matrices are written as

$$X^{a}=\mathsf{U}X_{\mathrm{gf}}^{a}\mathsf{U}^{\dagger}\,,$$

(2.7)

where the $X_{\mathrm{gf}}^{a}$ are not necessarily diagonalised. Sometimes we will also write $X_{\mathrm{gf}}^{d}=\lambda$ for convenience. The gauge-slice is parametrised by $\lambda$ and the $X_{\mathrm{gf}}^{a}$, while the gauge orbits are described by $\mathsf{U}$. Thus, we can write almost any vector $\ket{X^{a}}$ uniquely as

$$\ket{X^{a}}=\sqrt{\Delta(\lambda)}\ket{\mathsf{U}}\ket*{X_{\mathrm{gf}}^{1\dots d-1},\lambda}\,.$$

(2.8)

We have incorporated the Vandermonde measure factor

$$\Delta(\lambda)=\prod_{i<j}(\lambda_{j}-\lambda_{i})^{2}\,,$$

(2.9)

with $\lambda_{i}$ the diagonal entries of $\lambda$, into the state (2.8). Of course, the specific choice of gauge slice (2.6) is not essential; our discussion generalises to any gauge in which a matrix function $F(\{X^{a}\})$, transforming in the adjoint of $U(N)$, is taken to be diagonal and ordered.

A subtlety is that the decomposition (2.6) fails to be unique when $X^{d}$ has coincident eigenvalues. This leads to a residual gauge symmetry that, as we discuss below, renders our relational observables ill-defined on states with support on coincident eigenvalues of $X^{d}$ [58]. The states of main interest to us, such as the fuzzy sphere state, have vanishing support on these configurations.

The decomposition (2.8) shows how $\mathcal{H}_{\mathrm{ext}}$ contains many equivalent copies of each physical state. Wavefunctions in $\mathcal{H}_{\mathrm{inv}}$ are obtained by integrating (2.8) over $\mathsf{U}$. This demonstrates that the physical data of the state is parametrised by $\{X_{\mathrm{gf}}^{1\dots d-1},\lambda\}$. As evident from (2.8), this same data can also be accessed by making a choice of gauge-slice, such as setting $\mathsf{U}=\mathds{1}$ in (2.8). This gives a different embedding of physical states into $\mathcal{H}_{\mathrm{ext}}$, as illustrated in Fig. 3. We emphasise that there is no intrinsic sense in which any of the representatives of a physical state in $\mathcal{H}_{\mathrm{ext}}$ is more physical than the others (as long as one deals correctly with residual gauge symmetries). The best choice depends on the problem one wishes to address.

In gauge theories there is a tension between Gauss’s law on physical states and partitioning the degrees of freedom into local subsystems. In our case, different matrix components are related by the $U(N)$ Gauss’s law, preventing a naïve partition of the matrix into blocks. It is important to emphasise that the tension between partition and Gauss’s law is worse for matrix partitions than for spatial partitions of local gauge theories. In gauge theories, Gauss’s law results in a lack of local tensor factors of the physical Hilbert space but the theory still admits local subalgebras of physical operators [47]. These local subalgebras act on tensor factors of the analogue of our extended space $\mathcal{H}_{\mathrm{ext}}$ [45, 46, 48], but there are nevertheless ambiguities with defining entanglement entropy [59, 60]. This last observation was formalised into the statement that the definition of entropy requires a choice of factorisation map $J:\mathcal{H}_{\mathrm{phys}}\hookrightarrow\mathcal{H}_{\mathrm{fact}}$, where $\mathcal{H}_{\mathrm{fact}}$ admits a local tensor factorisation [61, 62]. The previous literature can then be interpreted as choosing $\mathcal{H}_{\mathrm{fact}}\cong\mathcal{H}_{\mathrm{ext}}$.

Since $U(N)$ gauge transformations move a given sub-block around the matrix, see Fig. 2, we do not obviously have gauge-invariant subalgebras that are local in the emergent space. We will thus bypass an algebraic description and specify the partition as a factorisation map: an isometric map from $\mathcal{H}_{\mathrm{phys}}$ to a subspace of $\mathcal{H}_{\mathrm{ext}}$. Recall from the discussion around Fig. 3 that $\mathcal{H}_{\mathrm{ext}}$ contains many equivalent copies of each physical state. The factorisation map will involve choosing a particular representative $\ket{\psi_{G}}\in\mathcal{H}_{\mathrm{ext}}$ of a physical state $\ket{\psi}\in\mathcal{H}_{\mathrm{phys}}$, and then using the tensor product structure of $\mathcal{H}_{\mathrm{ext}}$ to define a Hilbert space of the subsystem. We will now describe the embedding in more detail. In the following subsections we will interpret our construction as a choice of measure over quantum reference frames.

Firstly, we define a gauge-fixed state in $\mathcal{H}_{\mathrm{ext}}$, using the decomposition (2.8), as

$$\ket{\psi_{\mathds{1}}}\equiv\ket{\mathds{1}}\ket{\psi_{\mathrm{gf}}}\,,$$

(2.10)

where $\ket{\psi_{\mathrm{gf}}}$ is any normalisable superposition of the $\ket*{X_{\mathrm{gf}}^{1\dots d-1},\lambda}$ states in (2.8). The state $\ket{\psi_{\mathds{1}}}$ itself is only $\delta$-function normalisable because the unitary $\mathsf{U}$ in (2.8) has been fixed to be the identity. As in textbook quantum mechanics, we could make the state normalisable by constructing a wavepacket over unitaries $\mathsf{U}$ that is strongly peaked on $\mathsf{U}=\mathds{1}$.⁴⁴4 It is also possible to take care of this by modifying the inner product using group-averaging [63] or the Hamiltonian BRST formalism [64, 65]. We take the simplest approach available to us, expecting that different approaches differ only at subleading order. The details of this wavepacket will contribute to the entanglement at subleading orders in the semiclassical limit, and we will not keep track of these corrections. We take $\ket{\psi_{\mathds{1}}}$ to be invariant under all residual gauge symmetries, and similarly for the other representatives we consider below.

The state $\ket{\psi_{\mathds{1}}}$ in (2.10) is fully gauge-fixed. An invariant state is recovered by integrating $\ket{\psi_{\mathds{1}}}$ over the gauge orbits of the unitary group. However, we will see later that such an integration introduces too much microscopic information. In particular, in the fuzzy sphere model of interest below, it includes discontinuous volume-preserving transformations in which a region can be broken up into a large number of ‘Planck-sized’ disconnected regions. For this reason, we will wish to build a state in which the gauge transformations are coarse-grained. Thus we consider an intermediate case of partially gauge-fixed states, in which we integrate $\ket{\psi_{\mathds{1}}}$ over a subgroup $G\subseteq U(N)$. This is the third case illustrated in Fig. 3 above. That is, we consider

$$\ket{\psi^{\prime}_{G}}=\int_{G}\differential{g}\hat{\pi}(g)\ket{\psi_{\mathds{1}}}=\int_{G}\differential{g}\ket{g}\ket{\psi_{\mathrm{gf}}}\,.$$

(2.11)

Here, $\hat{\pi}$ is the representation of $U(N)$ on $\mathcal{H}_{\mathrm{ext}}$. For reasons to be made clear shortly, we call $G$ the frame transformation group. The state $\ket{\psi^{\prime}_{G}}$ has $U(N)/G$ gauge-fixed. The prime in $\psi^{\prime}_{G}$ indicates that we have one more step to perform.

Finally, there is one part of $U(N)$ that we will want to fully integrate over. We will be interested in $M\times M$ sub-blocks of the matrix and we wish to consider states that are fully invariant under the $U(M)$ associated to this sub-block, and the $U(N-M)$ associated to the complementary sub-block. This $U(M)\times U(N-M)$ does not move the sub-block around the matrix, as illustrated in Fig. 1 above. Correspondingly, in the fuzzy sphere model it does not move the subregion around the sphere and does not suffer from the same kind of microscopic sensitivity as the general $U(N)$ transformations discussed above. The $U(M)$ transformations, but not the $U(N-M)$ transformations, will instead induce ‘edge modes’ due to the fact that in the state $\ket{\psi_{\mathrm{gf}}}$ the degrees of freedom inside the subsystem carry a non-trivial $U(M)$ charge. These edge modes will be the source of our boundary area law. Thus we set

$$\ket{\psi_{G}}\equiv\int_{U(M)}\differential{U}\int_{U(N-M)}\differential{\widetilde{U}}\hat{\pi}(U\widetilde{U})\ket{\psi^{\prime}_{G}}=\int\differential{U}\differential{\widetilde{U}}\int\differential{g}\ket*{U\widetilde{U}g}\ket{\psi_{\mathrm{gf}}}\,.$$

(2.12)

Here, we are abusing notation by using $U$ to denote both an element of $U(M)$ as well as $U\oplus\mathds{1}_{N-M}\in U(N)$, and similarly with $\widetilde{U}$. We will continue with this practice henceforth. As we discuss further below, there may be a redundancy between the $U(M)\times U(N-M)$ integral and the $G$ integral in (2.12); this does not cause any difficulties. We will not need keep track of the normalisation of our states very carefully, as the normalisation will be accounted for automatically when we compute the von Neumann entropy.

Equation (2.12) defines a map $\mathcal{H}_{\mathrm{phys}}\to\mathcal{H}_{G,M}\subset\mathcal{H}_{\mathrm{ext}}$, where $\mathcal{H}_{G,M}$ is the Hilbert space spanned by states of the form given above. This is our factorisation map and these will be the states we work with. As we will explain in detail, there are two qualitatively different types of integral in (2.12): the integrals over $U,\widetilde{U}\in U(M)\times U(N-M)$ and the integral over $g\in G$. The latter integral is the key ingredient that is new relative to [36, 37]. We will return to the physical interpretation of these steps in the next subsection.

For the class of states in (2.12), we define the subsystem as a tensor factor of $\mathcal{H}_{\mathrm{ext}}$. Recall from (2.2) that $\mathcal{H}_{\mathrm{ext}}$ admits a tensor factorisation into $N^{2}$ copies of $L^{2}(\mathds{R})$, each one corresponding to an element of the matrix. This allows us to define our subsystem as a matrix sub-block, in the spirit of [41, 42]. To specify the sub-blocks of the matrices we introduce the projector and its complement

$$\Theta_{\Sigma}\equiv\Theta\equiv\left(\begin{matrix}\mathds{1}_{{\scriptscriptstyle M\times M}}&0_{{\scriptscriptstyle M\times(N-M)}}\\ 0_{{\scriptscriptstyle(N-M)\times M}}&0_{{\scriptscriptstyle(N-M)\times(N-M)}}\end{matrix}\right),\qquad\Theta_{\overline{\Sigma}}\equiv\overline{\Theta}\equiv 1-\Theta_{\Sigma}\,.$$

(2.13)

We can then break up each matrix into four blocks,

$$X^{a}_{\Sigma\Sigma}\equiv\Theta X^{a}\Theta,\qquad X^{a}_{\Sigma\overline{\Sigma}}\equiv\Theta X^{a}\overline{\Theta},\qquad X^{a}_{\overline{\Sigma}\Sigma}=\left(X^{a}_{\Sigma\overline{\Sigma}}\right)^{\dagger},\qquad X^{a}_{\overline{\Sigma}\overline{\Sigma}}\equiv\overline{\Theta}X^{a}\overline{\Theta}.$$

(2.14)

The extended Hilbert space thus decomposes as

$$\mathcal{H}_{\mathrm{ext}}=\mathcal{H}_{\Sigma\Sigma}\bigotimes\left(\mathcal{H}_{\Sigma\overline{\Sigma}}\otimes\mathcal{H}_{\overline{\Sigma}\overline{\Sigma}}\right)\equiv\mathcal{H}_{\mathrm{in}}\bigotimes\mathcal{H}_{\mathrm{out}}.$$

(2.15)

Here $\mathcal{H}_{\mathrm{in},\mathrm{out}}$ are defined as the Hilbert space on the corresponding side of the $\bigotimes$.

When $G=\left\{\mathds{1}\right\}$, and in regimes where $\ket{\psi_{\mathrm{gf}}}$ describes a semiclassical noncommutative geometry, the factorisation (2.15) of the Hilbert space corresponds to a geometric partition of the fuzzy geometry. The projector $\Theta$ in (2.13) is written in the same basis in which $\lambda$ was diagonalised in (2.6). When $\mathsf{U}=\mathds{1}$ in (2.6), then $\lambda=X^{d}$ and $\Sigma$ is therefore a geometric subregion with coordinates $x^{d}\leq\lambda_{M}$ (the $M$th diagonal entry of the matrix $X^{d}$) and $\overline{\Sigma}$ is its complement [41, 42]. A transformation by $\hat{\pi}(g)$ as in (2.11) effectively conjugates the projector with a volume-preserving diffeomorphism. Because we are integrating over all $g\in G$ in (2.11), the subsystem is then not a single subregion but an average over subregions (when $G\neq\{\mathds{1}\}$). This was illustrated in Fig. 2 above.

In this subsection and the following §2.4, we will interpret the construction above in the language of internal quantum reference frames [52, 53, 54, 55, 56, 57], which is closely related to the Page-Wootters formalism [50, 51].⁵⁵5 We thank Elliot Gesteau for alerting us to this language and Phillipp Höhn for detailed comments. The factorisation map (2.12) will be motivated using this framework. The reader that is happy with (2.12) as a technical starting point may wish to skim these sections. Technical developments towards our main result continue in §2.5.

The basic point is that what one might call gauge-fixed observables can be promoted to gauge-invariant relational observables: they are relational to a quantum reference frame. Our partition, from this point of view, corresponds to fixing some gauge-invariant data (the size of the matrix sub-block or the volume of a subregion) but being uncertain about the QRF.⁶⁶6 We use a slightly different language than the QRF literature. What we call different QRFs below are referred to as different orientations of the same QRF in the above cited works. This is why we call $G$ the frame transformation group.

A common approach to obtain gauge-invariant operators is to write down combinations of the basic operators $\hat{X}^{a}_{ij},\hat{\Pi}^{a}_{ij}$ that are invariant under gauge transformations, such as $\Tr\hat{X}^{a}\hat{X}^{a}$. An alternative approach is to define relational operators, as we now describe.

Consider a gauge, such as the choice above where the matrix $X^{d}$ is diagonal and ordered. Then, working in the basis (2.8), define a class of (generalised) projectors

$$\hat{P}_{\mathsf{U}}\equiv\ket{\mathsf{U}}\bra{\mathsf{U}}\,.$$

(2.16)

We may now define the (matrix-valued) reference frame operator

$$\hat{\mathsf{U}}\equiv\int_{U(N)}\differential{\mathsf{U}}\mathsf{U}\,\hat{P}_{\mathsf{U}}\,,$$

(2.17)

and then consider

$$\hat{X}_{\mathds{1},ij}^{a}\equiv\left(\hat{\mathsf{U}}^{\dagger}\hat{X}^{a}\hat{\mathsf{U}}\right)_{ij}.$$

(2.18)

To understand what these operators do we may act on a general basis state (2.8) of $\mathcal{H}_{\mathrm{ext}}$. From the definitions above, and using $\hat{X}^{a}\ket{\mathsf{U}}\ket{X_{\mathrm{gf}}}=\mathsf{U}X_{\mathrm{gf}}^{a}\mathsf{U}^{\dagger}\ket{\mathsf{U}}\ket{X_{\mathrm{gf}}}$, one has

$$\hat{X}_{\mathds{1},ij}^{a}\ket{\mathsf{U}}\ket{X_{\mathrm{gf}}}=X_{\mathrm{gf},ij}^{a}\ket{\mathsf{U}}\ket{X_{\mathrm{gf}}}\,.$$

(2.19)

That is to say, the relational operators pick out the values of the matrices on a particular gauge-slice. Gauge invariance is the fact that for $\mathsf{U}^{\prime}\in U(N)$,

$$\hat{\pi}(\mathsf{U}^{\prime})\hat{X}_{\mathds{1},ij}^{a}=\hat{X}_{\mathds{1},ij}^{a}\hat{\pi}(\mathsf{U}^{\prime})\,.$$

(2.20)

This may be verified by acting on basis vectors, similarly to (2.19).

The steps above demonstrate that the matrix elements of a fully gauge-fixed operator are gauge invariant data. We may think of this fact relationally: the gauge-fixed matrix elements are values taken by the operator given that $X^{d}$ is diagonal and ordered. Conditioning on the gauge-fixing provides an internal reference frame or, more pictorially, a series of ‘rods’ given by the diagonalised $X^{d}$.

The $\hat{X}_{\mathds{1},ij}^{a}$ operators, however, are not truly well-defined due to the possibility of coincident eigenvalues and the attendant residual gauge symmetries. If $X^{d}$ has coincident eigenvalues, then there are non-trivial unitaries $\mathsf{U}_{d}$ that commute with $\lambda$, so that $\lambda=\mathsf{U}_{d}\lambda\mathsf{U}_{d}^{\dagger}$. Defining $\widetilde{X}^{a}\equiv\mathsf{U}_{d}X^{a}\mathsf{U}_{d}^{{}^{\dagger}}$ for $a=1\dots d-1$, the states $\ket*{X^{1\dots d-1},\lambda}$ and $\ket*{\widetilde{X}^{1\dots d-1},\lambda}$ lie on the same gauge orbit; this is a residual gauge symmetry. The value of, say, $\hat{X}^{1}_{\mathds{1},ij}$ is therefore ambiguous on this gauge orbit, and the operator is not well-defined. It is possible that there is a more sophisticated gauge that does not leave these residual gauge symmetries; if so, the corresponding relational observables will be well-defined.⁷⁷7 The main difficulty is that an enlarged gauge orbit at certain points in configuration space implies a reduced number of independent relational operators. The number of such operators therefore has to vary along the gauge slice. This issue doesn’t arise when the gauge-invariant data is parametrised via traces. We should also note that there is no difficulty when all gauge orbits are enlarged. In particular, there is always a $U(1)^{N}$ worth of residual gauge symmetry, because if $\Phi$ is a diagonal matrix of phases, $\mathsf{U}\lambda\mathsf{U}^{\dagger}=\mathsf{U}\Phi\lambda\Phi^{\dagger}\mathsf{U}^{\dagger}$. But since this $U(1)^{N}$ exists for every configuration, it can be dealt with simply by restricting $\mathsf{U}$ in $\ket{\mathsf{U}}$ to lie in $U(N)/U(1)^{N}$. We ignore this subtlety, assuming that our wavefunctions do not have any support on orbits with coincident eigenvalues. Further discussion of this sort of issue can be found in [58].

Let us now construct relational momentum operators, which will again be well-defined only in the absence of residual gauge symmetry. In language similar to (2.18), they are

$$\hat{\Pi}_{\mathds{1},ij}^{a}\equiv\int_{U(N)}\differential{\mathsf{U}}\left(\mathsf{U}^{\dagger}\hat{P}_{\mathsf{U}}\hat{\Pi}^{a}\hat{P}_{\mathsf{U}}\mathsf{U}\right)_{ij}\,.$$

(2.21)

It is important that $\hat{\Pi}^{a}$ is flanked on both sides by the same projector $\hat{P}_{\mathsf{U}}$. This ensures that the components of the momentum along the gauge orbits are projected out. Explicitly, on a basis state of $\mathcal{H}_{\mathrm{ext}}$:

$$e^{i\Tr(A^{a}\hat{\Pi}_{\mathds{1}}^{a})}\ket*{\mathsf{U}X_{\mathrm{gf}}^{a}\mathsf{U}^{\dagger}}=\hat{P}_{\mathsf{U}}\ket*{\mathsf{U}(X_{\mathrm{gf}}^{a}+A^{a})\mathsf{U}^{\dagger}}=\ket*{\mathsf{U}}\ket*{X_{\mathrm{gf}}^{a}+A_{\mathrm{gf}}^{a}}.$$

(2.22)

The last equality defines $A_{\mathrm{gf}}^{a}$; it is the part of $A^{a}$ that doesn’t change the gauge. In particular, all the off-diagonal terms of $A_{\mathrm{gf}}^{d}$ are zero. Equivalently, the full momentum operator $\hat{\Pi}=-i\partial_{X}$ defines a vector field on configuration space, and the relational operator is the projection of the vector field onto the gauge slices.

The ordering of eigenvalues leads to another subtlety for the relational momentum operator. Consider the translation $\lambda_{1}\to\lambda_{1}+a$, generated by $\exp(ia\hat{\Pi}_{\mathds{1},11}^{d})$. If $a>\lambda_{2}-\lambda_{1}$ then the first eigenvalue is moved beyond the second eigenvalue, which is not allowed. This shift should instead be thought of as a shift of the first eigenvalue by $\lambda_{2}-\lambda_{1}$ and of the second eigenvalue by $a-\lambda_{2}$, which preserves the ordering. For our purposes, as shown in [66], one may treat the $\hat{\Pi}$ as standard translation operators, with the ordering imposed by the wavefunction. This is the perspective implicit in our factorisation map.

The gauge-invariant operators (2.18) and (2.21) allow the construction of a gauge-invariant algebra of observables associated to the sub-block

$$\widetilde{\mathcal{A}}_{\mathds{1}}=\left\{\hat{X}^{a}_{\mathds{1},ij},\hat{\Pi}^{a}_{\mathds{1},ij}\middle|i,j=1\dots M\right\}^{\prime\prime}\,,$$

(2.23)

where the ^′′ involves taking all sums and products. The algebra is not entirely well-defined due to the issues of residual gauge invariance described above. The tilde denotes that we have one more step left to go. We refer to this (or rather, the algebra $\mathcal{A}_{\mathds{1}}$ to be introduced shortly) as the algebra ‘in the QRF $X^{d}$.’

To the algebra $\widetilde{\mathcal{A}}_{\mathds{1}}$ we can associate a factorisation map, the embedding $\mathcal{H}_{\mathrm{phys}}\hookrightarrow\mathcal{H}_{\mathrm{in}}\otimes\mathcal{H}_{\mathrm{out}}$ given by $J_{\mathds{1}}:\ket{\psi}\mapsto\ket{\psi_{\mathds{1}}}$ of (2.10). Recall that the tensor factors $\mathcal{H}_{\mathrm{in/out}}$ were defined in (2.15), they are respectively the top-left block and the remaining matrix elements. The operators in the algebra $\widetilde{\mathcal{A}}_{\mathds{1}}$ are conjugated by $J_{\mathds{1}}$ under this map. The image of $\widetilde{\mathcal{A}}_{\mathds{1}}$ acts on $\mathcal{H}_{\mathrm{in}}$ while the image of its commutant $\widetilde{\mathcal{A}}_{\mathds{1}}^{\;\prime}$ acts on $\mathcal{H}_{\mathrm{out}}$. The conjugation is an important technicality and arises because $\ket{\psi_{\mathds{1}}}$ is a gauge-fixed state in $\mathcal{H}_{\mathrm{ext}}$.

We can now define the algebra $\mathcal{A}_{\mathds{1}}$, without the tilde. Since we are interested in partitioning the emergent space into two parts, specifying the whole QRF is too fine-grained: we don’t need to fix all the rods, we just need to know which region is covered by the first $M$ of them. This means that we want to construct operators that are ‘less relational’ than those in $\widetilde{\mathcal{A}}_{\mathds{1}}$. The rods inside the region are moved around by $U(M)$, while those outside are moved around by $U(N-M)$. These are transformations we do not wish to keep track of as they do not affect the partition; we want our QRFs to be incomplete [56]. Therefore, we average the projector $\hat{P}_{\mathsf{U}}$ over $U(M)\times U(N-M)$. The averaged projectors are labelled by $V\in U(N)/[U(M)\times U(N-M)]$:

$$\hat{P}_{V}\equiv\int\differential{U}\differential{\widetilde{U}}\hat{P}_{VU\widetilde{U}}\,.$$

(2.24)

And the averaged reference frame operator is now

$$\hat{V}\equiv\int\differential{V}V\,\hat{P}_{V}=\int\differential{\mathsf{U}}V\,\hat{P}_{\mathsf{U}}.$$

(2.25)

To define this operator, we must define a representative of $U(N)/[U(M)\times U(N-M)]$ in $U(N)$: we may choose it to be the space obtained by the action of the exponential map on the subspace $\mathfrak{u}(N)/[\mathfrak{u}(M)\oplus\mathfrak{u}(N-M)]$ of the Lie algebra $\mathfrak{u}(N)$.

In complete analogy to (2.18) and (2.21) above we may introduce the relational operators

$$\hat{X}_{M,ij}^{a}\equiv\left(\hat{V}^{\dagger}\hat{X}^{a}\hat{V}\right)_{ij}\,,\qquad\hat{\Pi}_{M,ij}^{a}\equiv\int_{U(N)}\differential{V}\left(V^{\dagger}\hat{P}_{V}\hat{\Pi}^{a}\hat{P}_{V}V\right)_{ij}\,.$$

(2.26)

The action of $\hat{X}_{M,ij}^{a}$ on vectors in the extended Hilbert space is then, cf. (2.19) above,

$$\hat{X}_{M,ij}^{a}\ket{\mathsf{U}}\ket{X_{\mathrm{gf}}}=(U{\widetilde{U}}X_{\mathrm{gf}}^{a}{\widetilde{U}}^{\dagger}U^{\dagger})_{ij}\ket{\mathsf{U}}\ket{X_{\mathrm{gf}}}\,.$$

(2.27)

The operators are ‘less relational,’ since they relate the matrix element not to a specific gauge (or system of rods) but to a $U(M)\times U(N-M)$ equivalence class of gauges.

The operators in (2.27) remember the location in $U(M)\times U(N-M)$. This means that, unlike for $\widetilde{\mathcal{A}}_{\mathds{1}}$ in (2.23), we must trace the matrices over the $M\times M$ sub-block to obtain a gauge-invariant algebra [56]. Thus we define

$$\mathcal{A}_{\mathds{1}}=\left\{\Tr F\left(\Theta\hat{X}_{M}^{a}\Theta,\Theta\hat{\Pi}_{M}^{b}\Theta\right)\right\},$$

(2.28)

for functions $F$ such that $F(U\hat{X}^{a}U^{\dagger},U\hat{\Pi}^{b}U^{\dagger})=UF(\hat{X}^{a},\hat{\Pi}^{b})U^{\dagger}$, such as polynomials. This is the same as the sub-block algebra written down in [42], at leading order.

The $\mathcal{A}_{\mathds{1}}$ algebra commutes with the generators of $U(M)$ gauge transformations — this is the algebraic counterpart of the central observation in [36] that the reduced density matrix carries $U(M)$ edge modes. This fact also means that it is natural to represent $\mathcal{A}_{\mathds{1}}$ on a Hilbert space annihilated by the $U(M)$ generators. Thus, the factorisation map we associate to this algebra is (2.12) with $G=\{\mathds{1}\}$:

$$J_{M}:\ket{\psi}\mapsto\int dUd{\widetilde{U}}\ket*{U{\widetilde{U}}}\ket{\psi_{\mathrm{gf}}}\,.$$

(2.29)

The crucial point is that $U(M)$ acts separately within $\mathcal{H}_{\mathrm{in}}$ and $\mathcal{H}_{\mathrm{out}}$ and does not mix degrees of freedom within these two spaces. Then, analogously to before, this embedding leads to a representation of the image under $J_{M}$ of $\mathcal{A}_{\mathds{1}}$ on $\mathcal{H}_{\mathrm{in}}$ and of the image of $\mathcal{A}_{\mathds{1}}^{\prime}$ on $\mathcal{H}_{\mathrm{out}}$. However, there will be additional entanglement because the action of $U(M)$ in the two factors is correlated. This is precisely the edge mode entanglement computed in [36, 37].

There was nothing preferred about the particular set of gauge-fixed states (2.10). We could have instead considered, for example, the states

$$\ket{\psi_{\mathsf{U}}}\equiv\ket{\mathsf{U}}\ket{\psi_{\mathrm{gf}}}\,,$$

(2.30)

for some fixed $\mathsf{U}\in U(N)$. This amounts to a different gauge-slice where $X^{d}=\mathsf{U}\lambda\mathsf{U}^{\dagger}$ is no longer diagonal. The construction of the previous subsection can be repeated, leading to a distinct algebra $\mathcal{A}_{\mathsf{U}}$ of an $M\times M$ sub-block of operators in the QRF $\mathsf{U}^{\dagger}X^{d}\mathsf{U}$. Of course, changing both the QRF and the sub-block by a $\mathsf{U}$ transformation will cancel each other, leading to the same subsystem. So, we change the QRF but not the sub-block to define $\mathcal{A}_{\mathsf{U}}$.

The fact that every QRF leads to a respectable algebra and subsystem — democratically so, from the perspective of the symmetries of the microscopic theory — means that choosing which frames to consider depends on the physical question one wishes to address. We will now make the argument, already outlined in the introduction and previously in this section, that it is natural to consider a coarse-grained average over QRFs.

In the introduction we recalled that, in semiclassical states supported on noncommutative geometries, a matrix sub-block corresponds to a subregion in the geometry. Reference frame transformations, i.e. unitary gauge transformations of the matrices, correspond to volume-preserving diffeomorphisms that move the subregion around. Our problem is thus seen to be similar to the difficulties encountered in defining gauge-invariant subregions in theories of gravity. It is instructive, then, to make an analogy to subregion-subregion duality in the AdS/CFT correspondence, see e.g. [67] and references therein. The gauge-invariant input that is fixed in that case is a boundary subregion $B$. The corresponding bulk subregion $b$ is then determined dynamically as the region bounded by the minimal extremal surface $X$ homologous to $B$, such that $\partial X=\partial B$.

In our setup, where there is no AdS/CFT-like asymptotic boundary, the natural gauge-invariant data associated to a subregion is its volume. This is determined by the size $M$ of the matrix sub-block — we recall the proof of this statement in (3.7) below.⁸⁸8At subleading semiclassical order, fixing the volume and fixing $M$ will give rise to different subsystems. The analogous procedure to AdS/CFT is then to fix the volume and let the theory determine the actual subregion. To this end it is natural to consider all subregions with a given fixed volume. As we have explained above, this amounts to considering a fixed size $M$ sub-block in all QRFs. Thus we are led to think about integrating over all QRFs. This will lead to an interesting sum over physically inequivalent subsystems, as the reduced state transforms non-linearly under changing QRFs [57]. Before we can set up the computation, however, there is one more issue to deal with: integrating over all QRFs introduces too much uncertainty.

We will show below that, in semiclassical models, the $U(N)$ transformations become volume preserving ‘diffeomorphisms’ — in quotes because the corresponding maps are typically not continuous, let alone differentiable. We illustrated this fact in Fig. 2 in the introduction. In the relational language, we can say that most transformations in $U(N)$ map a continuous system of rods into a highly discontinuous one. Suppose, for example, that the QRF $X^{d}$ is associated to a coordinate that increases along a certain direction of the emergent space. The QRF $\mathsf{U}^{\dagger}X^{d}\mathsf{U}$ is then instead organized in terms of the direction of increasing $\mathsf{U}^{\dagger}X^{d}\mathsf{U}$. This latter ‘direction’ will generically not have a simple geometric interpretation, as illustrated in Fig. 4.

The proliferation of non-geometric QRFs will have an undesirable technical consequence below: the many discontinuous QRFs dominate the average entanglement and, in particular, overwhelm the contribution of the minimal entropy surface. That is, they obscure the connection between entanglement and geometry.

In addition to the technical consequences of a sum over all QRFs, outlined above, one might expect that extracting sensible low-energy, semiclassical quantities should involve a coarse-graining of the microscopic QRF data. For instance, sufficiently short distance wiggles of the QRF should not change the effective state that a semiclassical observer (measuring with the help of the system of rods whose positions are also fluctuating) sees. This may suggest that the mathematical formalism of non-ideal QRFs, see e.g. [52, 68, 56], could be used to develop a consistent framework for a coarse-grained integral over QRFs. In this work we pursue a more low-tech solution, by considering a simple model for coarse-graining. Specifically, we restrict the frame transformations to a subgroup $G\subseteq U(N)$. This restriction clearly reduces the number of QRFs that will appear in the average. Within the extended Hilbert space formalism developed in §2.2, $G$ is the group introduced in (2.11) above.

One may ask whether there is an algebraic description of the average over frames. It is natural to consider a direct sum over the algebras for each of the possible subsystems

$$\mathcal{A}_{\mathrm{tot}}=\Big{\{}|g\rangle\langle g|\otimes a_{g}\;\Big{|}\;a_{g}\in\mathcal{A}_{g}\Big{\}}\,.$$

(2.31)

The $|g\rangle\langle g|$ factor is necessary to label the different elements. In §2.7 below we will argue that (2.31) can’t be quite right because it excludes replica symmetry-breaking effects in the entanglement. More physically, the observables associated to distinct — but in general overlapping — subregions should not be fully independent. It would be incredibly interesting to define the frame-averaged partition algebraically. Doing so could ultimately put our extended Hilbert space computations below on a firmer footing or perhaps refine the prescription. We leave this to future work, however.

In this subsection, we review how the matrix edge modes arise. Recall that we are interested in algebras invariant under $U(M)\times U(N-M)$, because these are the operators that depend only on the partition and not the entire QRF. This was implemented in our factorisation map by the explicit integral over this subgroup in (2.12).⁹⁹9 Mathematically, this is equivalent to making the reduced state invariant under the subgroup of the gauge symmetry that leaves the subsystem unchanged via a depolarisation channel, as in [57]. It is important to emphasise here that the two integrals in the factorisation map (2.12) have distinct raisons d’être: the $U(M)$ integral ensures that the subregion is not over-specified by a choice of interior coordinates while the $G$ integral implements a coarse-grained averaging over subregions.

In this subsection we describe the effects of the $U(M)$ integral in (2.12). The $U(M)\times U(N-M)$ subgroup of the gauge symmetry acts on the different blocks (2.14) of the matrices as, with $U\in U(M)$ and $\widetilde{U}\in U(N-M)$,

$$X^{a}_{\Sigma\Sigma}\to UX^{a}_{\Sigma\Sigma}U^{\dagger},\qquad X^{a}_{\overline{\Sigma}\overline{\Sigma}}\to\widetilde{U}X^{a}_{\overline{\Sigma}\overline{\Sigma}}\widetilde{U}^{\dagger},\qquad X^{a}_{\Sigma\overline{\Sigma}}\to UX^{a}_{\Sigma\overline{\Sigma}}\widetilde{U}^{\dagger}\,.$$

(2.32)

The transformation of each block in (2.32) involves only itself. This action therefore doesn’t mix the three Hilbert spaces in (2.15). The $U(N-M)$ subgroup acts entirely on $\mathcal{H}_{\mathrm{out}}$, and hence has no consequence for the reduced density matrix on $\mathcal{H}_{\mathrm{in}}$. The $U(M)$ subgroup, however, acts on both $\mathcal{H}_{\Sigma\Sigma}$ and $\mathcal{H}_{\Sigma\overline{\Sigma}}$. This was illustrated in Fig. 1. Thus, when we average over $U(M)$, we are adding back in entanglement that was destroyed by the gauge-fixing [47], while taking care not to disturb the physical region. This is the edge mode entanglement computed in [44, 36], that we will now recover.

The $U(M)$ generators in (2.4) can be written as a sum

$$\hat{G}_{\Sigma\Sigma}=2i\sum_{a=1}^{d}:[\hat{X}^{a}_{\Sigma\Sigma},\hat{\Pi}^{a}_{\Sigma\Sigma}]:+2i\sum_{a=1}^{d}:[\hat{X}^{a}_{\Sigma\overline{\Sigma}},\hat{\Pi}^{a}_{\overline{\Sigma}\Sigma}]:\,\equiv\hat{Q}_{\Sigma}+\hat{Q}_{\overline{\Sigma}}\,,$$

(2.33)

where the first term acts on $\mathcal{H}_{\Sigma\Sigma}$ and the second acts on $\mathcal{H}_{\Sigma\overline{\Sigma}}$. Furthermore, these two terms commute as quantum operators because they act on orthogonal parts of the extended Hilbert space. Given that $\hat{Q}_{\Sigma}$ and $\hat{G}_{\Sigma\Sigma}$ manifestly generate $U(M)$ actions, so does $\hat{Q}_{\overline{\Sigma}}$. That is, each of $\hat{Q}_{\Sigma}$ and $\hat{Q}_{\overline{\Sigma}}$ generate a $U(M)$ group that we may call $U(M)_{\Sigma}$ and $U(M)_{\overline{\Sigma}}$, respectively. The gauge symmetry is the ‘diagonal’ part of this $U(M)_{\Sigma}\times U(M)_{\overline{\Sigma}}$.

Any physical state in $\mathcal{H}_{\mathrm{phys}}$ can be decomposed into branches labelled by the irreps $\upmu$ of $U(M)_{\Sigma}$. The integral over $U(M)$ in the factorisation map (2.12) ensures that the corresponding state $\ket{\psi}\in\mathcal{H}_{\mathrm{ext}}$ is invariant under the diagonal $U(M)$ generated by $\hat{G}_{\Sigma\Sigma}$, so that $[\hat{Q}_{\Sigma}+\hat{Q}_{\overline{\Sigma}}]\ket{\psi}=0$. This condition implies that each branch also carries the same irrep $\upmu$ of $U(M)_{\overline{\Sigma}}$. Because $\hat{Q}_{\Sigma}$ and $\hat{Q}_{\overline{\Sigma}}$ act on separate degrees of freedom, the state is forced to take the form [46, 59]

$$\ket{\psi}=\sum_{\upmu\in\operatorname{irr}U(M)}\sqrt{p_{\upmu}}\ket{\psi_{\upmu}}\otimes\frac{1}{\sqrt{d_{\upmu}}}\sum_{m=1}^{d_{\upmu}}\ket{m}_{\Sigma\,\upmu}\otimes\ket{\overline{m}}_{\overline{\Sigma}\,\upmu}\,,$$

(2.34)

where $p_{\upmu}$ and $d_{\upmu}$ are the probability of finding the irrep $\upmu$ and the dimension of the irrep, respectively. The states $\ket{m}_{\Sigma\,\upmu},\ket{\overline{m}}_{\overline{\Sigma}\,\upmu}$ are bases for the $\upmu$ irrep of $U(M)_{\Sigma},U(M)_{\overline{\Sigma}}$, such that the superposition above is a singlet under $U(M)_{\Sigma}\times U(M)_{\overline{\Sigma}}$.¹⁰¹⁰10 As an example, the (unnormalised) singlet in the spin-$j$ irrep of $SU(2)$ is $\sum_{m=-j}^{j}(-1)^{j}\ket{j,m}\ket{j,-m}$. So, in this case, $\ket{\overline{j,m}}=(-1)^{j}\ket{j,-m}$. The states $\ket{\psi_{\upmu}}$ are singlets under $U(M)_{\Sigma}$ and $U(M)_{\overline{\Sigma}}$, one for each irrep. These singlet states live in the full Hilbert space and in particular also contain entanglement between $\mathcal{H}_{\mathrm{in}}$ and $\mathcal{H}_{\mathrm{out}}$. The decomposition (2.34) implies that the reduced density matrix is

$$\hat{\rho}_{\mathrm{in}}=\bigoplus_{\upmu}p_{\upmu}\left[\hat{\rho}_{\upmu,\mathrm{in}}\otimes\frac{\mathds{1}_{\upmu}}{d_{\upmu}}\right]\,,$$

(2.35)

with $\mathds{1}_{\upmu}=\sum_{m}\ket{m}_{\Sigma\,\upmu}\!\bra{m}_{\Sigma\,\upmu}$ and $\rho_{\upmu,\mathrm{in}}$ the reduced density matrix obtained from $\ket{\psi_{\upmu}}$. The entanglement entropy is therefore

$$S_{\mathrm{E}}=\sum_{\upmu}p_{\upmu}\left\{\log\frac{d_{\upmu}}{p_{\upmu}}+S_{\mathrm{vN}}\left(\rho_{\upmu,\mathrm{in}}\right)\right\}\,.$$

(2.36)

The above treatment of the $U(M)$ part of the matrix gauge symmetry — in particular equations (2.34) and (2.36) — is the same as in lattice gauge theories, e.g. [46]. Equations (2.34) and (2.36) have also appeared in interpretations of holography as a quantum error-correcting code [69], where the $\sum_{\upmu}p_{\upmu}\log d_{\upmu}$ term was identified as the source of the Ryu-Takayanagi area term. In matrix quantum mechanics too, an area law entanglement was obtained from the same term in [44, 36, 37]. The essential points of that computation are as follows. Firstly, semiclassical matrix states are strongly localised on the gauge orbit of a particular classical configuration, which we write as $X_{\mathrm{gf}}^{a}=X^{a}_{\mathrm{cl}}$. Secondly, in such states the reduced density matrix transforms under a single dominant irrep $\upmu_{\star}$. The dimension $d_{\upmu_{\star}}$ of this irrep can be computed from $X_{\mathrm{cl}}^{a}$, and the entanglement entropy is

$$S_{\mathrm{E}}\approx\log d_{\upmu_{\star}}\,.$$

(2.37)

The area law is then obtained by evaluating $d_{\upmu_{\star}}$, as we will recall below. The final term in (2.36) is the von Neumann entropy of the fluctuating fields in the emergent geometry, and is subleading in the semiclassical matrix limit.

The result (2.37) gives the gauge-theoretic entanglement entropy of the state (2.34). This is also the entanglement entropy of the state $\ket{\psi_{G}}$ in (2.12) when $G=\left\{\mathds{1}_{N}\right\}$. We must now incorporate the integral over a non-trivial $G$.

In this subsection we will average over QRFs by taking $G$ in (2.12) to be a non-trivial group. We will need to separate out the $G$ transformations that are also in $U(M)\times U(N-M)$, and those that are not. To that end, we can define two new subgroups and a quotient set

$$H\equiv G\cap U(M),\qquad H^{\prime}\equiv G\cap U(N-M),\qquad\mathsf{F}\equiv(H\times H^{\prime})\backslash G.$$

(2.38)

It is easy to check that $H$ and $H^{\prime}$ are subgroups of $G$. $\mathsf{F}$ is a left-quotient, which identifies $g\sim hh^{\prime}g$ for $h\in H,h^{\prime}\in H^{\prime}$. We will use this same notation for the left-quotient throughout. $\mathsf{F}$ is known as a generalised flag manifold; physically, it indexes the QRFs that lead to distinct subregions. Denote by $U,\widetilde{U}$ and $V$ elements of $U(M),U(N-M)$ and $\mathsf{F}$, respectively.

As we have noted, we will be focused on cases where the gauge-fixed wavefunction is sharply peaked on a certain classical configuration $X_{\mathrm{cl}}^{a}$. To leading semiclassical order we have $\innerproduct*{X_{\mathrm{gf}}^{a}}{\psi_{\mathrm{gf}}}\propto\delta(X^{a}_{\mathrm{gf}}-X^{a}_{\mathrm{cl}})$, so that the extended Hilbert space state (2.12) can be approximated by the gauge orbit

$$\ket{\psi_{G}}\approx\int\differential{U}\differential{\widetilde{U}}\int_{G}\differential{g}\ket*{U\widetilde{U}gX_{\mathrm{cl}}^{a}g^{\dagger}\widetilde{U}^{\dagger}U^{\dagger}}\equiv\int_{\mathsf{F}}\differential{V}\ket{\psi_{V}}.$$

(2.39)

We are not keeping track of the normalisation of the state at this point. In the final step we have defined the partially gauge-fixed state

$$\ket{\psi_{V}}=\Delta(V)\int\differential{U}\differential{\widetilde{U}}\ket*{U\widetilde{U}VX_{\mathrm{cl}}^{a}V^{\dagger}\widetilde{U}^{\dagger}U^{\dagger}}\,.$$

(2.40)

We have done the integral over $H\times H^{\prime}\subset G$ trivially by absorbing explicit factors of $H,H^{\prime}$ in the integrand into $U,\widetilde{U}$. The measure factor $\Delta(V)$ is associated to the quotient (2.38). The state $\ket{\psi_{V}}$ in (2.40) is invariant under, $U(M)$, $H$ and $H^{\prime}$, but not $\mathsf{F}$. The most important of these invariances will be $U(M)$, as this connects to the earlier discussion in §2.5.

The analysis in §2.5 above is valid for each $\ket{\psi_{V}}$ separately. In particular, as $\ket{\psi_{V}}$ is $U(M)$ invariant it can be decomposed into representations of $U(M)_{\Sigma}$ as in (2.34). We will argue in §5 that $\ket{\psi_{V}}$ is dominated by a single irrep $\upmu_{V}$ in two instances that together cover all cases of interest: $(i)$ generic $V$ and $(ii)$ $V$ corresponding to weakly curved and connected subregions. Thus the decomposition (2.34) becomes

$$\ket{\psi_{V}}\approx\ket{\psi_{\upmu_{V}}}\otimes\frac{1}{\sqrt{d_{\upmu_{V}}}}\sum_{m=1}^{d_{\upmu_{V}}}\ket{m}_{\Sigma\,\upmu_{V}}\ket{\overline{m}}_{\overline{\Sigma}\,\upmu_{V}}.$$

(2.41)

Approximating the state by a single irrep in this way is stronger than making the approximation in the entanglement entropy, as we did in (2.37). We will discuss the effects of the (here neglected) variance over irreps at the end of this subsection.

The reduced density matrix of $\ket{\psi_{G}}$ in $\mathcal{H}_{\mathrm{ext}}$ is, from (2.39),

$$\hat{\rho}_{\mathrm{in}}=\int\differential{V}\differential{V^{\prime}}\,\tr_{\mathrm{out}}\Big{[}\ket{\psi_{V}}\bra{\psi_{V^{\prime}}}\Big{]}.$$

(2.42)

In Appendix A we argue that we can approximate the integrand by something proportional to $\delta(V-V^{\prime})$ at leading semiclassical order. This is a non-trivial approximation with important physics content, as we discuss in §2.7 below. It means that the density matrix has no mixing between different subregions and hence two QRFs that lead to distinct subregions can be perfectly distinguished from the outside alone. With this approximation and using (2.41) in (2.42), we obtain the density matrix

$$\hat{\rho}_{\mathrm{in}}\approx{\mathcal{N}}\oplus\hskip-11.00008pt\displaystyle\int\differential{V}\,\left[\hat{\rho}_{\upmu_{V},\mathrm{in}}\otimes\frac{\mathds{1}_{\upmu_{V}}}{d_{\upmu_{V}}}\right]\,.$$

(2.43)

The (unimportant) normalisation ${\mathcal{N}}$ will be determined shortly. From (2.43):

$$\tr\hat{\rho}_{\mathrm{in}}^{n}\approx\delta(0)\,{\mathcal{N}}^{n}\int\differential{V}\,d_{\upmu_{V}}^{1-n}\tr\hat{\rho}_{\upmu_{V},\mathrm{in}}^{n}\,.$$

(2.44)

The prefactor of $\delta(0)$ here is not raised to the $n$th power and therefore cannot be absorbed into the normalisation of $\hat{\rho}_{\mathrm{in}}$; it leads to a $\log\delta(0)$ contribution to the entropy. In the following paragraph we elaborate on the meaning of this term.

The density matrix in (2.43) can also be written as $\hat{\rho}_{\mathrm{in}}\approx{\mathcal{N}}\int\differential{V}\,[\hat{\rho}_{V,\text{in}}\otimes\ket{V}\bra{V}]$, where $\ket{V}$ is just the position vector for the matrix $V$. The orthogonality of the terms with different $V$ is now explicit from $\bra{V}\ket{V^{\prime}}=\delta(V-V^{\prime})$. In computing $\tr\hat{\rho}_{\mathrm{in}}^{n}$ one then obtains the factor of $\delta(0)=\bra{V}\ket{V}$ in (2.44). This fact also determines the normalisation ${\mathcal{N}}=1/\delta(0)$. These delta functions will be smeared out by subleading corrections in which the $\ket{V}$ in the average over frames are replaced by states with an uncertainty $\Delta V$ in $V$. The prefactor $\delta(0)\sim\frac{1}{\Delta V}$. We will estimate $\Delta V$ in §5. The appearance of such prefactors is generic in continuum limits of probability distributions, as discussed recently in e.g. [70].¹¹¹¹11Consider a set of probabilities $p_{i}$ with $i\in\mathbb{Z}$. Let $x=\Delta x\,i$ and take the continuum limit $\Delta x\to 0$. In this limit $p_{i}\to\Delta x\,p(x)$ and the entropy $$s=-\sum_{i}p_{i}\log p_{i}\to-\int dx\,p(x)\log[\Delta x\,p(x)]=\widetilde{s}-\log\Delta x\,.$$ (2.45) Here $\widetilde{s}$ is the naïve continuum entropy, that can be negative, and $-\log\Delta x$ corresponds to the offset $\log\delta(0)$ in the text. Similarly in correspondence to the text: $\sum_{i}p_{i}^{n}\to\frac{1}{\Delta x}{\mathcal{N}}^{n}\int dx\,p(x)^{n}$, with ${\mathcal{N}}=\Delta x$.