but this is a contradiction to the statement that $U_{A}$ attains the minima,
and so one obtains that
$\min_{U_{A}}M_{\text{dist}}(U_{A}\psi_{A}U_{A}^{{\dagger}})\leq
M_{\text{dist}}^{(NL)}(\psi_{AB})$. This result combined with 3 concludes the
proof.

##  Appendix C Stabilizer relative entropies

###  C.1 Proof of Theorem 2

Let us start by proving the upper-bound to $M_{RS}^{(NL)}$. The proof is
similar to the proof done for the upper bound of the non-local trace distance
of magic. We choose $\sigma_{AB}=I/D$ where $D$ is the Schmidt rank of
$\rho_{A}$. Then the second term in the relative entropy expansion is simply

$-\mathrm{Tr}[\psi_{AB}\log\sigma_{AB}]=-\mathrm{Tr}[\sum_{i,j=1}^{D}\lambda_{i}\lambda_{j}|s_{i}\rangle\langle
s_{j}|\sum_{k=1}^{D}|s_{k}\rangle\langle s_{k}|\log(1/D)]=\log
D\mathrm{Tr}[\psi_{AB}]=\log D,$ (128)

which is the max entropy of the system $A$. Hence $M_{NL}(\rho_{AB})\leq
S_{max}(A)$ in this case. Concluding the proof for the upper bound. Shifting
our focus on the lower bound instead, let us note that for any $\rho$,
$M_{RS}(\rho)\geq F(\rho)$. This is because $\operatorname{STAB}_{0}\subset
FLAT$ as shown in the other note. Because $F(\rho)$ is isospectral under any
unitary conjugation, it must follow that $M_{RS}(U\rho U^{\dagger})\geq
F(U\rho U^{\dagger})=F(\rho)$. Therefore,

$\displaystyle\mathcal{F}_{R}(\rho_{A})\leq\min_{U_{A}}M_{RS}(U_{A}\rho_{A}U_{A}^{\dagger}).$
(129)

On the other hand, for any $\rho_{A}$, from monotonicity it follows that
$M_{RS}(\rho_{AB})\geq M_{RS}(\rho_{A})$ where
$\rho_{A}=\mathrm{Tr}_{B}[\rho_{AB}]$. Therefore, we must have

$\min_{U_{A}}M_{RS}(U_{A}\rho_{A}U_{A}^{\dagger})\leq\min_{U=V_{A}\otimes
V_{B}}M_{RS}(U\rho_{AB}U^{\dagger})\equiv M_{NL}(\rho_{AB}).$ (130)

We can see that this is true from a proof by contradiction. Suppose there
exists some $U_{A},U$ which attains the respective minima but has

$M_{RS}(U_{A}\rho_{A}U_{A}^{\dagger})>M_{RS}(U\rho_{AB}U^{\dagger}),$

then from monotonicity, we must have

$M_{RS}(U_{A}\rho_{A}U_{A}^{\dagger})>M_{RS}(U\rho_{AB}U^{\dagger})\geq
M_{RS}(\mathrm{Tr}_{B}[U\rho_{AB}U^{\dagger}])=M_{RS}(V_{A}\rho_{A}V_{A}^{\dagger})$

for some local unitary $V_{A}$ which yields a lower distance than $U_{A}$.
Since we assumed that $U_{A}$ attains the minimum, this violates our
assumption, concluding the proof for the lower bound.

###  C.2 Proof of Proposition 5

Consider a state $\psi_{AB}$ where $\psi_{A}$, $\psi_{B}$ are isospectral (up
to truncation of 0 eigenvalues) to that of a subsystem $\rho_{A},\rho_{B}$.
Suppose they are states where we have removed the local magic such that both
$\psi_{A},\psi_{B}$ are diagonal in the computational (or another complete
stabilizer basis). Again, this can be done by first rewriting the true state
in the Schmidt basis, which are orthonormal. Then we replace the Schmidt basis
by an orthonormal stabilizer basis. Since the mixture of stabilizer states is
in the convex hull of stabilizer states, each $\psi_{A},\psi_{B}$ must have
zero local magic. Note that there are also other basis choices such that the
basis state need not be a stabilizer, such states can also be in the convex
hull of the stabilizer group as long as they are not pure states. By
definition, $M^{(NL)}_{R}(\rho_{AB})\leq M_{R}(\psi_{AB})$ because we have
chosen a particular instance of the local unitary $U_{A}\otimes U_{B}$ on the
right hand side whereas the left hand side is minimized over all possible
instances. Since $\psi_{AB}$ is pure, then $S(\psi_{AB})=0$. While for the
other term, if $\sigma_{AB}$ is pure, then the relative entropy is either $0$
when $\sigma=\psi$ or $\infty$ for any other $\sigma$ that’s mixed. We pick a
stabilizer state $\sigma_{AB}=\lambda_{i}^{2}|s_{i}\rangle\langle s_{i}|_{AB}$
where $\lambda_{i}$ are the Schmidt coefficients of
$\psi_{AB}=\sum_{i}\lambda_{i}|s_{i}\rangle_{AB}$ where $|s_{i}\rangle$ are
the stabilizer basis we chose. For convenience, we can choose them to be the
computational basis which is orthonormal. Then for the second term, we write

$\displaystyle M_{R}(\psi_{AB})$
$\displaystyle=-\mathrm{Tr}[\psi_{AB}\log\sigma_{AB}]$ (131)
$\displaystyle=-\mathrm{Tr}[\sum_{ij}\lambda_{i}\lambda_{j}|s_{i}\rangle\langle
s_{j}|\sum_{k}\log(\lambda_{k}^{2})|s_{k}\rangle\langle s_{k}|]$ (132)
$\displaystyle=-\sum_{i,j,k}\delta_{ij}\delta_{ik}\lambda_{i}\lambda_{j}\log(\lambda_{k}^{2})$
(133) $\displaystyle=-\sum_{k}p_{k}\log p_{k}$ (134)

where we have set $\lambda_{k}^{2}=p_{k}$ because each Schmidt coefficient is
real. $\delta_{ij}$ are Kronecker deltas because we have chosen the basis
$\\{|s_{k}\rangle\\}$ to be orthonormal. Note that $\sum_{k}p_{k}=1$. In this
case, the second term is nothing but $S(A)=S(B)$ which is the von Neumann
entropy of a subsystem. Since we have chosen a particular stabilizer state
$\sigma_{AB}$, this serves as an upper bound of the relative entropy of magic.
Hence

$\displaystyle M^{(NL)}_{R}(\rho_{AB})\leq M^{(NL)}_{R}(\psi_{AB})\leq
S(A)=S(B).$ (135)

###  C.3 Proof of Theorem 4

Let $\rho_{AB}^{\epsilon}$ represent the state that minimizes the non-local
magic. Therefore
$M^{(NL,\epsilon)}_{RS}(\rho_{AB})=M_{(RS)}^{NL}(\rho^{\epsilon}_{AB})$.
Drawing from Theorem 2, we understand that:

$\begin{split}M^{(NL,\epsilon)}_{RS}(\rho_{AB})\geq&S_{max}(\rho^{\epsilon}_{A})-S(\rho^{\epsilon}_{A})\geq\min_{\|\chi-\rho_{A}\|<\epsilon}\left(S_{max}(\chi)-S(\chi)\right).\end{split}$
(136)

On the right-hand side, our goal is to identify a state $\chi$ within the
$\epsilon$-ball of $\rho_{A}$ that minimizes the difference between
$S_{max}(\chi)$ and $S(\chi)$. Interestingly, the state that minimizes this
difference also reduces $S_{max}(\chi)$ to its lowest value
$S_{max}^{\epsilon}$. To illustrate, denote $\chi_{A}^{\epsilon}$ as the state
that minimizes $S_{max}$ within the $\epsilon$-ball. Then consider increasing
$S_{max}$ by modifying one eigenvalue of $\chi_{A}^{\epsilon}$ from zero to
$\delta$. This adjustment results in an increase $\Delta
S_{max}=S_{max}^{-1}$, while the change in entropy is capped at $\Delta
S\leq\delta\left|\log\delta\right|$. Such a modification invariably elevates
the entropy gap, i.e. $\Delta(S_{max}-S)\geq
S_{max}^{-1}-\delta\left|\log\delta\right|>0$, since $\delta$ can be
arbitrarily small. To evaluate the von Neumann entropy of the state
$\chi_{A}^{\epsilon}$, as a modification from $\rho_{A}$ by dropping some
eigenvalues whose total contribution to the trace is smaller than $\epsilon$.
Let’s denote their contribution to the von Neumann entropy as $S_{\epsilon}$.
Then the entropy of the new state $\chi_{A}^{\epsilon}$ is given by
$S(\chi_{A}^{\epsilon})=\frac{S(\rho_{A})-S_{\epsilon}}{1-\epsilon}\leq\frac{S(\rho_{A})}{1-\epsilon}$.
Therefore, we get the following inequality:

$\begin{split}M^{(NL,\epsilon)}_{RS}(\rho_{AB})\geq&S_{max}(\chi_{A}^{\epsilon})-S(\chi_{A}^{\epsilon})\\\
\geq&S_{max}^{\epsilon}(\rho_{A})-(1-\epsilon)^{-1}S(\rho_{A}).\end{split}$
(137)

Regarding the upper bound, since $\chi_{A}^{\epsilon}$ minimizes the maximal
entropy, it satisfies the following condition:

$S_{max}^{\epsilon}(\rho_{A})=S_{max}(\chi_{A}^{\epsilon}).$ (138)

While this condition specifies the spectrum of $\chi_{A}^{\epsilon}$, we
retain the flexibility to select a purification $\chi_{AB}^{\epsilon}$,
ensuring its deviation from $\rho_{AB}$ remains within an $\epsilon$ bound.
Consequently, the process of minimizing the non-local magic leads us to the
following inequality:

$\begin{split}M^{(NL,\epsilon)}_{RS}(\rho_{AB})\leq&M_{RS}^{(NL)}(\chi_{AB}^{\epsilon})\\\
\leq&\operatorname{rank}{\chi_{AB}^{\epsilon}}=S_{max}(\chi_{A}^{\epsilon}).\end{split}$
(139)

where the second step is a result from Theorem 2.

##  Appendix D Estimate by Stabilizer-Renyi-entropy

###  D.1 Proof of Theorem 3

In this section, we provide an estimation of the second Stabilizer-Renyi-
entropy measure of the non-local magic. It is defined in stabrenyi  as the
second Renyi-entropy of a probability distribution,
$p_{a}=\frac{1}{d}|\langle\psi|P_{a}|\psi\rangle|^{2}$, over all the Pauli-
string basis $P_{a}$.

$\mathcal{M}_{2}(|\psi\rangle):=-\log(\sum_{a}p_{a}^{2})-\log{d}.$ (140)

Given the entanglement spectrum $\\{\lambda_{i}\\}$, we construct a state
$|\psi^{\prime}\rangle$ with small local magic,

$|\psi^{\prime}\rangle_{AB}=\sum_{i=0}^{r-1}\sqrt{\lambda_{i}}|s_{i}\rangle_{A}|s_{i}\rangle_{B}.$
(141)

where the rank $r$ is taken to be $2^{n}$ for integer $n$. The Pauli operators
on the Hilbert space $\mathcal{H}_{AB}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$
can be factorized as product of Pauli operators on $\mathcal{H}_{A}$ and
$\mathcal{H}_{B}$ respectively, $P^{ab}=P^{a}\otimes P^{b}$. We denote their
matrix elements as $P^{a,b}_{ij}:=\langle s_{i}|P^{a,b}|s_{j}\rangle$, and
compute the magic measure $\mathcal{M}_{2}$ as follows,

$\begin{split}&\mathcal{M}_{2}(|\psi^{\prime}\rangle)=-\log\left(\sum_{a=1}^{r^{2}}\sum_{b=1}^{r^{2}}\left|\sum_{i,j=0}^{r-1}\sqrt{\lambda_{i}}\sqrt{\lambda_{j}}P^{a}_{ij}P^{b}_{ij}\right|^{4}\right).\end{split}$
(142)

The result is complicated and depends on specific choice of the basis
$|s_{i}\rangle$’s. We simplify the analysis by assuming that the orthonormal
basis $|s_{i}\rangle$’s are common eigenstates of a stabilizer group
$\mathcal{S}=\\{S_{1},S_{2},\cdots,S_{n}\\}$. This condition allows us to
write the Pauli matrices $P^{a}_{ij}$ in computational basis. Substituting the
matrix representation of Pauli operators, we find that

$\begin{split}\mathcal{M}_{2}=&-\log\left(\sqrt{\lambda_{i_{1}}}\sqrt{\lambda_{i_{2}}}\sqrt{\lambda_{i_{3}}}\sqrt{\lambda_{i_{4}}}\sqrt{\lambda_{i_{5}}}\sqrt{\lambda_{i_{6}}}\sqrt{\lambda_{i_{7}}}\sqrt{\lambda_{i_{8}}}(\sum_{a}P^{a}_{i_{1}i_{2}}P^{a}_{i_{3}i_{4}}P^{a}_{i_{5}i_{6}}P^{a}_{i_{7}i_{8}})^{2}\right)\\\
=&-\log\left(\sum_{i_{1},i_{2},i_{3},i_{4}=0}^{r-1}\sqrt{\lambda_{i_{1}}\lambda_{i_{2}}\lambda_{i_{3}}\lambda_{i_{4}}\lambda_{i_{3}\wedge
i_{2}\wedge i_{1}}\lambda_{i_{4}\wedge i_{2}\wedge i_{1}}\lambda_{i_{1}\wedge
i_{3}\wedge i_{4}}\lambda_{i_{2}\wedge i_{3}\wedge i_{4}}}\right).\end{split}$
(143)

where $\wedge$ denotes the bitwise XOR operation. This expression depends on
order of eigenvalues and takes minimum when the eigenvalues are ordered,
$\lambda_{i}>\lambda_{j}$ for $i<j$. If we take all the eigenvalues to be the
same, then each term in the summation is equal to $\frac{1}{r^{4}}$. The
number of terms is $r^{4}$ since we are summing over four indices. The
argument is equal to 1 in this case. Therefore, the non-local Stabilizer Renyi
entropy vanishes when the spectrum is flat. 12 gives a comparison of the
direct SRE calculation against the estimation given by Eq. (143). As can be
observed in the plots, the approximation in Eq. (143) is correct up to
numerical imprecision.

(a) $\mathcal{M}_{2}$ computed from stabilizer Renyi entropy, compared to the
estimation in Eq. (143).

(b) Residual from computational and analytic calculations of
$\mathcal{M}_{2}$, accurate to one part in $10^{16}$.

Figure 12:

We can derive an upper bound for $\mathcal{M}_{2}$, by averaging over the
permutations of eigenvalues, this gives us the expression,

$\begin{split}\mathcal{M}_{2}\leq\overline{\mathcal{M}}_{2}=-\log\left(\sum_{i=0}^{r-1}\lambda_{i}^{4}+7\sum_{0\leq
i\neq j\leq r-1}\lambda_{i}^{2}\lambda_{j}^{2}+\frac{7}{r-3}\sum_{0\leq i\neq
j\neq k\neq l\leq r-1}\lambda_{i}\lambda_{j}\lambda_{k}\lambda_{l}+\right.\\\
\left.\frac{\sum_{0\leq i_{1}\neq i_{2}\neq\cdots\neq i_{8}\leq
r-1}\prod_{a=1}^{8}\sqrt{\lambda_{i_{a}}}}{(r-3)(r-5)(r-6)(r-7)}\right).\end{split}$
(144)

In Eq. (144), the sum inside the logarithm is taken over products of distinct
eigenvalues. Computing this sum explicitly, and expressing the result in terms
of different Renyi entropies $S_{\alpha}$, we obtain

$\begin{split}\overline{\mathcal{M}}_{2}=-\log&\left(7e^{-2S_{2}}-6e^{-3S_{4}}+7e^{-S_{0}}(1-6e^{-S_{2}}+8e^{-2S_{3}}+3e^{-2S_{2}})+\right.\\\
&\left.e^{-4S_{0}}(e^{4S_{1/2}}+105e^{-3S_{4}}-420e^{S_{1/2}}+\cdots)\right)\\\
=-\log&\left(7e^{-2S_{2}}-6e^{-3S_{4}}+e^{4S_{1/2}-4S_{0}}\right)+O(e^{-S_{1/2}}).\end{split}$
(145)

The averaged magic $\overline{\mathcal{M}}_{2}$ is a complicated combination
of Renyi entropies, ranging from $S_{1/2}$ to $S_{4}$. However, in the large
Hilbert dimension limit, where $S_{1/2}\gg 1$, the averaged magic
$\overline{\mathcal{M}}_{2}$ simplifies to the final expression in Eq. (144).
It provides straightforward estimate of $\mathcal{M}_{2}$ based on a few Rényi
entropy terms. To establish a rigorous bound for $\mathcal{M}_{2}$, we start
with Eq. (145), leading to:

$\begin{split}\overline{\mathcal{M}}_{2}&\leq-\log\left(\sum_{i=0}^{r-1}\lambda_{i}^{4}+7\sum_{0\leq
i\neq j\leq r-1}\lambda_{i}^{2}\lambda_{j}^{2}\right)\\\
&\leq-\log\left(\left(\sum_{i}\lambda_{i}^{2}\right)^{2}\right)=2S_{2}.\end{split}$
(146)

This holds for all spectrum distributions. Rewriting Eq. (145) in terms of the
entropy difference $\delta_{1/2}=S_{0}-S_{1/2}$, we obtain the following
expansion;

$\begin{split}\overline{\mathcal{M}}_{2}=&-\log\left(e^{-4\delta_{1/2}}+e^{-S_{0}}(7+21e^{-4\delta_{1/2}}-28e^{-3\delta_{1/2}})+O(e^{-2S_{0}})\right)\\\
\leq&4\delta_{1/2}.\end{split}$ (147)

Note that the coefficient associated with $e^{-S_{0}}$ in the expansion
remains non-negative for any value of $\delta_{1/2}$ and vanishes when
$\delta_{1/2}=0$. Verifying that these coefficients are non-negative for every
order of $e^{-S_{0}}$ supports the inequality. Finally, combining this with
the previously established bound finishes our proof that:

$\mathcal{M}_{2}(\\{\lambda_{i}\\})\leq\overline{\mathcal{M}}_{2}\leq\min\\{2S_{2},4(S_{0}-S_{1/2})\\}.$
(148)

###  D.2 Proof of 8

Let $|\phi\rangle$ denotes an entangled pair of qubits, with the entanglement
spectrum given by $\\{\lambda,1-\lambda\\}$. We show that the non-local
stabilizer Renyi entropy $\mathcal{M}_{2}(\lambda)$ of $|\phi\rangle$ is
bounded by the non-flatness $\partial_{n}\tilde{S}_{n}$. From Eq. (63), we
find that $\mathcal{M}_{2}(\lambda)$ is equal to,

$\mathcal{M}_{2}(\lambda)=-\log\left(1-4\lambda+20\lambda^{2}-32\lambda^{3}+16\lambda^{4}\right).$
(149)

By definition Eq. (36), the non-flatness is

$-\partial_{n}\tilde{S}_{n}=n\frac{\lambda^{n}(1-\lambda)^{n}\left(\log\frac{\lambda}{1-\lambda}\right)^{2}}{\left(\lambda^{n}+(1-\lambda)^{n}\right)^{2}}.$
(150)

Both functions are zero at $\lambda=0,\ \frac{1}{2},\ 1$. So let’s make a
Taylor expansion around these value. Around $\lambda=\frac{1}{2}$, we have
that

$\begin{split}\mathcal{M}_{2}(\lambda)&=4(\lambda-1/2)^{2}-8(\lambda-1/2)^{2}+O((\lambda-1/2)^{3})\\\
-\partial_{n}\tilde{S}_{n}|_{n=1}&=4(\lambda-1/2)^{2}-\frac{16}{3}(\lambda-1/2)^{2}+O((\lambda-1/2)^{3})\\\
-\frac{1}{2}\partial_{n}\tilde{S}_{n}|_{n=2}&=4(\lambda-1/2)^{2}-\frac{160}{3}(\lambda-1/2)^{2}+O((\lambda-1/2)^{3}).\end{split}$
(151)

Therefore for $\lambda$ close to $1/2$, the following inequility holds:

$-\frac{1}{2}\partial_{n}\tilde{S}_{n}|_{n=2}\leq\mathcal{M}_{2}(\lambda)\leq-\partial_{n}\tilde{S}_{n}|_{n=1}.$
(152)

$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.2$$0.4$$0.6$$\lambda$$\mathcal{M}_{2}$Plot of $\mathcal{M}_{2}$ and $\partial_{n}\tilde{S}_{n}$ v.s. $\lambda$ |  | $\mathcal{M}_{2}$ |
---|---|---|---
|  | $-\partial_{n}\tilde{S}_{n}|_{n=1}$ |
|  | $-\frac{1}{2}\partial_{n}\tilde{S}_{n}|_{n=2}$ |
Figure 13: The non-local stabilizer Renyi entropy $\mathcal{M}_{2}$ is bounded
by the anti-flatness $\partial_{n}\tilde{S}_{n}$

Similarly, one can ahow that this inequility holds for $\lambda$ close to $0$
and $1$, where the functions are,

$\begin{split}\mathcal{M}_{2}(\lambda)&=4\lambda-12\lambda^{2}+O(\lambda^{3})\\\
-\partial_{n}\tilde{S}_{n}|_{n=1}&=\lambda\log^{2}\lambda+(2\log\lambda-\log^{2}\lambda)\lambda^{2}+O(\lambda^{3})\\\
-\frac{1}{2}\partial_{n}\tilde{S}_{n}|_{n=2}&=\lambda^{2}\log^{2}\lambda+2(\log\lambda+\log^{2}\lambda)\lambda^{3}+O(\lambda^{4}).\end{split}$
(153)

For other value of $\lambda$, we justify this inequility by the plot in 13.
Both the Stabilizer Renyi entropy and anti-flatness are additive. Therefore
for state $|\psi\rangle$ that can be distilled into product of entangled pairs
$U_{A}\otimes
U_{B}|\psi\rangle_{AB}=\otimes_{i=1}^{k}|\phi\rangle_{a_{i}b_{i}}$, we have
the inequility,

$\frac{1}{2}\left|\frac{\partial_{n}\mathcal{A}_{n}|_{n=2}}{4G}(|\psi\rangle_{AB})\right|\leq\mathcal{M}_{2}(|\psi\rangle_{AB})\leq\left|\frac{\partial_{n}\mathcal{A}_{n}|_{n=1}}{4G}(|\psi\rangle_{AB})\right|.$
(154)

###  D.3 Distillation of Matrix Product State

We further elaborate our discussions from Section 4.1. Building on the MERA
representation of CFT, we transform the state on the boundary of the past
light-cone, $\partial A$, into a Matrix Product State (MPS) using local
unitaries, as defined in Eq. (71) and illustrated in Fig. 2. In this section,
we further contend that this MPS state can approximately be distilled into a
tensor product of entangled pairs:

$|\chi\rangle_{AB}\approx U_{A}\otimes
U_{B}\left(\otimes_{i=1}^{k}|\phi_{i}\rangle_{a_{i}b_{i}}\right).$ (155)

where $k$ is the size of MPS state. It’s clear that this approximation does
not hold in general due to the disparity in the number of free parameters
between the most general entanglement spectrum (contains $2^{k-1}$ parameters)
and that of the tensor product of entangled pairs ($k$ parameters). However,
for translationally invariant MPS states characterized by short correlation
lengths, this approximation is valid.

$4$$5$$6$$7$$8$$9$$10$$11$$-6$$-5$$-4$$-3$$-2$$-1$$0$$k$$\log{\epsilon_{k}}$Plot of $\log\epsilon_{k}$ v.s. $k$ |  | Identical EPR |
---|---|---|---
|  | Non-Identical EPR |
Figure 14: Scaling of error with system size.

$2$$3$$4$$5$$6$$7$$8$$0$$0.5$$1$$1.5$$2$$2.5$$k$$\mathcal{M}_{2}$Plot of $\mathcal{M}_{2}$ and $\partial_{n}\tilde{S}_{n}$ v.s. $k$ |  | $\partial_{n}\tilde{S}_{n}|_{n=1}$ |
---|---|---|---
|  | $M_{2}$ |
|  | $\frac{1}{2}\partial_{n}\tilde{S}_{n}|_{n=2}$ |

(a)

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$0$$1$$2$$3$$4$Samples$M_{2}$$\mathcal{M}_{2}$ and $\partial_{n}S_{n}$ for random samples |  | $\partial_{n}\tilde{S}_{n}|_{n=1}$ |
---|---|---|---
|  | $M_{2}$ |
|  | $\frac{1}{2}\partial_{n}\tilde{S}_{n}|_{n=2}$ |

(b)

Figure 15: (a) Scaling of $\mathcal{M}_{2}$ and $\partial_{n}\tilde{S}_{n}$
with state size $k$ for a particular sample of random matrix in MPS. (b) For
randomly generated samples of MPS states with a fixed size $k=7$,
$\mathcal{M}_{2}$ is bounded by anti-flatness $\partial_{n}\tilde{S}_{n}$.

To substantiate this approximation, we simulate several MPS states using $k$
number of identical random matrices to construct the reduced state
$\rho_{A}=\operatorname{tr}_{B}(|\chi\rangle\langle\chi|)$ and evaluate its
entanglement spectrum. We then approximate this spectrum by fitting it to the
tensor product of individual entangled pair spectra:

$\min_{\\{\lambda_{i}\\}}\left|\mathrm{Spec}(\rho_{A})-\bigotimes_{i=1}^{k}\begin{pmatrix}\lambda_{i}&0\\\
0&1-\lambda_{i}\end{pmatrix}\right|=\epsilon_{k}$ (156)

where $\epsilon_{k}$ quantifies the approximation error. Our numerical
analysis up to $k=11$ reveals an exponential decrease in $\epsilon_{k}$ with
increasing $k$. We present two distinct scenarios in 14: In the first
scenario, we require all EPR pairs in the tensor product to be identical,
yielding an error trend of $\epsilon_{k}\sim 0.1\times 1.2^{-k}$. In the
second scenario, we relax this constraint, allowing for variability among the
EPR pairs, which results in a more pronounced error reduction, following
$\epsilon_{k}\sim 0.05\times 1.4^{-k}$. With the distillation assumption
justified we expect the inequility Eq. (154) to be true for general MPS state
and therefore for a CFT. We plot the magic and the Renyi entropy (dual to
brane area) for a set of randomly generated samples of MPS states in 15 and
verify the validity of the bound Eq. (154).

##  Appendix E Validity of various bound for magic

In the main text, we introduced several approximations for non-local magic,
noting its proportional relationship to anti-flatness in certain regimes and
its closeness to entropy in others. This section delineates the conditions
under which these approximations hold true.

### Near flat limit

We begin by examining the approximation between non-local magic and anti-
flatness, specifically:

$\mathcal{M}_{2}(|\psi\rangle_{AB})\approx\frac{\mathcal{F}(\rho_{A})}{\operatorname{Pur}^{2}(\rho_{A})}$
(157)

which is applicable primarily in the near-flat limit of the entanglement
spectrum. This is because the left-hand side (LHS) is additive and scales
linearly with $n$, while the right-hand side (RHS) can be expressed as:

$\frac{\mathcal{F}(\rho_{A})}{\operatorname{Pur}^{2}(\rho_{A})}=\frac{\operatorname{tr}(\rho_{A}^{3})-\operatorname{tr}(\rho_{A}^{2})^{2}}{\operatorname{tr}(\rho_{A}^{2})^{2}}=e^{2(S_{2}(A)-S_{3}(A))}-1,$
(158)

which becomes additive only at the linear order of the Taylor expansion in the
entropy difference. Hence, the condition $S_{2}(A)-S_{3}(A)<\frac{1}{2}$ must
be met, indicating an almost flat spectrum or a small system size.
Additionally, this regime aligns with where the two anti-flatness measures
defined previously converge, particularly when:

$\langle(\delta\log\rho)^{2}\rangle_{\rho}\approx\frac{\langle(\delta\rho)^{2}\rangle_{\rho}}{\langle\rho\rangle_{\rho}^{2}}$
(159)

under the condition
$\sum_{i}\delta\lambda_{i}^{2}\ll\sum_{i}\bar{\lambda}_{i}^{2}$. In the flat
limit $\sum_{i}\bar{\lambda}_{i}^{2}\sim e^{-S_{0}}$. and
$\sum_{i}\delta\lambda_{i}^{2}\sim e^{-S_{2}}-e^{-S_{0}}$. So the
approximation requires $e^{S_{0}-S_{2}}-1<1$, that is $S_{0}-S_{2}\sim O(1)$.
This also corresponds to near-flat regime or small system size.

### Far from flat limit

In contrast, for quantum states with a far-from-flat entanglement spectrum,
where the entropy differences across Rényi indices are comparable to the
entropy itself, the scenario changes. Referring to Theorem 3, the upper bound
for the second Stabilizer Rényi entropy measure of non-local magic is:

$\mathcal{M}_{2}^{NL}(|\psi\rangle_{AB})\leq\mathcal{M}_{2}(\\{\lambda_{i}\\})\leq\min\\{2S_{2}(A),4(S_{0}(A)-S_{1/2}(A))\\},$
(160)

indicating a transitional crossover around
$S_{0}(A)-S_{1/2}(A)\sim\frac{1}{2}S_{2}(A)$. Beyond this point, non-local
magic transitions from being proportional to anti-flatness to being
proportional to entropy. Our numerical analyses within the Ising model confirm
this transition: in the disordered phase and at critical points, non-local
magic correlates with entropy $S$ both when varying the model parameter and
the subsystem size. However, in the symmetry-breaking phase (refer to F.1), it
deviates and becomes anti-correlated with entropy, as shown in Fig. (17).

##  Appendix F Supplemental results for Ising Model

###  F.1 Symmetry breaking phase

In the $g<0$ regime, the Ising model enters the symmetry-breaking phase in the
thermodynamic limit. However, our analysis is conducted on a finite-size
lattice, where the ground state remains symmetric to spin flipping.
Heuristically, we can think of this ground state being approximated by
something similar to the GHZ state:

$|G\rangle_{sym}\approx\frac{1}{\sqrt{2}}(|00\cdots 0\rangle+|11\cdots
1\rangle).$ (161)

$-0.2$$-0.1$$0$$0.1$$0.2$$0$$0.2$$0.4$$0.6$$0.8$$g$$\mathcal{M}_{2}$$\mathcal{M}_{2}$ v.s. $g$ |  | $\mathcal{M}_{2}$ |
---|---|---|---
|  | $S$ |
Figure 16: Plot of $\mathcal{M}_{2}$ v.s. $g$, at $b=10^{-4}$ and $|A|=9$.

$2$$4$$6$$8$$0.15$$0.2$$0.25$$|A|$$\mathcal{M}_{2}$$\mathcal{M}_{2}$ v.s. $|A|$ |  | $g=-0.09$ |
---|---|---|---
|  | $g=-0.08$ |
|  | $g=-0.06$ |
|  | $g=-0.04$ |
|  | $g=-0.02$ |

(a)

$2$$4$$6$$8$$0.5$$0.6$$0.7$$0.8$$|A|$$S$$S$ v.s. $|A|$ |  | $g=-0.09$ |
---|---|---|---
|  | $g=-0.08$ |
|  | $g=-0.06$ |
|  | $g=-0.04$ |
|  | $g=-0.02$ |

(b)

Figure 17: (a)Non-local magic $\mathcal{M}_{2}$ v.s. $|A|$. (b) Entropy $S$
v.s. $|A|$, at $b=10^{-5}$ and $g>-0.1$.

To approximate the true ground state achievable in the thermodynamic limit
within our finite lattice model, we introduce a small bias field in the
$z$-direction: H=H_Ising(g)+b∑_i Z_i. As the bias $b$ increases, the ground
state transitions towards one of the two symmetry-broken states: &|G ⟩_↑=|↑↑⋯↑
⟩
|G ⟩_↓=|↓↓⋯↓ ⟩. Exploring how non-local magic $\mathcal{M}_{2}$ behaves as we
adjust different parameters led to some fascinating results that are
particularly noticeable when a non-zero bias field is applied. As shown in 16,
a distinctive “valley” emerges in the $\mathcal{M}_{2}$ plot within the $g<0$
regime. We juxtapose entropy and non-local magic in our plots to underscore
their divergent behaviors and the unique information conveyed by non-local
magic. This valley can be understood as arising from the competition between
two types of ground states. Within the valley, the system’s ground state
approximates the symmetric GHZ-like state $|G\rangle_{sym}$, as defined in Eq.
(161). In this region, non-local magic values are minimized because the
reduced density matrix of $|G\rangle_{sym}$ resembles that of a maximally
mixed single qubit state, leading to a flat spectrum and, consequently, lower
$\mathcal{M}_{2}$ follows from Corollary 3.1. Additionally, we observe
diminished $\mathcal{M}_{2}$ values in regions far from the critical point,
where $|g|$ is sufficiently large, as indicated by the plateau beyond $g<-0.2$
in 16. Here, the ground state transitions to a symmetry-broken state
$|G\rangle_{\uparrow/\downarrow}$, which lacks non-local magic due to its
tensor product structure. Despite the low non-local magic values associated
with both $|G\rangle_{sym}$ and $|G\rangle_{\uparrow/\downarrow}$, the
transition between these states has to past through a regime of non-trivial
non-local magic. This occurs because continuous parameter changes cannot be
approximated by discrete Clifford transformations, resulting in a notable
increase in non-local magic. The $\mathcal{M}_{2}$ measure captures this as a
pronounced peak, delineating the transition between the two ground states near
$g\sim-0.1$ in 16. An additional noteworthy aspect of non-local magic inside
the valley is its counterintuitive decrease with increasing subregion size
$|A|$, as depicted in 17(a). This phenomenon is unique to the valley. In
contrast, entropy consistently increases with $|A|$. This unusual trend in
$\mathcal{M}_{2}$ is also linked to the proximity to the symmetric state
$|G\rangle_{sym}$, which results in an almost flat entanglement spectrum
within the valley. Consequently, $\mathcal{M}_{2}$ aligns more closely with
the entropy differential $S_{0}-S$ rather than the entropy itself, as
discussed in Section 3.3.3, offering an explanation for the inverse
relationship observed between $S$ and $\mathcal{M}_{2}$ in this region. It’s
also important to note that the competition between $|G\rangle_{sym}$ and
$|G\rangle_{\uparrow/\downarrow}$ is a manifestation of finite-size effects.
As demonstrated in 18, the valley tends to diminish with increasing lattice
size $n$. Specifically, when we set $b=10^{-4}$ (see 18(a)), the peak of non-
local magic shifts closer to $g=0$ with larger lattice sizes. Similarly, with
$g=-0.11$ (see 18(b)), the peak moves towards $b=0$ as the lattice size
expands. This suggests that the parameter space favoring the symmetric state
narrows in both dimensions with increasing lattice size.

$-0.2$$-0.1$$0$$0.1$$0.2$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$g$$\mathcal{M}_{2}$$\mathcal{M}_{2}$ v.s. $g$ |  | $n=18$ |
---|---|---|---
|  | $n=24$ |
|  | $n=26$ |

(a)

$-7$$-6$$-5$$-4$$-3$$-2$$0$$0.1$$0.2$$0.3$$0.4$$\log b$$\mathcal{M}_{2}$$\mathcal{M}_{2}$ v.s. $\log b$ |  | n=18 |
---|---|---|---
|  | n=24 |

(b)

Figure 18: (a) Plot of $\mathcal{M}_{2}$ v.s. $g$, at $b=10^{-4}$. The lattice
size takes $n=18$, $n=24$ and $n=26$; (b) Plot of $\mathcal{M}_{2}$ v.s. $\log
b$, at $g=-0.11$, with lattice size taking $n=18$ and $n=24$ Figure 19:
Comparison between $\mathcal{M}_{2}$ and $|\partial_{n}\tilde{S}_{n}|$ at
various magnetic field $b$ and model parameter $g$

Expanding our analysis, 19 explores the non-local magic across a broader range
of the bias field $b$. We find that beyond $b>0.01$ the valley disappears, and
the $g<0$ phase transitions to being governed by the symmetry-broken ground
state $|G\rangle_{\uparrow/\downarrow}$. As $b$ decreases towards zero, the
peak is pushed to the left where the valley widens, signifying the growing
significance of the symmetric ground state $|G\rangle_{sym}$, which becomes
dominant for all $g<0$ in the absence of $b$. Fig. (20) depicts the
$\mathcal{M}_{2}$ surface as a function of subregion size $|A|$ and critical
angle $\theta$. As we decrease the magnitude of the bias field, the symmetry-
breaking peak is pushed towards lower and lower $\theta$ values.

Figure 20: Non-local magic $\mathcal{M}_{2}$ surface, as a function of
critical angle $\theta$ and subregion size $|A|$, for different bias offset
fields. The bias magnetic field decreases, the peak indicating a symmetry-
breaking effect in the system is pushed further away from criticality.

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