in Eq. 81 into on-site operators, so both states are SPTs).

The first unitary layer is given by

$\displaystyle\mathcal{U}^{\vphantom{\dagger}}_{0}$
$\displaystyle=\Big{(}\prod\limits_{n=1}^{k}\,\mathrm{SWAP}^{\vphantom{\dagger}}_{2n-1,2n}\,\Big{)}\,\Bigg{[}\prod\limits_{j=k+1}^{(m+1)k}\,\mathcal{B}^{\vphantom{\dagger}}_{2j-1,2j}\,\Bigg{]}\,\mathds{1}\,,$
(83)

corresponding to (_i_) the $2\,k$ sites on the left, (_ii_) the $2\,m\,k$
sites in the bulk, and (_iii_) the $k$ sites on the right, respectively, where
$\mathrm{SWAP}\,\lvert a,b\rangle=\lvert b,a\rangle$ and, for convenience, we
define the Bell encoding channel $\mathcal{B}$ as

$\displaystyle\mathcal{B}^{\vphantom{\dagger}}_{a,b}\equiv\mathrm{CNOT}\left(a\to
b\right)\,H^{\vphantom{\dagger}}_{a}\,,~{}~{}$ (84)

where $\mathrm{CNOT}\left(a\to b\right)$ flips the $Z^{\vphantom{*}}$ state of
qubit $b$ if qubit $a$ is in the $Z^{\vphantom{*}}$ state $\lvert 1\rangle$,
and acts trivially otherwise, so that $\mathcal{B}^{\,}_{a,b}\,\lvert
00\rangle^{\,}_{a,b}=\lvert\mathrm{Bell}\rangle^{\,}_{a,b}$. Note that the
SWAP gates in $\mathcal{U}^{\,}_{0}$ (83) can be omitted if we swap the states
of sites $2n-1$ and $2n$ in $\lvert\Psi^{\,}_{0}\rangle$ (82), at the cost of
decreasing $L$ (13) by one. When preparing the valence-bond state $\lvert
V^{\,}_{k}\rangle$ (80) directly, $\mathcal{U}^{\,}_{0}$ only contains the
bulk term in Eq. 83.

The next layers of $\mathcal{U}$ are given by

$\mathcal{U}^{\vphantom{\dagger}}_{p}=\begin{cases}\prod\limits_{j=1}^{(m+1)k^{\vphantom{\dagger}}}\,\mathrm{SWAP}^{\vphantom{\dagger}}_{2j+2p-2,2j+2p-1}\,&\,p~{}{\rm
odd}\\\\[2.0pt]
\prod\limits_{j=1}^{(m+1)k}\,\mathrm{SWAP}^{\vphantom{\dagger}}_{2j+2p-3,2j+2p-2}\,&\,p~{}{\rm
even}\end{cases}\,,~{}~{}$ (85)

which can be thought of more simply as alternating layers of SWAP gates on
even versus odd bonds, where the first layer of SWAP gates acts on even
bonds—i.e., the SWAP gates in $\mathcal{U}^{\,}_{1}$ (85) are staggered by one
site with respect to the Bell encoding channels $\mathcal{B}$ in
$\mathcal{U}^{\,}_{0}$ (83).

The product of the untiaries
$\mathcal{U}^{\,}_{k-1}\cdots\mathcal{U}^{\,}_{0}$ map the state
$\lvert\boldsymbol{0}\rangle$ to the valence-bond state $\lvert
V^{\,}_{k}\rangle$ (80). The analogous state $\lvert\Psi^{\,}_{T-1}\rangle$
for teleportation is captured by Fig. 8 for $k=1,2,3$; however, writing a
general expression is inconvenient. The final layer of unitary evolution
applies the Bell _decoding_ channel to the qubits in ovals in Fig. 8, and SWAP
gates to all qubits on the right end, i.e.,

$\displaystyle\mathcal{U}^{\vphantom{\dagger}}_{k}=\prod\limits_{j=1}^{km}\mathcal{B}^{\dagger}_{k+2j-1,k+2j}\,\prod\limits_{n=0}^{k-1}\mathrm{SWAP}^{\vphantom{\dagger}}_{N-2n-1,N-2n},$
(86)

where the SWAP gates on the final $2k$ sites can be omitted at the cost of
decreasing $L$ (13) by one. Following the application of all unitaries, one
measures $Z^{\vphantom{*}}_{j}$ on all sites $[k+1,(2m+1)k]$, where every $2k$
consecutive outcomes are determine recovery channels according to
$(m_{x,1},m_{z,1},m_{x,2},m_{z,2},\dots,m_{x,k},m_{z,k})$, where
$\sigma^{z/x}_{f_{n}}$ is applied to $f_{n}$ if $\sum_{s=1}^{m}m_{x/z,s}\cong
1$ mod $2$.

##  7 Outlook

We have shown that all “standard” teleportation protocols that teleport $k$
logical states a distance $L>vT$ (13) are equivalent to preparing a resource
state with nontrivial SPT order protected by the symmetry
$\mathcal{G}=\mathbb{Z}_{2}^{2k}$ (65) using local unitary evolution for time
$T\ll L$, measuring the $2k$ string order parameters (63) in the bulk, and
applying Pauli recovery gates (34) based on the outcomes. These protocols are
physical in that the logical states $\lvert\psi^{\,}_{n}\rangle$ are realized
on particular physical qubits. We require that no unitaries are applied to a
site after it is measured, though we expect that this requirement can be
relaxed.

The crucial feature of standard teleportation protocols from the perspective
of constraining the resource state is that all outcome-dependent operations
are Pauli gates. The recovery group—generated by all physical error-correction
operators $\chi$ in $\mathcal{R}$ (18), modulo complex phases—is simply the
$k$-qubit Pauli group
$\mathbf{P}^{\vphantom{*}}_{*}{k}/\mathsf{U}\left(1\right)$ for standard
teleportation, which forms a projective representation of the symmetry group
$\mathcal{G}=\mathbb{Z}_{2}^{2k}$ (65). We expect that our analysis can be
extended, _mutatis mutandis_ , to more general recovery operations.

In this sense, our results (namely, Theorem 18) represent a first step toward
classifying all quantum teleportation protocols based solely on their recovery
(or decoding) channels (18). There are numerous recovery operations to
consider beyond the Paulis of standard teleportation: The corresponding group
can be continuous and non-Abelian (i.e.,
$\chi^{\,}_{g}\,\chi^{\,}_{h}\neq\omega(g,h)\,\chi^{\,}_{h}\,\chi^{\,}_{g}$
for some complex phase $\omega$). We can also condition measurements (or the
basis of a measurement) on prior outcomes, and condition operations on the
outcomes of measurements in other ways (e.g., nonlinearly). For example, Ref.
66 considers Pauli recovery operations but does not require that they be
conditioned on outcome parities, in which case the resource states can be
chosen to have SPT order, but the symmetry generators $U(g)$ are no longer “on
site” (49). Alternatively, one might consider non-Abelian—and even
_continuous_ —recovery groups, whose corresponding resource states are matrix
product states Gross and Eisert (2007, 2010) that generally do not have string
order Popp  _et al._ (2005). We expect that our methods using the Stinespring
formalism will be crucial in analyzing these more exotic protocols and their
potential relation to exotic forms of SPT order.

## Acknowledgments

We thank Andy Lucas, Charles Stahl, and Chao Yin for useful discussions and
collaboration on related work. This work was supported in part by the U.S. Air
Force Office of Scientific Research under Grant No. FA9550-21-1-0195 (YH), the
Office of Naval Research via Grant No. N00014-23-1-2533 (YH), the Simons
Collaboration on Ultra-Quantum Matter via Award No. 651440 (DTS), and the US
Department of Energy via Grant No. DE-SC0024324 (AJF).

##  Appendix A Proof of Proposition 1

###### Proof.

For the teleportation conditions 8 to be necessary and sufficient for quantum
state transfer (see Def. 1), they must be equivalent to (5) for the initial
state $\lvert\Psi^{\,}_{0}\rangle$ (6) and final state
$\lvert\Psi^{\,}_{T}\rangle$ (7). We prove both directions of implication in
turn, using the fact that all operators acting on a qubit can be decomposed
onto the Pauli operators
$\mathds{1},X^{\vphantom{*}},Y^{\vphantom{*}},Z^{\vphantom{*}}$, which are
spanned by $X^{\vphantom{*}}$ and $Z^{\vphantom{*}}$. The coefficients
$\alpha$ and $\beta$ of a logical state $\lvert\psi\rangle$ can be extracted
from the expectation values of the Pauli operators (4).

We first establish that Eqs. 5, 6, and 7 imply Eq. 8—i.e., Eq. 8 is necessary
for state transfer. First, Eq. 7 implies that $X^{\vphantom{*}}_{f_{n}}$ is
the $X^{\vphantom{*}}$-type logical operator for the $n$th logical qubit
acting on the final state $\lvert\Psi^{\,}_{T}\rangle$ (7), as it interchanges
$\alpha^{\,}_{n}$ and $\beta^{\,}_{n}$. In the Heisenberg picture, this
logical operator evolves under $\mathcal{W}$ according to

$X^{(n)}_{{\rm
L}}(T)\,=\,X^{\vphantom{*}}_{f_{n}}(T)\,=\,\mathcal{W}^{\dagger}\,X^{\vphantom{*}}_{f_{n}}\,\mathcal{W}\,,~{}~{}$
(A.87)

and now, using the fact that $X^{2}_{i_{n}}=\mathds{1}$ is involutory, we
introduce an arbitrary operator $\mathsf{S}^{\,}_{n,x}$ via

$\displaystyle X^{(n)}_{{\rm L}}(T)\,$
$\displaystyle=\,\mathds{1}\,\mathcal{W}^{\dagger}X^{\vphantom{*}}_{f_{n}}\mathcal{W}\,=\,X^{2}_{i_{n}}\,\mathcal{W}^{\dagger}X^{\vphantom{*}}_{f_{n}}\mathcal{W}$
$\displaystyle=\,X^{\vphantom{*}}_{i_{n}}\,\left(X^{\vphantom{*}}_{i_{n}}\,\mathcal{W}^{\dagger}X^{\vphantom{*}}_{f_{n}}\mathcal{W}\right)\,\equiv\,X^{\vphantom{*}}_{i_{n}}\,\mathsf{S}^{\,}_{n,x}\,,~{}~{}$
(A.88)

which implicitly defines the (unitary) operator $\mathsf{S}^{\,}_{n,x}$ as

$\mathsf{S}^{\,}_{n,x}\,\equiv\,X^{\vphantom{*}}_{i_{n}}\,\mathcal{W}^{\dagger}X^{\vphantom{*}}_{f_{n}}\mathcal{W}\,.$
(A.89)

Leaving the coefficients $\alpha^{\,}_{j},\beta^{\,}_{j}$ for $j\neq n$
implicit for convenience, we apply $\mathsf{S}^{\,}_{n,x}$ (A.89) to the
initial state $\lvert\Psi^{\,}_{0}(\alpha^{\,}_{n},\beta^{\,}_{n})\rangle$ (6)
and use the first part of Eq. 5 to find

$\displaystyle\mathsf{S}^{\,}_{n,x}\,\lvert\Psi^{\vphantom{\prime}}_{0}(\alpha^{\,}_{n},\beta^{\,}_{n})\rangle\,$
$\displaystyle=\,X^{\vphantom{*}}_{i_{n}}\,\mathcal{W}^{\dagger}X^{\vphantom{*}}_{f_{n}}\mathcal{W}\,\lvert\Psi^{\vphantom{\prime}}_{0}(\alpha^{\,}_{n},\beta^{\,}_{n})\rangle$
$\displaystyle=\,X^{\vphantom{*}}_{i_{n}}\,\mathcal{W}^{\dagger}X^{\vphantom{*}}_{f_{n}}\lvert\Psi^{\vphantom{\prime}}_{T}(\alpha^{\,}_{n},\beta^{\,}_{n})\rangle$
$\displaystyle=\,X^{\vphantom{*}}_{i_{n}}\,\mathcal{W}^{\dagger}\lvert\Psi^{\vphantom{\prime}}_{T}(\beta^{\,}_{n},\alpha^{\,}_{n})\rangle$
$\displaystyle=\,X^{\vphantom{*}}_{i_{n}}\lvert\Psi^{\vphantom{\prime}}_{0}(\beta^{\,}_{n},\alpha^{\,}_{n})\rangle$
$\displaystyle=\,\lvert\Psi^{\vphantom{\prime}}_{0}(\alpha^{\,}_{n},\beta^{\,}_{n})\rangle\,,~{}~{}$
(A.90)

meaning that $\mathsf{S}^{\,}_{n,x}$ (A.89) is an element of the initial-state
stabilizer group
$\operatorname*{stab}\left(*\right){\lvert\Psi^{\,}_{0}\rangle}$ (3), and is
guaranteed to be unitary. Next, applying the Heisenberg-evolved operator
$X^{\vphantom{*}}_{f_{n}}(T)$ (A.87) to the initial state
$\lvert\Psi^{\,}_{0}\rangle$ (6) gives

$\displaystyle X^{(n)}_{{\rm
L}}(T)\,\lvert\Psi^{\vphantom{\prime}}_{0}(\alpha^{\,}_{n},\beta^{\,}_{n})\rangle\,$
$\displaystyle=\,X^{\vphantom{*}}_{i_{n}}\,\mathsf{S}^{\,}_{n,x}\,\lvert\Psi^{\vphantom{\prime}}_{0}(\alpha^{\,}_{n},\beta^{\,}_{n})\rangle$
$\displaystyle=\,X^{\vphantom{*}}_{i_{n}}\,\lvert\Psi^{\vphantom{\prime}}_{0}(\alpha^{\,}_{n},\beta^{\,}_{n})\rangle\,,~{}~{}$
(A.91)

which holds for _any_ choice of $\alpha^{\,}_{n},\beta^{\,}_{n}$. This implies
that $X^{(n)}_{{\rm
L}}(T)\,=\,\mathcal{W}^{\dagger}\,X^{\vphantom{*}}_{f_{n}}\,\mathcal{W}\,$
realizes an $X^{\vphantom{*}}$-type logical operator for the $n$th logical
qubit acting on the _initial_ state $\lvert\Psi^{\,}_{0}\rangle$ (6)—i.e., it
acts as $X^{\vphantom{*}}_{i_{n}}$ times some stabilizer
$\mathsf{S}^{\,}_{n,x}$ (A.89), where the latter acts trivially on
$\lvert\Psi^{\,}_{0}\rangle$ (6).

An analogous procedure applies to the $Z^{\vphantom{*}}$-type logical operator
for the $n$th logical qubit. That operator evolves in the Heisenberg picture
according to

$Z^{(n)}_{{\rm
L}}(T)\,=\,Z^{\vphantom{*}}_{f_{n}}(T)\,=\,\mathcal{W}^{\dagger}\,Z^{\vphantom{*}}_{f_{n}}\,\mathcal{W}\,,~{}~{}$
(A.92)

in analogy to Eq. A.87, and implied by Eq. 7. We then define another
stabilizer $\mathsf{S}^{\,}_{n,z}\equiv
Z^{\vphantom{*}}_{i_{n}}\,\mathcal{W}^{\dagger}Z^{\vphantom{*}}_{f_{n}}\mathcal{W}$,
implying that $\mathsf{S}^{\,}_{n,z}$ is also an element of
$\operatorname*{stab}\left(*\right){\lvert\Psi^{\,}_{0}\rangle}$ (3). Applying
the logical $Z^{\vphantom{*}}$ operator (A.92) to $\lvert\Psi^{\,}_{0}\rangle$
(6) reveals that

$Z^{(n)}_{{\rm
L}}(T)=\mathcal{W}^{\dagger}\,Z^{\vphantom{*}}_{f_{n}}\,\mathcal{W}=Z^{\vphantom{*}}_{i_{n}}\,\mathsf{S}^{\,}_{n,z}\,,~{}~{}$
(A.93)

is the $Z^{\vphantom{*}}$-type logical operator for the $n$th logical qubit
acting on the initial state $\lvert\Psi^{\,}_{0}\rangle$ (6), in analogy to
Eq. A.91.

Thus, the fact that $\mathcal{W}$ teleports $k$ logical states in the
Schrödinger picture via EQ. 5 implies that (_i_) $\mathcal{W}$ teleports $k$
pairs of logical _operators_ in the Heisenberg picture; (_ii_) $X^{(n)}_{{\rm
L}}(T)\,=\,\mathcal{W}^{\dagger}\,X^{\vphantom{*}}_{f_{n}}\,\mathcal{W}\,=\,X^{\vphantom{*}}_{i_{n}}\,\mathsf{S}^{\vphantom{\prime}}_{n,x}$
and $Z^{(n)}_{{\rm
L}}(T)\,=\,\mathcal{W}^{\dagger}\,Z^{\vphantom{*}}_{f_{n}}\,\mathcal{W}\,=\,Z^{\vphantom{*}}_{i_{n}}\,\mathsf{S}^{\vphantom{\prime}}_{n,z}$
are valid $X^{\vphantom{*}}$\- and $Z^{\vphantom{*}}$-type logical operators
for the $n$th logical qubit acting on the _initial_ state
$\lvert\Psi^{\,}_{0}\rangle$ (6); and (_iii_) the operators
$\mathsf{S}^{\,}_{n,x}$ and $\mathsf{S}^{\,}_{n,z}$ are elements of the
initial-state stabilizer group
$\operatorname*{stab}\left(*\right){\lvert\Psi^{\,}_{0}\rangle}$ (3). Hence,
Eq. 5 implies Eq. 8.

Now, we prove the reverse implication: Any protocol $\mathcal{W}$ satisfying
Eq. 8 with the initial state $\lvert\Psi^{\,}_{0}\rangle$ (6) implies state
transfer (5) with final state $\lvert\Psi^{\,}_{T}\rangle$ (7)—i.e., Eq. 8 is
sufficient for Eq. 5. For convenience, we define

$\displaystyle\rho^{\vphantom{\dagger}}_{n}\,$
$\displaystyle=\,\lvert\psi^{\,}_{n}\rangle\langle\psi^{\,}_{n}\rvert$
$\displaystyle=\,\frac{1}{2}\mathds{1}+c^{(n)}_{1}\,X^{\vphantom{*}}_{i_{n}}++c^{(n)}_{2}\,Y^{\vphantom{*}}_{i_{n}}+c^{(n)}_{3}\,Z^{\vphantom{*}}_{i_{n}}$
$\displaystyle=\,\frac{1}{2}\left(\mathds{1}+\sum_{\nu=1}^{3}c^{(n)}_{\nu}\sigma^{\nu}_{i_{n}}\right)\,,~{}$
(A.94)

where the coefficients $c^{(n)}_{\nu}$ are given by

$\displaystyle c^{(n)}_{\nu}\,$
$\displaystyle\equiv\,\left\langle\psi^{\,}_{n}\middle|\sigma^{\nu}_{i_{n}}\middle|\psi^{\,}_{n}\right\rangle\,,~{}~{}$
(A.95)

for the $n$th logical qubit, as in Eq. 4. We can then write the initial
density matrix associated with $\lvert\Psi^{\,}_{0}\rangle$ (6) as

$\displaystyle\rho^{\,}_{0}$
$\displaystyle=\lvert\Psi^{\vphantom{\prime}}_{0}\left(\boldsymbol{\alpha},\boldsymbol{\beta}\right)\rangle\langle\Psi^{\vphantom{\prime}}_{0}\left(\boldsymbol{\alpha},\boldsymbol{\beta}\right)\rvert\,$
$\displaystyle=\left[\prod\limits_{n=1}^{k}\,\rho^{\vphantom{\dagger}}_{n}\right]\otimes\lvert\Phi^{\vphantom{\prime}}_{i}\rangle\langle\Phi^{\vphantom{\prime}}_{i}\rvert^{\vphantom{\dagger}}_{\rm
rest}\,,$ (A.96)

which realizes
$\rho^{\,}_{n}=\lvert\psi^{\,}_{n}\rangle\langle\psi^{\,}_{n}\rvert$ on each
initial logical qubit $i_{n}$. Using Eqs. A.96 and A.94, we write

$\displaystyle\mathcal{W}\,\lvert\Psi^{\vphantom{*}}_{i}\rangle$
$\displaystyle=\mathcal{W}\,\lvert\psi^{\,}_{n}\rangle\langle\psi^{\,}_{n}\rvert^{\vphantom{*}}_{i_{n}}\lvert\Psi^{\vphantom{*}}_{i}\rangle$
$\displaystyle=\mathcal{W}\,\frac{1}{2}\left(\mathds{1}+\sum_{\nu=1}^{3}c^{(n)}_{\nu}\sigma^{\nu}_{i_{n}}\right)\lvert\Psi^{\vphantom{*}}_{i}\rangle$
$\displaystyle=\mathcal{W}\,\frac{1}{2}\left(\mathds{1}+\sum_{\nu=1}^{3}c^{(n)}_{\nu}\sigma^{\nu}_{i_{n}}\mathsf{S}^{\,}_{n,\nu}\right)\lvert\Psi^{\vphantom{*}}_{i}\rangle$
$\displaystyle=\mathcal{W}\,\frac{1}{2}\left(\mathds{1}+\sum_{\nu=1}^{3}c^{(n)}_{\nu}\mathcal{W}^{\dagger}\sigma^{\nu}_{f_{n}}\,\mathcal{W}\right)\lvert\Psi^{\vphantom{*}}_{i}\rangle$
$\displaystyle=\frac{1}{2}\left(\mathds{1}+\sum_{\nu=1}^{3}c^{(n)}_{\nu}\sigma^{\nu}_{f_{n}}\right)\mathcal{W}\,\lvert\Psi^{\vphantom{*}}_{i}\rangle$
$\displaystyle=\lvert\psi^{\,}_{n}\rangle\langle\psi^{\,}_{n}\rvert^{\vphantom{*}}_{f_{n}}\,\mathcal{W}\,\lvert\Psi^{\vphantom{*}}_{i}\rangle\,,~{}~{}$
(A.97)

where the second line follows from Eq. A.94; the third line uses
$\mathsf{S}^{\,}_{n,\nu}\in\operatorname*{stab}\left(*\right){\lvert\Psi^{\,}_{0}\rangle}$
to insert stabilizers inside the parenthetical expression; the fourth line
uses Eqs. 8 and 9; and the last line uses Eq. A.94 once again. Thus, the final
state $\lvert\Psi^{\,}_{T}\rangle=\mathcal{W}\,\lvert\Psi^{\,}_{0}\rangle$ (7)
is stabilized by the $n$th logical qubit’s density matrix on site
$f^{\,}_{n}$. For brevity, we represent the final density matrix as
$\rho^{\,}_{T}=\mathcal{W}\,\lvert\Psi^{\,}_{0}\rangle\langle\Psi^{\,}_{0}\rvert\mathcal{W}^{\dagger}$.
Since $\lvert\psi^{\,}_{n}\rangle\langle\psi^{\,}_{n}\rvert^{\,}_{f_{n}}$ only
acts nontrivially on site $f_{n}$, we have

$\displaystyle 1$ $\displaystyle={\rm
tr}\left[\,\rho^{\vphantom{*}}_{T}\,\right]={\rm
tr}\left[\,\lvert\psi^{\,}_{n}\rangle\langle\psi^{\,}_{n}\rvert^{\vphantom{*}}_{f_{n}}\,\rho^{\vphantom{*}}_{T}\,\right]$
$\displaystyle={\rm
tr}\left[\,\lvert\psi^{\,}_{n}\rangle\langle\psi^{\,}_{n}\rvert\,\rho^{\vphantom{*}}_{f_{n}}\,\right]=F\left(\rho^{\vphantom{*}}_{f_{n}},\lvert\psi^{\vphantom{*}}_{n}\rangle\right)\,,~{}$
(A.98)

where $F$ is the _fidelity_ Jozsa (1994), and in second line we used Eq. A; in
the third line we have defined
$\rho^{\,}_{f_{n}}\equiv\operatorname*{tr}^{\,}_{f^{c}_{n}}\left[\,\rho^{\,}_{T}\,\right]$
as the reduced final density matrix on site $f_{n}$; in the last line we have
used the fact that the fidelity obeys
$F(\rho,\lvert\psi\rangle)=\left\langle\psi\middle|\rho\middle|\psi\right\rangle={\rm
tr}\left[\,\lvert\psi\rangle\langle\psi\rvert\,\rho\,\right]$ when one of the
states is pure Jozsa (1994). Note that, in the second line the trace is over
the entire system, whereas in the third line, it is only over $f_{n}$. Since
$F(\rho^{\vphantom{*}}_{f_{n}},\lvert\psi^{\vphantom{*}}_{n}\rangle)=1$, we
must have that

$\displaystyle\rho^{\vphantom{*}}_{f_{n}}=\operatorname*{tr}^{\,}_{f^{c}_{n}}\left[\,\rho\,\right]=\lvert\psi^{\,}_{n}\rangle\langle\psi^{\,}_{n}\rvert\,,~{}$
(A.99)

i.e., the final reduced density matrix on site $f_{n}$ is the logical $n$th
(pure) state. Since the above arguments hold for all $n=1,\dots,k$
independently, we conclude that
$\lvert\Psi^{\,}_{T}\rangle=\mathcal{W}\,\lvert\Psi^{\,}_{0}\rangle$ satisfies
(5) for the final state $\lvert\Psi^{\,}_{T}\rangle$ (7). Thus, Eq. 8 is both
necessary and sufficient for Eq. 5. ∎

##  Appendix B Proof of Proposition 4

###### Proof.

A generic and nontrivial single-qubit observable $A^{\,}_{j}$ is guaranteed to
have exactly two unique eigenvalues $a^{\,}_{0}>a^{\,}_{1}$, with spectral
decomposition (15)

$\displaystyle A^{\vphantom{\dagger}}_{j}$
$\displaystyle=a^{\,}_{0}\,P^{(0)}_{j}+a^{\,}_{1}\,P^{(1)}_{j}\,,~{}~{}$
(B.100)

which we decompose onto the Pauli basis via

$\displaystyle A^{\vphantom{\prime}}_{j}$
$\displaystyle=\frac{1}{2}\,\sum\limits_{\nu=0}^{3}\,\operatorname*{tr}\limits_{j}\,\left[\,A^{\vphantom{\prime}}_{j}\,\sigma^{\nu}_{j}\,\right]\,\sigma^{\nu}_{j}$
$\displaystyle=\frac{1}{2}\left(a^{\,}_{0}+a^{\,}_{1}\right)\,\mathds{1}+\vec{\alpha}^{\vphantom{\prime}}_{j}\cdot\vec{\sigma}^{\vphantom{\prime}}_{j}\,,~{}~{}$
(B.101)

where
$\vec{\sigma}^{\,}_{j}=(X^{\vphantom{*}}_{j},Y^{\vphantom{*}}_{j},Z^{\vphantom{*}}_{j})$
is a vector of Pauli matrices and $\vec{\alpha}^{\,}_{j}$ is a vector of
coefficients as determined by the Pauli decomposition. Importantly, using the
properties of $2\times 2$ matrices and the fact that
$\det\,\left[A^{\,}_{j}\right]=a^{\,}_{0}\,a^{\,}_{1}$ we find

$\lvert\vec{\alpha}\rvert=\frac{1}{2}\lvert
a^{\,}_{0}-a^{\,}_{1}\rvert\,,~{}~{}$ (B.102)

for the magnitude of $\vec{\alpha}$. This provides for the definition of a
unit vector $\hat{\alpha}$ that is parallel to the original vector
$\vec{\alpha}$.

We rewrite the decomposed observable $A^{\,}_{j}$ (B.101) as

$\displaystyle A^{\vphantom{\dagger}}_{j}$
$\displaystyle=\frac{1}{2}\left(a^{\,}_{0}+a^{\,}_{1}\right)\,\mathds{1}+\frac{1}{2}\left(a^{\,}_{0}-a^{\,}_{1}\right)\,\sum\limits_{n=1}^{3}\,\hat{\alpha}^{\vphantom{\prime}}_{j,n}\,\sigma^{n}_{j}$
$\displaystyle=\frac{1}{2}\operatorname*{tr}\limits_{j}\left[A^{\vphantom{\prime}}_{j}\right]\,\mathds{1}+\frac{1}{2}\,\left(a^{\,}_{0}-a^{\,}_{1}\right)\,\bar{A}^{\vphantom{\prime}}_{j}\,,~{}~{}$
(B.103)

where the vector
$\hat{\alpha}^{\,}_{j}=(\hat{\alpha}^{\,}_{j,1},\hat{\alpha}^{\,}_{j,2},\hat{\alpha}^{\,}_{j,3})$
is real valued and has unit length. This expression agrees with Eq. 33.

We next recover simple expressions for the projectors $P^{(n)}_{j}$ (B.100)
onto the states (or spaces of states) with eigenvalues $a^{\,}_{n}$. We first
note that these two projectors must be _orthogonal_ for the eigenvalue
equation $A^{\,}_{j}\,P^{(n)}_{j}=a^{\,}_{n}\,P^{(n)}_{j}$ to hold. Second, we
note that these two projectors must be _complete_ (i.e., sum to the identity),
or else evaluating $\det\,\left[A^{\,}_{j}\right]$ in the eigenbasis of
$A^{\,}_{j}$ would return zero, independent of the values of $a^{\,}_{0,1}$.
As a result,

$\displaystyle A^{\vphantom{\dagger}}_{j}$
$\displaystyle=a^{\,}_{n}\,P^{(n)}_{j}+a^{\,}_{1-n}\left(\mathds{1}-P^{(n)}_{j}\right)$
$\displaystyle=a^{\,}_{1-n}\,\mathds{1}+\left(a^{\,}_{n}-a^{\,}_{1-n}\right)\,P^{(n)}_{j}\,,~{}~{}$
(B.104)

for either $n=0,1$. Rearranging this expression gives

$\displaystyle P^{(n)}_{j}$
$\displaystyle=\left(a^{\,}_{n}-a^{\,}_{1-n}\right)^{-1}\,\left(A^{\vphantom{\prime}}_{j}-a^{\,}_{1-n}\,\mathds{1}\right)$
(B.105)
$\displaystyle=\frac{1}{a^{\,}_{n}-a^{\,}_{1-n}}\left(\frac{1}{2}\operatorname*{tr}\limits_{j}\left[A^{\vphantom{\prime}}_{j}\right]\mathds{1}+\lvert\vec{\alpha}\rvert\,\bar{A}^{\vphantom{\prime}}_{j}-a^{\,}_{1-n}\mathds{1}\right)$
$\displaystyle=\frac{1}{a^{\,}_{n}-a^{\,}_{1-n}}\left(\frac{1}{2}(a^{\,}_{n}-a^{\,}_{1})\mathds{1}+\frac{1}{2}(a^{\,}_{0}-a^{\,}_{1})\,\bar{A}^{\vphantom{\prime}}_{j}\right)$
$\displaystyle=\frac{1}{2}\left(\mathds{1}+(-1)^{n}\,\bar{A}^{\vphantom{\prime}}_{j}\right)\,,~{}~{}$
(B.106)

where we used Eq. B.101 to write $A^{\,}_{j}$ in terms of its involutory part
$\bar{A}^{\,}_{j}$ (33), along with the relations
$a^{\,}_{0}-a^{\,}_{1}=(-1)^{n}(a^{\,}_{n}-a^{\,}_{1-n})=2\lvert\vec{\alpha}\rvert$
(B.102) and
$\operatorname*{tr}_{j}\left[A^{\,}_{j}\right]=a^{\,}_{0}+a^{\,}_{1}$.

Because the involutory operator $A^{\,}_{j}$ squares to the identity, its
eigenvalues are $\pm 1$. Accordingly, the projectors onto eigenstates of
$A^{\,}_{j}$ (which are also eigenstates of the original observable
$A^{\,}_{j}$) are given by Eq. B.106. Intuitively, these observables must
share eigenstates because they commute: They differ only by a scalar factor
and an overall, universally commuting identity component. Finally, because
only the projectors onto eigenstates of $A^{\,}_{j}$ appear in the unitary
measurement channel (16), measurement of $A^{\,}_{j}$ is equivalent to
measurement of $\bar{A}^{\,}_{j}$ (33). ∎

##  Appendix C Proof of Proposition 5

###### Proof.

At Heisenberg time $\tau=0$, the initial operator acts as
$\Gamma\otimes\widetilde{\mathds{1}}^{\vphantom{*}}_{\rm ss}$ (i.e., trivially
on all Stinespring registers). The outcome-averaged Heisenberg update to
$\Gamma$ is given by

$\displaystyle\Gamma^{\prime}$ $\displaystyle=\operatorname*{tr}\limits_{{\rm
ss},j}\,\left[\,\mathcal{M}^{\dagger}_{\left[A^{\,}_{j}\right]}\,\Gamma\otimes\widetilde{\mathds{1}}^{\vphantom{*}}_{j}\,\mathcal{M}^{\vphantom{\dagger}}_{\left[A^{\,}_{j}\right]}\,\widetilde{P}^{(0)}_{j}\,\right]\,,~{}~{}$
(C.107)

since the $j$th Stinespring qubit is prepared in the $\lvert 0\rangle$ state
by default. Using Eqs. 16 and B.106, the above becomes

$\displaystyle\Gamma^{\prime}$
$\displaystyle=\sum\limits_{m,n=0,1}\frac{1}{4}\left(\mathds{1}+(-1)^{m}\,\bar{A}^{\vphantom{\prime}}_{j}\right)\,\Gamma\,\left(\mathds{1}+(-1)^{m}\,\bar{A}^{\vphantom{\prime}}_{j}\right)$
$\displaystyle~{}~{}~{}\quad~{}~{}\times~{}{\rm
tr}\left[\,\widetilde{X}^{m}_{j}\,\widetilde{\mathds{1}}^{\vphantom{*}}_{j}\,\widetilde{X}^{n}_{j}\,\widetilde{P}^{(n)}_{j}\,\right]$
$\displaystyle=\frac{1}{4}\sum\limits_{m=0,1}\,\left(\Gamma+(-1)^{m}\,\left\\{\bar{A}^{\,}_{j},\,\Gamma\right\\}+\bar{A}^{\,}_{j}\,\Gamma\,\bar{A}^{\,}_{j}\right)$
$\displaystyle=\frac{1}{4}\sum\limits_{m=0,1}\,\left(\bar{A}^{\,}_{j}+(-1)^{m}\,\mathds{1}\right)\,\left\\{\bar{A}^{\,}_{j},\,\Gamma\right\\}\,,~{}~{}$
(C.108)

and if $\Gamma$ acts trivially (i.e., as the identity $\mathds{1}$) on qubit
$j$, then we find $\Gamma^{\prime}=\Gamma$, meaning that the measurement did
nothing, on average. However, if $\Gamma$ acts on qubit $j$ as
$\sigma^{\nu}_{j}$, then Eq. C.108 can be rewritten

$\displaystyle\Gamma^{\prime}$
$\displaystyle=\frac{1}{2}\sum\limits_{m=0,1}\,(-1)^{m}\,P^{(m)}_{j}\,\sum\limits_{\mu=1}^{3}\,\hat{\alpha}^{\vphantom{\prime}}_{j,\mu}\,\left\\{\sigma^{\mu}_{j},\,\sigma^{\nu}_{j}\right\\}$
$\displaystyle=\sum\limits_{m=0,1}\,(-1)^{m}\,P^{(m)}_{j}\,\hat{\alpha}^{\vphantom{\prime}}_{j,\nu}=\operatorname*{tr}\limits_{j}\,\left[\,\bar{A}^{\vphantom{\prime}}_{j}\,\sigma^{\nu}_{j}\,\right]\,\bar{A}^{\vphantom{\prime}}_{j}\,,~{}~{}$
(C.109)

which is neither involutory nor unitary, and thus incompatible with the
teleportation conditions (8) imposed by, e.g., Def. 6 and Prop. 1. To see
this, note that $\Gamma^{\prime}$ squares to ${\rm
tr}\left[\bar{A}^{\,}_{j}\,\sigma^{\nu}_{j}\,\right]\,\mathds{1}$, which
generically has operator norm (11) less than one. However, if
$\bar{A}^{\,}_{j}=\sigma^{\mu}_{j}$, then $\Gamma^{\prime}$ (C.109) is either
identically zero or equal to $\Gamma$ (if $\mu=\nu$).

In other words, if we evolve any logical operator $\Gamma$ (or Pauli-string
observable of interest) under a dilated unitary channel $\mathcal{M}$
corresponding to projective measurement of a single-qubit observable
$A^{\,}_{j}$ _without_ outcome-dependent recovery operations, then the
Heisenberg-evolved operator $\Gamma^{\prime}$ (C.107) realizes one of the
following:

  1. 1.

If the operator $\Gamma$ and $A^{\,}_{j}$ share no support, then
$\Gamma^{\prime}=\Gamma$, meaning that $\mathcal{M}$ acts trivially.

  2. 2.

If the involutory part of $A^{\,}_{j}$ and the Pauli content of $\Gamma$ on
qubit $j$ are both $\sigma^{\mu}_{j}$, then $\Gamma^{\prime}=\Gamma$, so that
$\mathcal{M}$ acts trivially.

  3. 3.

If the involutory part of $A^{\,}_{j}$ and the Pauli content of $\Gamma$ on
qubit $j$ correspond to _distinct_ Pauli operators, then $\Gamma^{\prime}=0$,
meaning that teleportation fails because the logical operator is destroyed.

  4. 4.

If the involutory part of $A^{\,}_{j}$ realizes a superposition of Pauli
operator (33), then $\Gamma^{\prime}$ is no longer involutory nor unitary, and
thus cannot reproduce the logical operator algebra (8) acting on the initial
state, which is required by Prop. 1.

In all cases above, either the measurement channel $\mathcal{M}$ acts
trivially on a Pauli string $\Gamma$ of interest, or it causes teleportation
to fail (because $\Gamma$ fails to be unitary). Hence, measurement without
outcome-dependent recovery operations is not useful to quantum teleportation.
∎

##  Appendix D Proof of Proposition 6

###### Proof.

By construction, the logical reduced density matrix

$\rho^{\vphantom{\prime}}_{\rm
L}\equiv\operatorname*{tr}_{F^{c}}\left[\mathcal{W}\,\rho^{\vphantom{\prime}}_{0}\otimes\lvert\boldsymbol{0}\rangle\langle\boldsymbol{0}\rvert^{\vphantom{\prime}}_{\rm
ss}\,\mathcal{W}^{\dagger}\right]\,,~{}~{}$ (D.110)

is a pure state, and separable with respect to the partition $F|F^{c}$. In
particular, $\lvert F\rvert=k$ is the number of logical qubits, and
$\rho^{\,}_{\rm L}$ (D.110) is guaranteed to be unitarily equivalent to the
pure state
$\bigotimes_{n=1}^{k}\lvert\psi^{\,}_{n}\rangle\langle\psi^{\,}_{n}\rvert^{\,}_{n}$.
Because $F^{c}$ includes all Stinespring qubits, $\rho^{\,}_{\rm L}$ (D.110)
is implicitly averaged over the outcomes of all measurements in $\mathcal{W}$.

Additionally, the logical reduced density matrix $\rho^{\,}_{\rm L}$ (D.110)
after realizing the sequence of measurement outcomes
$\boldsymbol{n}=(n^{\,}_{1},\dots,n^{\,}_{M})$ is given by Friedman  _et al._
(2022b); Barberena and Friedman (2023); Friedman  _et al._ (2022a)

$\displaystyle\rho^{(\boldsymbol{n})}_{\rm
L}\equiv\frac{\operatorname*{tr}_{F^{c}}\left[\,\rho^{\vphantom{\prime}}_{T}\,\widetilde{P}^{(\boldsymbol{n})}_{\rm
ss}\,\right]}{\operatorname*{tr}_{\rm
dil}\left[\,\rho^{\vphantom{\prime}}_{T}\,\widetilde{P}^{(\boldsymbol{n})}_{\rm
ss}\,\right]}\,,~{}~{}$ (D.111)

where the denominator is equal to the the probability
$p^{\,}_{\boldsymbol{n}}$ of the “outcome trajectory” $\boldsymbol{n}$.

Thus, the outcome-averaged logical reduced density matrix $\rho^{\,}_{\rm L}$
(D.110) is automatically a convex sum over trajectories $\boldsymbol{n}$ of
the reduced density matrix $\rho^{(\boldsymbol{n})}_{\rm L}$ (D.111) weighted
by the scalar $p^{\,}_{\boldsymbol{n}}$. However, such a density matrix is
pure _if and only if_ every term in the sum is proportional to $\rho^{\,}_{\rm
L}$ (D.110). Because $\rho^{\,}_{\rm L}$ (D.110) and
$\rho^{(\boldsymbol{n})}_{\rm L}$ (D.111) are proportional and normalized for
all $\boldsymbol{n}$, we must have that $\rho^{(\boldsymbol{n})}_{\rm
L}=\rho^{\vphantom{\prime}}_{\rm L}$ for all $\boldsymbol{n}$. Hence, the
output state $\rho^{\,}_{\rm L}$ (D.110) is realized on _all_ outcome
trajectories. ∎

##  Appendix E Proof of Lemma 7

###### Proof.

Let $\mathcal{R}^{\,}_{s}$ be a recovery channel (34) and $\Gamma$ to be be
any logical operator. By construction (and Def. 5), both $\chi^{\,}_{s}$ (34)
and $\Gamma$ are Pauli-string operators (2). Evolving $\Gamma$ under
$\mathcal{R}^{\,}_{s}$ in the Heisenberg picture gives

$\displaystyle\mathcal{R}^{\dagger}_{s}\,\Gamma\otimes\widetilde{\mathds{1}}^{\vphantom{*}}\,\mathcal{R}^{\vphantom{\dagger}}_{s}$
$\displaystyle=\sum\limits_{m,n=0,1}\,\chi^{m}_{s}\,\Gamma\,\chi^{n}_{s}\otimes\widetilde{P}^{(m)}_{r^{\,}_{s}}\,\widetilde{\mathds{1}}^{\vphantom{*}}\,\widetilde{P}^{(n)}_{r^{\,}_{s}}$
$\displaystyle=\sum\limits_{m=0,1}\,\chi^{m}_{s}\,\Gamma\,\chi^{m}_{s}\otimes\widetilde{P}^{(m)}_{r^{\,}_{s}}$
$\displaystyle=\sum\limits_{m=0,1}\,\left(-1\right)^{m\,\lambda^{\,}_{\Gamma,s}}\,\Gamma\otimes\widetilde{P}^{(m)}_{r^{\,}_{s}}$
$\displaystyle=\Gamma\otimes\left(\delta^{\,}_{\lambda^{\,}_{\Gamma,s},0}\,\widetilde{\mathds{1}}^{\vphantom{*}}+\delta^{\,}_{\lambda^{\,}_{\Gamma,s},1}\,\widetilde{Z}^{\vphantom{*}}_{r^{\,}_{s}}\right)$
$\displaystyle=\Gamma\otimes\widetilde{Z}^{\lambda^{\,}_{\Gamma,s}}_{r^{\,}_{s}}\,,~{}~{}$
(E.112)

where $\widetilde{Z}^{\vphantom{*}}_{r^{\,}_{s}}$ is the product of
$\widetilde{Z}^{\vphantom{*}}_{j}$ for $j\in r^{\,}_{s}$, where

$\chi^{\vphantom{\dagger}}_{s}\,\Gamma\,\chi^{\vphantom{\dagger}}_{s}=\left(-1\right)^{\lambda^{\vphantom{\prime}}_{\Gamma,s}}\,\Gamma\,,~{}~{}$
(E.113)

defines $\lambda^{\vphantom{\prime}}_{\Gamma,s}\in\\{0,1\\}$ for each $\Gamma$
and $\mathcal{R}^{\,}_{s}$.

Next, we prove Eq. 38 for a given Pauli string $\Gamma$. The result (E.112)
establishes that evolving $\Gamma$ under any $\mathcal{R}$ multiplies $\Gamma$
by Stinespring operators $\widetilde{Z}^{\vphantom{*}}_{j}$ for $j\in
r^{\,}_{s}$. Hence, any recovery channel $\mathcal{R}^{\,}_{\Gamma}$ (38)
attaches $\widetilde{Z}^{\vphantom{*}}_{r}$ to $\Gamma$ provided that
$\left\\{\chi^{\,}_{\Gamma},\,\Gamma\right\\}=0$. The particular choice

$\chi^{\vphantom{\dagger}}_{\Gamma}=\prod\limits_{s=1}^{n}\chi^{\lambda^{\vphantom{\prime}}_{\Gamma,s}}_{s}\,,~{}~{}$
(E.114)

always fulfils this condition, where
$\lambda^{\vphantom{\prime}}_{\Gamma,s}\in\\{0,1\\}$ is defined in Eq. E.113
and $n$ is the number of Clifford recovery channels $\mathcal{R}^{\,}_{s}$.
However, any choice of Pauli string $\chi^{\,}_{\Gamma}$ that anticommutes
with $\Gamma$ leads to a channel $\mathcal{R}^{\,}_{\Gamma}$ (38) that is
equivalent to $\mathcal{R}$ in its action on $\Gamma$.

Next, consider a set of logical operators $\boldsymbol{\Gamma}$, consisting
of, e.g., all $X^{\vphantom{*}}$\- and $Z^{\vphantom{*}}$-type logical
operators for the $k$ logical qubits. Again, Eq. E.112 establishes that
$\mathcal{R}$ acts by attaching (possibly different)
$\widetilde{Z}^{\vphantom{*}}_{j}$ operators to different logical operators
$\Gamma\in\boldsymbol{\Gamma}$. However, by the previous result (38),
$\mathcal{R}$ is equivalent in its action on each
$\Gamma\in\boldsymbol{\Gamma}$ to some $\mathcal{R}^{\,}_{\Gamma}$ (38) where
$\chi^{\,}_{\Gamma}$ is _any_ Pauli string that anticommutes with $\Gamma$.
Hence, we can write $\mathcal{R}$ as a product over
$\Gamma\in\boldsymbol{\Gamma}$ of channels $\mathcal{R}^{\,}_{\Gamma}$ (38)
provided that $\chi^{\,}_{\Gamma}$ anticommutes only with the logical operator
$\Gamma$, and not its counterpart $\Gamma^{\prime}$, which acts on the same
logical qubit. Since
$\left\\{\Gamma,\,\Gamma^{\prime}\right\\}=\left[\Gamma^{\prime},\,\Gamma^{\prime}\right]=0$,
the choice $\chi^{\,}_{\Gamma}=\Gamma^{\prime}$ is sufficient to ensure that
Eq. 37 is equivalent to $\mathcal{R}$ for all logical operators. We note that
the decomposition (37) is not unique. ∎

##  Appendix F Proof of Lemma 8

###### Proof.

The canonical-form protocol $\mathcal{W}=\mathcal{R}\mathcal{M}\mathcal{U}$
(31) involves dilated channels of the form

$\displaystyle\mathcal{R}^{\vphantom{\dagger}}_{j,k}$
$\displaystyle=\sum\limits_{n=0,1}\,\chi^{n}_{j,k}\otimes\widetilde{P}^{(n)}_{j}$
(F.115) $\displaystyle\mathcal{M}^{\vphantom{\dagger}}_{j}\,$
$\displaystyle=\frac{1}{2}\sum\limits_{m=0,1}\left(\mathds{1}+(-1)^{m}\,\bar{A}^{\prime}_{j}\right)\otimes\widetilde{X}^{m}_{j}\,,~{}~{}$
(F.116)

where $\chi^{\,}_{j,k}\in\mathbf{P}^{\vphantom{*}}_{*}{\rm ph}$ is a physical
Pauli string, $\bar{A}_{j}^{\prime}$ is the involutory part (33) of an
operator of the form

$A_{j}^{\prime}=U^{\vphantom{\dagger}}_{j}\,A^{\,}_{j}U^{\dagger}_{j}$ (F.117)

where $A^{\,}_{j}$ is a single-qubit observable, and $U^{\,}_{j}$ is a generic
physical unitary. Note that Eq. F.117 captures modifications due to writing
$\mathcal{W}^{\prime}$ in canonical form (31).

The above form (F.116) of the measurement channels is guaranteed by Prop. 4.
Additionally, by Defs. 5 and 6, all outcome-dependent recovery operations are
captured by Eq. F.115. All other channels in $\mathcal{W}$ correspond to
physical unitaries, captured by $\mathcal{U}$ in Eq. 37.

Consider the Heisenberg evolution of the Pauli string

$\Gamma(0)=\Gamma=\sigma^{\nu}_{f_{n}}\otimes\widetilde{\mathds{1}}^{\vphantom{*}}_{\rm
ss}\in\mathbf{P}^{\vphantom{*}}_{\rm ph}\,,~{}~{}$ (F.118)

which acts as a logical operator on the final state
$\lvert\Psi^{\,}_{T}\rangle$ (7). Evolving this operator in the Heisenberg
picture, projecting onto the default Stinespring initial state
$\lvert\boldsymbol{0}\rangle^{\,}_{\rm ss}$, and averaging over outcomes leads
to Eq. 8, i.e.,

$\Gamma^{\vphantom{\prime}}_{\rm ph}(T)=\operatorname*{tr}\limits_{\rm
ss}\left[\,\mathcal{W}^{\dagger}\,\Gamma(0)\,\mathcal{W}\,\widetilde{P}^{(\boldsymbol{0})}_{\rm
ss}\,\right]=\sigma^{\nu}_{i_{n}}\mathsf{S}^{\vphantom{\dagger}}_{n,\nu}\,,~{}~{}$
(F.119)

which is the same logical operator $\Gamma$ acting on the initial state
$\lvert\Psi^{\,}_{0}\rangle$ (6), where $\mathsf{S}^{\,}_{n,\nu}$ is an
element of the initial-state stabilizer group
$\operatorname*{stab}\left(\lvert\Psi^{\,}_{0}\rangle\right)\subset\operatorname*{Aut}\left(*\right){\mathcal{H}^{\,}_{\rm
ph}}$ (3).

Importantly, the naïve protocol $\mathcal{W}$ need not be in canonical form
(31). Without loss of generality, we have omit any measurements without
corresponding feedback and convert trivial recovery operations into physical
unitaries, as described above. However, for convenience, we also pull all non-
Clifford physical unitaries into $\mathcal{U}$, and if the combination of a
measurement channel $\mathcal{M}^{\,}_{j}$ is conditioned on a prior outcome
in the naïve protocol $\mathcal{W}$, then we pull prior recovery channels
through $\mathcal{M}^{\,}_{j}$ as needed so that it realizes Eq. F.116. This
is not necessary if $\mathcal{M}^{\,}_{j}$ commutes with all prior recovery
channels (in the Schrödinger picture) or if $A^{\,}_{j}$ is a Pauli string.

As a result, all operations after the first measurement channel (in the
Schrödinger picture) correspond either to measurements or Clifford channels.
Hence, in the Heisenberg picture, any logical operator
$\Gamma(\tau^{\,}_{j,k})$ prior to the recovery channel $\mathcal{R}^{\,}_{s}$
is guaranteed to be a Pauli string, provided that no measurement channels have
been encountered. Note that any number of recovery operations conditioned on
various measurement outcomes can be factorized into the desired form (F.115),
up to Clifford gates $U\in\mathbf{C}^{\vphantom{*}}_{*}{\rm ph}$, by Def. 5
and Lemma 7.

Suppose that $\mathcal{M}^{\,}_{j}$ is the first measurement channel
encountered in the Heisenberg picture. By Lemma 7,

$\mathcal{R}^{\dagger}_{j,k}\,\Gamma(\tau^{\vphantom{\prime}}_{j,k})\,\mathcal{R}^{\vphantom{\dagger}}_{j,k}=\Gamma(\tau^{\vphantom{\prime}}_{j,k})\,\widetilde{Z}^{\lambda^{\vphantom{\prime}}_{\Gamma,j,k}}_{j}\,,~{}~{}$
(F.120)

where $\lambda^{\,}_{\Gamma,j,k}$ is defined via

$\chi^{\vphantom{\dagger}}_{j,k}\,\Gamma(\tau^{\vphantom{\prime}}_{j,k})\,\chi^{\vphantom{\dagger}}_{j,k}=\left(-1\right)^{\lambda^{\vphantom{\prime}}_{\Gamma,j,k}}\,\Gamma(\tau^{\vphantom{\prime}}_{j,k})\,,~{}~{}$
(E.113)

so that the physical part of $\Gamma(\tau^{\,}_{j,k})$ is unaffected by the
recovery channel, while the Stinespring part is multiplied by the operator
$\widetilde{Z}^{\vphantom{*}}_{j}$ if and only if the recovery Pauli
$\chi^{\,}_{j,k}$ anticommutes with $\Gamma(\tau^{\,}_{j,k})$ immediately
prior to the recovery channel $\mathcal{R}^{\,}_{j,k}$ (in the Heisenberg
picture). If $\lambda^{\,}_{\Gamma,j,k}=0$ in Eq. E.113 then Eq. F.120 is
trivial, as claimed.

We then observe that recovery channels conditioned on other measurement
outcomes also act on $\Gamma$ via Eq. F.120. Additionally, only the recovery
channels $\mathcal{R}^{\,}_{j,k}$ encountered prior to the measurement channel
$\mathcal{M}^{\,}_{j}$ (in the Heisenberg picture) are actually sensitive to
the measurement outcome, and hence we convert all other “trivial” recovery
channels into physical unitaries. We then define

$\Lambda^{\,}_{\Gamma,j}=\sum_{k}\lambda^{\,}_{j,k}~{}{\rm mod}~{}2\,,~{}~{}$
(F.121)

which captures whether or not the combination of _all_ recovery channels
$\mathcal{R}^{\,}_{j,k}$ conditioned on the $j$th outcome act nontrivially on
the logical operator $\Gamma$ via Eq. F.120.

We can then write the _physical part_ of the time-evolved logical operator
$\Gamma(t^{\,}_{j})$ immediately prior to the measurement channel
$\mathcal{M}^{\,}_{j}$ (in the Heisenberg picture) as

$\Gamma(t^{\vphantom{\prime}}_{j})=U^{\dagger}_{M}\,\Gamma\,U^{\vphantom{\dagger}}_{M}\,,~{}~{}$
(F.122)

where $U^{\,}_{M}$ is the product of all physical Clifford gates $U$ and the
outcome-independent parts of all recovery-channels in $\mathcal{W}$ after the
measurement channel $\mathcal{M}^{\,}_{j}$ in the Schrödinger picture. Hence,
it is the same unitary $U^{\,}_{j}$ that appears in Eq. F.117. The Stinespring
part of $\Gamma$ is

$\widetilde{\Gamma}(t^{\vphantom{\prime}}_{j})=\widetilde{Z}^{\Lambda^{\vphantom{\prime}}_{\Gamma,j}}_{j}\otimes\cdots\,,~{}~{}$
(F.123)

where $\cdots$ captures $\widetilde{Z}^{\vphantom{*}}$ operators attached by
recovery channels conditioned on the outcomes of other measurements.

Evolving $\Gamma$ (F.122) through the measurement channel
$\mathcal{M}^{\,}_{j}$ results in a logical operator $\Gamma$ with physical
part

$\displaystyle\Gamma(t^{\prime}_{j})$
$\displaystyle=\operatorname*{tr}\limits_{{\rm
ss},j}\left[\,\mathcal{M}^{\dagger}_{j}~{}\Gamma(t^{\vphantom{\prime}}_{j})\otimes\widetilde{Z}^{\Lambda^{\vphantom{\prime}}_{\Gamma,j}}_{j}~{}\mathcal{M}^{\vphantom{\dagger}}_{j}\,\widetilde{P}^{(0)}_{j}\,\right]$
$\displaystyle=\frac{1}{4}\sum\limits_{m,n=0,1}\left(\mathds{1}+(-1)^{m}\,\bar{A}^{\vphantom{\prime}}_{j}\right)\Gamma(t^{\vphantom{\prime}}_{j})\left(\mathds{1}+(-1)^{n}\,\bar{A}^{\vphantom{\prime}}_{j}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\left\langle*\middle|0\middle|\widetilde{X}^{m}_{j}\,\widetilde{Z}^{\Lambda^{\vphantom{\prime}}_{\Gamma,j}}_{j}\,\widetilde{X}^{n}_{j}\right\rangle{0}$
$\displaystyle=\sum\limits_{n=0,1}\frac{\left(-1\right)^{n\,\Lambda^{\vphantom{\prime}}_{\Gamma,j}}}{4}\left(\bar{A}^{\vphantom{\prime}}_{j}+\left(-1\right)^{n}\mathds{1}\right)\left\\{\bar{A}^{\vphantom{\prime}}_{j},\,\Gamma(t^{\vphantom{\prime}}_{j})\right\\}$
$\displaystyle=\frac{1}{2}~{}\bar{A}^{\,\Lambda^{\vphantom{\prime}}_{\Gamma,j}-1}_{j}~{}\left\\{\bar{A}^{\vphantom{\prime}}_{j},\,\Gamma(t^{\vphantom{\prime}}_{j})\right\\}\,,~{}~{}$
(F.124)

where we have traced out the $j$th Stinespring qubit to average over
measurement outcomes and projected onto the default initial state $\lvert
0\rangle$, since no subsequent channels act nontrivially on this qubit. The
remaining Stinespring content of $\Gamma(t^{\prime}_{j})$ corresponds to the
$\cdots$ terms in Eq. F.123.

Importantly, the operator $\Gamma(t_{j}^{\prime})$ (F.124) must be
_involutory_ , which leads to the condition

$\displaystyle\mathds{1}$
$\displaystyle=\Gamma^{2}(t_{j}^{\prime})=\frac{1}{2}\mathds{1}+\frac{1}{4}\left\\{\Gamma(t^{\vphantom{\prime}}_{j}),\,\bar{A}^{\vphantom{\prime}}_{j}\,\Gamma(t^{\vphantom{\prime}}_{j})\,\bar{A}^{\vphantom{\prime}}_{j}\right\\}\,,~{}$
(F.125)

independent of $\Lambda^{\,}_{\Gamma,j}$, leading to

$\displaystyle 0$
$\displaystyle=\Gamma(t^{\vphantom{\prime}}_{j})\,\bar{A}^{\vphantom{\prime}}_{j}\,\Gamma(t^{\vphantom{\prime}}_{j})\,\bar{A}^{\vphantom{\prime}}_{j}+\bar{A}^{\vphantom{\prime}}_{j}\,\Gamma(t^{\vphantom{\prime}}_{j})\,\bar{A}^{\vphantom{\prime}}_{j}\,\Gamma(t^{\vphantom{\prime}}_{j})-2\,\mathds{1}$
$\displaystyle=\left[\Gamma(t^{\vphantom{\prime}}_{j}),\,\bar{A}^{\vphantom{\prime}}_{j}\right]^{2}\,,~{}~{}$
(F.126)

and denoting this commutator as

$B=\left[\Gamma(t^{\vphantom{\prime}}_{j}),\,\bar{A}^{\vphantom{\prime}}_{j}\right]=-B^{\dagger}\,,~{}~{}$
(F.127)

we see that $B^{2}=B^{\dagger}B=BB^{\dagger}=0$, meaning that $B$ is a normal
matrix; since $B^{\dagger}B\geq 0$, the only solution to $B^{\dagger}B=0$ is
$B=0$. Alternatively, we note that $B$ is both skew-Hermitian and nilpotent;
the former implies that $B$ has purely imaginary eigenvalues, while the latter
implies that all complex eigenvalues of $B$ are zero.

Either way, the result is the same, and

$\left[A^{\vphantom{\prime}}_{j},\,\Gamma(t^{\vphantom{\prime}}_{j})\right]=0\,,~{}~{}$
(39)

for _all_ logical operators $\Gamma$ evolved to the time $t^{\,}_{j}$ (F.122)
immediately prior to the measurement of $A^{\,}_{j}$, in the Heisenberg
picture. Given Eq. 39, Eq. F.124 becomes

$\Gamma(t^{\prime}_{j})=\bar{A}^{\,\Lambda^{\vphantom{\prime}}_{\Gamma,j}}_{j}~{}\Gamma(t^{\vphantom{\prime}}_{j})\,,~{}~{}$
(F.128)

and the same result recovers from Eq. F.124 in the case where
$\Gamma(t^{\,}_{j})$ has no support on site $j$. We note that the latter
scenario also satisfies the commutation condition (39); hence, both Eqs. 39
and F.128 hold generally.

Thus, successful teleportation requires that the measured observable
$A^{\,}_{j}$ commutes with _all_ logical operators $\Gamma(t^{\,}_{j})$
evolved in the Heisenberg picture to the time $t^{\,}_{j}$ immediately prior
to the corresponding measurement (39). Then, the combination of measuring
$A^{\,}_{j}$ and all recovery operations $\mathcal{R}^{\,}_{j,k}$ (F.115)
conditioned on the outcome attaches the involutory part $\bar{A}^{\,}_{j}$
(33) of the measured observable $A^{\,}_{j}$ to $\Gamma(t^{\,}_{j})$ (F.122)
via Eq. F.128, provided that an odd number of recovery Pauli operators
$\chi^{\,}_{j,k}$ anticommute with the corresponding Heisenberg-evolved
logical operators $\Gamma(\tau^{\,}_{j,k})$ (E.113) immediately prior to the
$k$th recovery channel conditioned on the $j$th outcome—i.e.,
$\Lambda^{\,}_{\Gamma,j}=1$ (F.121). This holds for each logical operator
$\Gamma$.

We now restore the Stinespring operators in the $\cdots$ term in Eq. F.123,
and repeat this procedure for the remaining measurements. We note that
subsequent recovery operations (in the Heisenberg picture) generally commute
with $\bar{A}^{\,}_{j}$, as described below Eq. F.119. The other possibility
is that $\bar{A}^{\,}_{j}$ is a Pauli string that anticommutes with one or
more $\chi^{\,}_{i,\ell}$ recovery operators (F.115); however, this remains
compatible with the update in Eq. F.120, since the operator conjugated by the
channel $\mathcal{R}^{\,}_{i,\ell}$ is still a Pauli string (until all
recovery operations have concluded, all physical unitaries are elements of
$\mathbf{C}^{\vphantom{*}}_{*}{\rm ph}$). Thus, all subsequent recovery
channels $\mathcal{R}^{\,}_{i,\ell}$ (in the Heisenberg picture) either act
trivially on $\bar{A}^{\,}_{j}$ in Eq. F.119, or $\bar{A}^{\,}_{j}$ is a Pauli
string that can be absorbed into $\Gamma(t^{\,}_{j})$.

These properties are preserved by all subsequent gates and measurements. When
all recovery channels have been encountered, we allow generic (i.e., non-
Clifford) physical unitaries to appear in $\mathcal{W}$, which may be absorbed
into the unitary operators $U$ in Eq. F.122. Since only the conjugation by
recovery channels $\mathcal{R}^{\,}_{i,\ell}$ is sensitive to having a Pauli-
string input, this does not pose an obstacle to evolving under any remaining
measurement channels. It is straightforward to verify that the same conditions
that applied to prior recovery and measurement channels also apply to all
remaining dilated channels.

Finally, we note that in a canonical-form protocol
$\mathcal{W}=\mathcal{R}\,\mathcal{M}\,\mathcal{U}$, by Lemma 7,
$\Gamma(\tau^{\,}_{j})=\Gamma$ for all $j\in[1,M]$. However, the operators
$\Gamma(t^{\,}_{j})$ may be equal to $\Gamma$ times up to $M-j$ involutory
operators, per Eq. F.128. ∎

##  Appendix G Proof of Proposition 9

###### Proof.

We first consider the $k=1$ case described in the statement of Prop. 9. Since
$\mathcal{W}$ only contains a single projective measurement of an effectively
involutory observable $\bar{A}$ (33) by Prop. 4, there is only one associated
Stinespring qubit. There can also be no outcome-dependent measurement
channels. Written in canonical form
$\mathcal{W}=\mathcal{R}\mathcal{M}\mathcal{U}$ (31), we have

$\displaystyle\mathcal{R}=\mathds{1}\otimes\widetilde{P}^{(0)}+\chi\otimes\widetilde{P}^{(1)}$
(G.129)

for some Pauli-string $\chi$ (2). By Lemma 8, acting on a logical operator
$\Gamma$ in the Heisenberg picture, the combination of $\mathcal{R}$ and
$\mathcal{M}$ either attaches $\bar{A}$ or acts trivially.

Consider the logical operators $X^{\vphantom{*}}_{{\rm L}}$ and
$Z^{\vphantom{*}}_{{\rm L}}$, and Eq. 8 of Prop. 1. If the dilated channels
attach $\bar{A}$ to $X^{\vphantom{*}}_{{\rm L}}$ alone, then
$Z^{\vphantom{*}}_{{\rm L}}$ only grows under $\mathcal{U}$, so that $L\leq
vT$ for that operator, and hence the logical qubit. The same holds if we
switch $X^{\vphantom{*}}_{{\rm L}}$ and $Z^{\vphantom{*}}_{{\rm L}}$. If the
dilated channels attach $\bar{A}$ to _both_ $X^{\vphantom{*}}_{{\rm L}}$ and
$Z^{\vphantom{*}}_{{\rm L}}$, then they attach $\bar{A}^{2}=\mathds{1}$ to
$Y^{\vphantom{*}}_{{\rm L}}$, which therefore obeys $L\leq vT$. Hence, a
single measurement cannot be used to teleport all three logical basis
operators a distance $L>vT$. For $k=M=1$, this can also be proven for _any_
physical teleportation protocol by showing that Hermiticity of
$\mathsf{S}^{\,}_{\nu}$ is equivalent to
$\left[\sigma^{\nu}_{i},\,\mathcal{U}^{\dagger}\,\bar{A}\,\mathcal{U}\right]=0$
for all $\nu$, which implies that at most two logical operators can exceed the
Lieb-Robinson bound (28).

Next, consider a standard teleportation protocol $\mathcal{W}$ with $k\geq 1$
with $M<2k$. For $\mathcal{W}$ to succeed with fidelity one, the teleportation
conditions (8) of Prop. 1 must be satisfied for all $k$ _pairs_ of logical
operators. To exceed the Lieb-Robinson bound $L\leq vT$ (28), distinct
stabilizers $\mathsf{S}^{\,}_{n,\nu}$ must realize for all $2k$ logical basis
operators—equivalently, a different involutory operator $\bar{A}^{\,}_{j}$
must be attached to all $2k$ logical basis operators. Otherwise, one can
define another valid set of logical (basis) operators by taking products of
logical operators and/or their associated stabilizers
$\mathsf{S}^{\,}_{n,\nu}$ to find one or more logical operators that satisfy
$\sigma^{\nu}_{i_{n}}=\mathcal{U}^{\dagger}\,\sigma^{\nu}_{f_{n}}\,\mathcal{U}$
(8), which can only be satisfied if $L\leq d(i_{n},f_{n})\leq vT$, as with
$k=1$ Friedman  _et al._ (2022a).

Alternatively, since Lemma 10 _independently_ proves that two measurements are
sufficient to teleport a logical qubit a distance $L>vT$, a protocol with
$M<2k$ would require that _at least_ one logical qubit is teleported a
distance $L>vT$ using a single measurement, which violates the $k=1$ result
proven above. ∎

##  Appendix H Proof of Lemma 10

###### Proof.

Again, we first consider the $k=1$ case, and generalize to $k>1$ afterward.
For concreteness, consider the Heisenberg evolution of the logical operators
$X^{\vphantom{*}}_{{\rm L}}$ and $Z^{\vphantom{*}}_{{\rm L}}$, which act on
the final state $\lvert\Psi^{\,}_{T}\rangle$ (7) as $X^{\vphantom{*}}_{f}$ and
$Z^{\vphantom{*}}_{f}$, respectively. In this context, Lemma 7 guarantees that

$\mathcal{W}=\mathcal{R}^{\vphantom{\dagger}}_{x}\,\mathcal{R}^{\vphantom{\dagger}}_{z}\,\mathcal{M}^{\vphantom{\dagger}}_{2}\,\mathcal{M}^{\vphantom{\dagger}}_{1}\,\mathcal{U}\,,~{}~{}$
(H.130)

where the order of the _effective_ recovery channels $\mathcal{R}^{\,}_{x,z}$
is inconsequential, and $\mathcal{M}$ and $\mathcal{U}$ are unmodified.

Evolving the logical operators through the recovery channels $\mathcal{R}$ in
the Heisenberg picture leads to

$\displaystyle\sigma^{\nu}_{\rm L}(\tau)$
$\displaystyle=\mathcal{R}^{\dagger}\,\sigma^{\nu}_{f}\,\mathcal{R}=\sigma^{\nu}_{f}\,\widetilde{Z}^{\lambda^{\vphantom{\prime}}_{\nu,1}}_{1}\,\widetilde{Z}^{\lambda^{\vphantom{\prime}}_{\nu,2}}_{2}\,,~{}~{}$
(H.131)

where $\lambda^{\vphantom{\prime}}_{\nu,i}=1$ if an odd number of recovery
operators $\chi^{\,}_{i,k}$ based on the $i$th measurement outcome anticommute
with $\sigma^{\nu}_{f}$, and is zero otherwise (for $\nu=x,y,z$). Evolving the
operator $\sigma^{\nu}_{\rm L}(\tau)$ through the two measurement of
$A^{\,}_{2}$ (and projecting onto the default state $\lvert 0\rangle$ of the
corresponding Stinespring register) leads to

$\displaystyle\sigma^{\nu}_{\rm L}(t^{\,}_{2})$
$\displaystyle=\operatorname*{tr}\limits_{{\rm
ss},2}\left[\,\mathcal{M}^{\dagger}_{2}\,\sigma^{\nu}_{\rm
L}(\tau)\,\mathcal{M}^{\vphantom{\dagger}}_{2}\,\widetilde{P}^{(0)}_{2}\,\right]$
$\displaystyle=\sigma^{\nu}_{f}\,A^{\lambda^{\vphantom{\prime}}_{\nu,2}}_{2}\,\widetilde{Z}^{\lambda^{\vphantom{\prime}}_{\nu,1}}_{1}\,,~{}~{}$
(H.132)

where Lemma 8 imposes the condition

$\left[A^{\vphantom{\prime}}_{2},\,\sigma^{\nu}_{\rm
L}(\tau)\right]=\left[A^{\vphantom{\prime}}_{2},\,\sigma^{\nu}_{f}\right]=0\,,~{}~{}$
(H.133)

and evolving through the remaining measurement gives

$\displaystyle\sigma^{\nu}_{\rm L}(t^{\,}_{1})$
$\displaystyle=\operatorname*{tr}\limits_{{\rm
ss},2}\left[\,\mathcal{M}^{\dagger}_{1}\,\sigma^{\nu}_{\rm
L}(t^{\,}_{2})\,\mathcal{M}^{\vphantom{\dagger}}_{1}\,\widetilde{P}^{(0)}_{1}\,\right]$
$\displaystyle=\sigma^{\nu}_{f}\,A^{\lambda^{\vphantom{\prime}}_{\nu,2}}_{2}\,A^{\lambda^{\vphantom{\prime}}_{\nu,1}}_{1}\,,~{}~{}$
(H.134)

where, now, Lemma 8 imposes the condition

$\left[A^{\vphantom{\prime}}_{1},\,\sigma^{\nu}_{\rm
L}(t^{\,}_{2})\right]=\left[A^{\vphantom{\prime}}_{1},\,\sigma^{\nu}_{f}\,A^{\lambda^{\vphantom{\prime}}_{\nu,2}}_{2}\right]=0\,,~{}~{}$
(H.135)

and we now consider the conditions of the Lemma.

First, $\lambda^{\,}_{\nu,i}=1$ must hold for at least one logical operator
labelled $\nu$, for both $i=1,2$; otherwise, one of the measurements acts
trivially by Prop. 5, and the other is then useless to $\mathcal{W}$ by Prop.
9. Second, if $\lambda^{\,}_{\nu,i}=1$ for both $i=1,2$ and $\nu=x,z$, e.g.,
then $X^{\vphantom{*}}_{{\rm L}}(t^{\,}_{1})$ and $Z^{\vphantom{*}}_{{\rm
L}}(t^{\,}_{1})$ are both multiplied by $A^{\,}_{2}\,A^{\,}_{1}$; however,
this is equivalent to a _single_ measurement of the involutory operator
$B=A^{\,}_{2}\,A^{\,}_{1}$, which cannot be useful to teleportation by Prop.
9. Third, if $\lambda^{\,}_{\nu,i}=0$ for $i=1,2$ for _any_ $\nu$, then that
logical operator must be teleported by $\mathcal{U}$ alone, in which case
$L\leq vT$, violating the conditions of the Lemma. Hence, $\mathcal{W}$ must
attach one of the observables $A^{\,}_{1,2}$ to $X^{\vphantom{*}}_{{\rm L}}$
and the other to $Z^{\vphantom{*}}_{{\rm L}}$ (or any pair of logical
operators).

Suppose that $\mathcal{W}$ attaches $A^{\,}_{2}$ to $Z^{\vphantom{*}}_{{\rm
L}}$ and $A^{\,}_{1}$ to $X^{\vphantom{*}}_{{\rm L}}$—i.e.,
$\lambda^{\,}_{z,2}=\lambda^{\,}_{x,1}=1$ while
$\lambda^{\,}_{x,2}=\lambda^{\,}_{z,1}=0$. The commutation conditions imposed
by Lemma 8 take the form

$\displaystyle\left[A^{\vphantom{\prime}}_{2},\,X^{\vphantom{*}}_{f}\right]$
$\displaystyle=\left[A^{\vphantom{\prime}}_{1},\,X^{\vphantom{*}}_{f}\right]=0$
(H.136a)
$\displaystyle\left[A^{\vphantom{\prime}}_{2},\,Z^{\vphantom{*}}_{f}\right]$
$\displaystyle=\left[A^{\vphantom{\prime}}_{1},\,Z^{\vphantom{*}}_{f}A^{\vphantom{\prime}}_{2}\right]=0\,,~{}~{}$
(H.136b)

and we focus on the final relation in Eq. H.136b, which, by linearity of the
commutator can be written

$\left[A^{\vphantom{\prime}}_{1},\,Z^{\vphantom{*}}_{f}A^{\vphantom{\prime}}_{2}\right]=\left[A^{\vphantom{\prime}}_{1},\,Z^{\vphantom{*}}_{f}\right]\,A^{\vphantom{\prime}}_{2}+Z^{\vphantom{*}}_{f}\,\left[A^{\vphantom{\prime}}_{1},\,A^{\vphantom{\prime}}_{2}\right]=0\,.~{}~{}$
(H.137)

We note that $A^{\vphantom{\prime}}_{1}$ _only_ fails to commute with an
operator on site $f$ if, in the bare protocol $\mathcal{W}^{\prime}$ (which
need not be in canonical form), the single-qubit observable $A_{1}^{\prime}$
either (_i_) is supported only on site $f$ or (_ii_) is supported on some site
$j\neq f$ where both $j$ and $f$ lie in the support of some physical unitary
$U$ that acts after $\mathcal{M}_{1}^{\prime}$ (in the Schrödinger picture).
In either case, $A_{1}^{\prime}$ lies in the light cone of site $f$ generated
by time evolution alone. Hence, attaching such an operator to any logical
operator $\sigma^{\nu}_{\rm L}$ cannot be more useful to teleportation than
time evolution alone, which obeys $L\leq vT$ (28). However, Prop. 9
establishes that the measurement of $A^{\,}_{2}$ cannot lead to $L>vT$ on its
own.

Hence, both $A^{\,}_{1}$ and $A^{\,}_{2}$ must act trivially on $f$ (in
canonical form or otherwise), so that Eq. H.137 becomes

$\left[A^{\vphantom{\prime}}_{1},\,A^{\vphantom{\prime}}_{2}\right]=0\,,~{}~{}$
(41)

meaning that two observables $A^{\,}_{1}$ and $A^{\,}_{2}$ are mutually
compatible if they commute. This also implies that the channels
$\mathcal{M}^{\,}_{1}$ and $\mathcal{M}^{\,}_{2}$ commute, so their order in
Eq. H.130 is inconsequential. Moreover, all of the arguments above hold if we
replace $X^{\vphantom{*}}_{{\rm L}}$ and $Z^{\vphantom{*}}_{{\rm L}}$ with any
pair of logical operators. Note that Eq. 41 must hold in the bare protocol and
in canonical form (H.130).

Having proven the Lemma for $k=1$ (i.e., as stated), we now establish that
$M=2k$ measurements is sufficient to teleport $k\geq 1$ logical qubits a
distance $L>vT$. Again, we assume a standard teleportation protocol
$\mathcal{W}$ in canonical form (31). Lemma 7 guarantees that

$\displaystyle\mathcal{W}=\Big{[}\prod\limits_{n=1}^{k}\,\mathcal{R}^{\vphantom{\dagger}}_{n,x}\,\mathcal{R}^{\vphantom{\dagger}}_{n,x}\Big{]}\,\mathcal{M}^{\vphantom{\dagger}}_{2k}\,\cdots\mathcal{M}^{\vphantom{\dagger}}_{1}\,\mathcal{U}\,,~{}$
(H.138)

acting on logical operators, where the order of the _effective_ recovery
channels $\mathcal{R}^{\,}_{n,\nu}$ (38) is unimportant, and both
$\mathcal{M}$ and $\mathcal{U}$ are unchanged compared to the true protocol.

We next repeat the analysis used in the $k=1$ case, noting that the
combination of $\mathcal{R}$ and $\mathcal{M}$ attaches observables to logical
operators, per Lemma 8. By Prop. 9, the same observable cannot be attached to
both logical basis operators for a given logical qubit $n$. Moreover, the
$k=1$ case above establishes that the attached operator $\bar{A}^{\,}_{s}$
(the measured observable) evolves under $\mathcal{U}$ into the initial logical
operator $\sigma^{\nu}_{i_{n}}$ (and some stabilizer element). Hence, if the
same operator $\bar{A}^{\,}_{s}$ is attached to logical operators for the
logical qubits $n$ _and_ $n^{\prime}$, then at least one of the logical
operators $n$ and $n^{\prime}$ acts nontrivially on _both_ $i_{n}$ and
$i_{n^{\prime}}$, violating Eq. 8. Hence, we assign a unique observable
$A^{\,}_{s}$ to each logical basis operator.

The first measured observable encountered in the Heisenberg picture is
$A^{\,}_{2k}$; by Lemma 8, we have that

$\displaystyle\left[A^{\vphantom{\prime}}_{2k},\,\sigma^{\nu}_{f_{n}}\right]=0~{}~{}\forall\,n,\nu\,,~{}~{}$
(H.139)

and since the label $n$ is meaningless, suppose that $\bar{A}^{\,}_{2k}$ is
attached to $X^{\vphantom{*}}_{f_{k}}$ (i.e., $k$th logical $X^{\vphantom{*}}$
operator).

We next encounter the observable $A^{\,}_{2k-1}$, finding

$\displaystyle\left[A^{\vphantom{\prime}}_{2k-1},\,X^{\vphantom{*}}_{f_{k}}\,A^{\vphantom{\prime}}_{2k}\right]=0\,,~{}~{}$
(H.140)

and note that $A^{\,}_{2k-1}$ commutes with all other logical operators.
Following Eq. H.137, this implies that

$\left[A^{\vphantom{\prime}}_{2k-1},\,A^{\vphantom{\prime}}_{2k}\right]=0\,,~{}~{}$
(H.141)

in analogy to Eq. 41. The logical operator to which $A^{\,}_{2k-1}$ is
attached is unimportant, as long as each measured observable is attached to a
unique logical operator.

As a result, every subsequent observable obeys one or more expressions like
Eq. H.140 by Lemma 8. Since the measured observables always commute with the
logical operators (which is required for $L>vT$), the observable
$A^{\,}_{2k-s}$ obeys Eq. H.140 for the $s$ logical operators to which prior
observables have been attached, and commutes with all other logical operators.
Each of the $s$ expressions of the form of Eq. H.140 reduce to Eq. H.141 for
each of the measured observables already encountered. Repeating this for all
$2k$ measurement channels, we find that

$\displaystyle\left[A^{\vphantom{\prime}}_{j},\,A^{\vphantom{\prime}}_{j^{\prime}}\right]=0~{}~{}\forall\,j,j^{\prime}\,,~{}~{}$
(H.142)

i.e., all measured observables mutually commute. The proof that a protocol
$\mathcal{W}$ with $2k$ measurements satisfying Eq. H.142 can teleport $k$
logical qubits a distance $L>vT$ is straightforward: One simply invokes the
remainder of the proof for $k=1$ for each logical qubit individually. ∎

##  Appendix I Proof of Lemma 11

###### Proof.

The proof of Eq. 43 involves several steps, each of which utilizes the limits
on operator growth imposed by the Lieb-Robinson bound (28) in Theorem 2, and
builds on the teleportation conditions established in Prop. 1,

$\displaystyle\sigma^{\nu}_{\rm L}(T)$
$\displaystyle\equiv\operatorname*{tr}\limits_{\rm
ss}\left[\,\mathcal{W}^{\dagger}\,\sigma^{\nu}_{f}\,\mathcal{W}\,\widetilde{P}^{(\boldsymbol{0})}_{\rm
ss}\,\right]$
$\displaystyle=\mathcal{U}^{\dagger}\,A^{\vphantom{\prime}}_{\nu,j}\,\sigma^{\nu}_{f}\,\mathcal{U}=\sigma^{\nu}_{i}\,\mathsf{S}^{\vphantom{\prime}}_{\nu}\,,~{}~{}$
(I.143)

where $\sigma^{\nu}_{\rm L}(T)$ is physical part of the $\nu$-type logical
operator acting on the _initial_ state, which we take to be a product state,
the final equality follows from Eq. 8, and Lemma 10 guarantees that exactly
one of the two observables is attached to the $\sigma^{\nu}_{\rm L}$, which we
denote by

$A^{\vphantom{\prime}}_{\nu,j}\equiv
U^{\vphantom{\dagger}}_{\nu}\,\bar{A}_{\nu
j}^{\prime}\,U^{\dagger}_{\nu}\,,~{}~{}$ (32)

where $\bar{A}_{j}^{\prime}$ is the involutory part (33) of the single-qubit
observable $A^{\,}_{j}$ acting on qubit $j$ (which depends on $\nu$), and
$U^{\,}_{\nu}$ comprises all physical unitaries $U$ _after_ the measurement of
$A_{\nu j}^{\prime}$ in the naïve protocol $\mathcal{W}^{\prime}$, in the
Schrödinger picture. We can then write
$\mathcal{U}=U^{\,}_{\nu}\,\mathcal{U}^{\,}_{\nu}$, so that
$\mathcal{U}^{\,}_{\nu}$ comprises all physical unitaries in
$\mathcal{W}^{\prime}$ _prior_ to the measurement of $A_{\nu j}^{\prime}$,
where $U^{\,}_{\nu}$ has depth $\tau^{\,}_{\nu}$, $\mathcal{U}^{\,}_{\nu}$ has
depth $T^{\,}_{\nu}$, and $T=\tau^{\,}_{\nu}+T^{\,}_{\nu}$. We then rewrite
Eq. I as

$\sigma^{\nu}_{i}\,\mathsf{S}^{\vphantom{\prime}}_{\nu}=\left(\mathcal{U}^{\dagger}_{\nu}\,\bar{A}_{\nu
j}^{\prime}\,\mathcal{U}^{\vphantom{\dagger}}_{\nu}\right)\,\left(\mathcal{U}^{\dagger}\,\sigma^{\nu}_{f}\,\mathcal{U}\right)\,,~{}~{}$
(I.144)

and it will prove convenient to define

$\Sigma^{\nu}_{\ell}\equiv\mathcal{U}^{\dagger}\,\sigma^{\nu}_{f}\,\mathcal{U}\,,~{}~{}$
(I.145)

and the Lemma’s requirement that $L>vT$ implies that $d(\ell,i)\geq 1$, so
$\Sigma^{\nu}_{\ell}$ cannot be $\sigma^{\nu}_{\rm L}(T)$ (I).

Next, we establish by contradiction that the light cones of $\bar{A}_{\nu
j}^{\prime}$ and $\sigma^{\nu}_{f}$ in Eq. I.144 have to meet. If the light
cones do not meet, then since $\Sigma^{\nu}_{\ell}$ cannot realize
$\sigma^{\nu}_{i}$, it must realize part of the stabilizer element
$\mathsf{S}^{\,}_{\nu}$ in Eq. I.144, while
$\mathcal{U}^{\dagger}_{\nu}\,\bar{A}_{\nu
j}^{\prime}\,\mathcal{U}^{\vphantom{\dagger}}_{\nu}$ realizes
$\sigma^{\nu}_{i}$ times another part of $\mathsf{S}^{\,}_{\nu}$, so that

$\displaystyle\mathsf{S}^{\vphantom{\prime}}_{\nu}$
$\displaystyle=\mathsf{S}^{(1)}_{\nu}\otimes\mathsf{S}^{(2)}_{\nu}=\sigma^{\nu}_{i}\,\mathcal{U}^{\dagger}_{\nu}\,\bar{A}_{\nu
j}^{\prime}\,\mathcal{U}^{\vphantom{\dagger}}_{\nu}\otimes\Sigma^{\nu}_{\ell}\,,~{}~{}$
(I.146)

where
$\mathsf{S}^{(1)}_{\nu},\mathsf{S}^{(2)}_{\nu}\in\operatorname*{stab}\left(\lvert\Psi^{\,}_{0}\rangle\right)$
(3) have disjoint support.

Note that any two elements $\mathsf{S}^{\,}_{1},\mathsf{S}^{\,}_{2}$ of
$\operatorname*{stab}\left(*\right){\lvert\Psi\rangle}$ satisfy

$\displaystyle\lVert\left\\{\mathsf{S}^{\vphantom{*}}_{1},\,\mathsf{S}^{\vphantom{*}}_{2}\right\\}\rVert=2\,,~{}~{}$
(I.147)

which we prove for all
$\mathsf{S}^{\,}_{1},\mathsf{S}^{\,}_{2}\in\operatorname*{stab}\left(*\right){\lvert\Psi\rangle}$.
First, note that
$\left\\{\mathsf{S}^{\,}_{1},\,\mathsf{S}^{\,}_{2}\right\\}\,\lvert\Psi\rangle=2\,\lvert\Psi\rangle$,
which implies that
$\lVert\left\\{\mathsf{S}^{\,}_{1},\,\mathsf{S}^{\,}_{2}\right\\}\rVert\geq 2$
(since there exists an eigenvalue of the anticommutator with eigenvalue at
least two). Second, note that unitarity of the stabilizer elements—along with
the triangle inequality—ensure that
$\lVert\left\\{\mathsf{S}^{\,}_{1},\,\mathsf{S}^{\,}_{2}\right\\}\rVert\leq
2\,\lVert\mathsf{S}^{\,}_{1}\,\mathsf{S}^{\,}_{2}\rVert=2$.

Crucially, the initial state $\lvert\Psi^{\,}_{0}\rangle$ (6) is either a
product state, or is separable with respect to the support of the two
stabilizer elements. However, Eq. I.146 implies that

$\left\\{\mathsf{S}^{(2)}_{\mu},\,\mathsf{S}^{(2)}_{\nu}\right\\}=\mathcal{U}^{\dagger}\,\left\\{\sigma^{\mu}_{f},\,\sigma^{\nu}_{f}\right\\}\,\mathcal{U}=2\,\delta^{\,}_{\mu,\nu}\,,~{}~{}$
(I.148)

which is incompatible with Eq. I.147 for $\mu\neq\nu$, contradicting the claim
that $\mathsf{S}^{(2)}_{\nu}$ are elements of
$\operatorname*{stab}\left(\lvert\Psi^{\,}_{0}\rangle\right)$ (3). Hence, the
light cones of the operators $\bar{A}_{\nu j}^{\prime}$ and $\sigma^{\nu}_{f}$
must meet.

Moreover, the operators $\sigma^{\nu}_{\rm L}(T)$ (I) must realize the Pauli
algebra on site $i$ only, and commute everywhere else. The most efficient
strategy to ensure this is to evolve the commuting operators $A_{\nu
j}^{\prime}$ (for $\nu=x,z$) to nearby sites (using a circuit with depth
$\tau^{\,}_{0}$, and apply a unitary “decoding” channel $\mathcal{Q}$ that
maps, e.g.,

$A^{\prime}_{\nu j}\to\sigma^{\nu}_{j}\,\sigma^{\nu}_{j+v}\,.~{}~{}$ (I.149)

As established in Ref. 36, the most efficient strategy is to evolve
$\sigma^{\nu}_{j}\to\sigma^{\nu}_{i}$ and
$\sigma^{\nu}_{j+v}\to\sigma^{\nu}_{\ell-v}$, with

$\mathcal{U}^{\dagger}\,\sigma^{\nu}_{f}\,\mathcal{U}=\Sigma^{\nu}_{\ell}=\sigma^{\nu}_{\ell}\,,~{}~{}$
(I.150)

so that $\sigma^{\nu}_{\ell}\,\sigma^{\nu}_{\ell-v}$ can be converted into
stabilizers of $\lvert\Psi^{\,}_{0}\rangle$ (6) for both $\nu=x,z$ by the
“encoding” channel $\mathcal{Q}^{\dagger}$.

Using the constraints above, we bound the teleportation distance $L=d(i,f)$
(13). We first note that

$f-\ell\leq v\left(T-1\right)\,,~{}~{}$ (I.151)

since at least one layer of the circuit must convert $\sigma^{\nu}_{\ell-v}$
and $\sigma^{\nu}_{\ell}$ into a stabilizer. We then note that

$\displaystyle j-i$ $\displaystyle\leq
v\,\left(T-\max_{\nu}\tau^{\vphantom{\prime}}_{\nu}-\tau^{\vphantom{\prime}}_{0}-1\right)$
(I.152a) $\displaystyle\ell-j$ $\displaystyle\leq
v\left(T-\max_{\nu}\tau^{\vphantom{\prime}}_{\nu}-\tau^{\vphantom{\prime}}_{0}\right)\,,~{}~{}$
(I.152b)

where $\max_{\nu}\tau^{\,}_{\nu}$ is the total depth of all physical unitary
channels following the first measurement (in the Schrödinger picture). By
summing the foregoing inequalities, we obtain the following bound on $L$,

$L=d(i,f)\leq
3vT-2v\left(\max_{\nu}\tau^{\,}_{\nu}+\tau^{\vphantom{\prime}}_{0}+1\right)\,,$
(I.153)

where the depth $\tau^{\,}_{0}\geq 0$ increases if the measurements of
$A_{\nu,j}^{\prime}$ are either (_i_) far from one another (i.e.,
$d(j^{\,}_{1},j^{\,}_{2})\gg 1$) or (_ii_) far from the optimal position
$j^{*}\approx i+T/3$. Taking $\tau^{\,}_{0}\to 0$ in Eq. I.153 gives the
(potentially loose) bound

$L=d(i,f)\leq
3vT-2v\left(\max_{\nu}\tau^{\vphantom{\prime}}_{\nu}+1\right)\,.~{}$ (43)

∎

##  Appendix J Proof of Lemma 12

###### Proof.

The proof of Lemma 12 closely resembles that of Lemma 11 (see App. I). Again,
the light cone emanating from one pair of measurements must intersect the
light cone of the final-state logical operators $\sigma^{\nu}_{f}(T)$ in the
Heisenberg picture, while the light cone emanating from a (generically
distinct) pair of measurements must reach the initial logical site $i$. For
$M=4$, the light cones of these two measurement regions must intersect each
other; for $M=2m$ with $m>2$, we instead “daisy chain” the various light cones
of the measured observables, in series.

As in the $M=2$ case described by Lemma 12, we need only consider the
_optimal_ scenario. This means that pairs of measurements are well separated,
and all of the measured observables commute with one another and all logical
operators. This applies both to the canonical-form observables $A^{\,}_{j}$
(32) and the single-qubit observables $A_{j}^{\prime}$ of the naïve protocol
$\mathcal{W}^{\prime}$. Evolving the logical through the dilated channels
$\mathcal{R}$ and $\mathcal{M}$ leads to

$\sigma^{\nu}_{f}(t)=\sigma^{\nu}_{f}\,\prod\limits_{s=1}^{M}\,U^{\vphantom{\dagger}}_{s,\nu}\,\bar{A}^{\vphantom{\dagger}}_{s,\nu}\,U^{\dagger}_{s,\nu}\,,~{}~{}$
(J.154)

where $s$ runs over pairs of measurements (one of which is assigned to
$X^{\vphantom{*}}_{{\rm L}}$ and the other to $Z^{\vphantom{*}}_{{\rm L}}$),
and $U^{\,}_{s,\nu}$ captures all unitary evolution following (in the
Schrödinger picture) the measurement of the single-qubit operator $A_{s,\nu}$
in the _naïve_ protocol $\mathcal{W}^{\prime}$ (and has depth
$\tau^{\,}_{s,\nu}$).

Evolving through $\mathcal{W}$ again leads to Eq. I.150 for the
$\sigma^{\nu}_{f}$ term in Eq. J.154, which is moved to some site $\ell$
obeying

$f-\ell\leq v\left(T-1\right)\,,~{}~{}$ (I.151)

as before. Again, it is optimal to measure nearby sites—e.g.,
$j^{\,}_{s,z}=j^{\,}_{s,x}+v$, as in Fig. 2 and Eq. I.149. We denote by
$j^{\,}_{s}$ the left site of the pair $s$. As in Lemma 12, we have

$\displaystyle d(j^{\vphantom{\prime}}_{1},i)$ $\displaystyle\leq
v\,\left(T-\max_{\nu}\tau^{\vphantom{\prime}}_{1,\nu}-1\right)$ (I.152a)
$\displaystyle d(\ell,j^{\vphantom{\prime}}_{m})$ $\displaystyle\leq
v\left(T-\max_{\nu}\tau^{\vphantom{\prime}}_{m,\nu}-1\right)\,,~{}~{}$
(I.152b)

where $m=M/2$ is the number of pairs of nearby measurements, and we take
$\tau^{\vphantom{\prime}}_{0}=0$ compared to Eq. I.152. The new ingredient
compared to Lemma 12 is

$\displaystyle d(j^{\vphantom{\prime}}_{s+1},j^{\vphantom{\prime}}_{s})\leq
v\left(T-\max_{\nu}\max(\tau^{\vphantom{\prime}}_{s,\nu},\tau^{\vphantom{\prime}}_{s+1,\nu})-1\right)\,,$
(J.155)

and adding these all up—and taking $\tau^{\,}_{s,\nu}=0$ to reflect optimal
protocols—we find that

$\displaystyle L$ $\displaystyle=d(i,f)\leq
3v(T-1)+2v\sum\limits_{s=1}^{m-1}(T-1)\,,~{}$ (J.156)

and noting that adding a single additional measurement cannot alter this
bound, we replace $m=\lfloor M/2\rfloor$ to recover

$\displaystyle L\leq
vT+2v\left\lfloor\frac{M}{2}\right\rfloor\left(T-1\right)\,.~{}$ (46)

∎

##  Appendix K Proof of Theorem 13

###### Proof.

To prove Eq. 47, we derive the optimal spacing of measurements for a given
logical qubit $n$, as in the proofs of Lemmas 11 and 12. This is
straightforward, as Lemmas 7 and 8 jointly guarantee that the combination of
measurements and outcome-dependent operations do not “mix” between logical
operators. Moreover, Lemma 10—whose proof in App. H applies to all $k\geq
1$—guarantees that all of the measured observables $A^{\,}_{j}$ (32)—and their
involutory parts $\bar{A}^{\,}_{j}$ (33)—commute with one another and with all
final-state logical operators, by Lemma 8.

Evolving all logical operators through the channels $\mathcal{R}$ and
$\mathcal{M}$ in the Heisenberg picture, projecting onto the default
Stinespring initial state $\lvert\boldsymbol{0}\rangle^{\,}_{\rm ss}$, and
tracing out the Stinespring degrees of freedom leads to

$\displaystyle\sigma^{\nu}_{f_{n}}(t)$
$\displaystyle=\operatorname*{tr}\limits_{\rm
ss}\left[\,\mathcal{M}^{\dagger}\,\mathcal{R}\,\sigma^{\nu}_{f_{n}}\,\mathcal{R}\,\mathcal{M}\,\widetilde{P}^{(\boldsymbol{0})}_{\rm
ss}\,\right]$
$\displaystyle=\bar{A}^{\vphantom{\prime}}_{1,n,\nu}\cdots\bar{A}^{\vphantom{\prime}}_{m,n,\nu}\,\sigma^{\nu}_{f_{n}}\,,~{}~{}$
(K.157)

where the first label on $\bar{A}^{\,}_{s,n,\nu}$ corresponds to the
“measurement region” $s\in[1,m]$ Friedman  _et al._ (2022a), $n$ labels the
logical qubit, and $\nu$ labels the logical basis operator (e.g.,
$\nu\in\\{x,z\\}$).

We now work out the optimal spacing between measurement sites
$j^{\,}_{s,n,\nu}$ for different $s$. Following the proofs in Apps. I and J,
we note that the logical basis operators $\sigma^{\nu}_{f_{n}}(t)$ (K.157)
still realize the Pauli algebra on site $f_{n}\gg i_{n}$. As before, the
strategy is for the pair observables $A^{\,}_{s,n,x}$ and $A^{\,}_{s,n,z}$ to
be measured on nearby sites $j^{\,}_{s,n}$ and $j^{\,}_{s,n}+v$, and then
converted into a pair of operators of the same type ($\nu$) under the next
unitary channel encountered in the Heisenberg picture—i.e.,

$\displaystyle\bar{A}^{\vphantom{\prime}}_{s,n,\nu}\to\sigma^{\nu}_{j^{\vphantom{\prime}}_{s,n}}\,\sigma^{\nu}_{j^{\vphantom{\prime}}_{s,n}+v}~{}~{}~{}{\rm
and}~{}~{}~{}\sigma^{\nu}_{f_{n}}\to\sigma^{\nu}_{f_{n}-v}\,,~{}~{}$ (K.158)

which can always be achieved in duration $\Delta T\geq 1$ Friedman  _et al._
(2022a).

As before, the next step is to send the pair of $\sigma^{\nu}$ operators
created on nearby sites via (K.158) in opposite directions. Note that we need
to “save” $\Delta T\geq 1$ of the total unitary evolution to “undo” Eq. K.158.
The next segment of the Heisenberg evolution of the logical operator (K.157),
with duration $T-2$, sends $f_{n}-v\to f_{n}-v(T-1)$, $j^{\,}_{s,n}\to
j^{\,}_{s,n}-v(T-1)$, and $j^{\,}_{s,n}+v\to j^{\,}_{s,n}+v(T-1)$. At this
point, the sites of anticommutation must be “undone” (since the initial state
is a product state, by Def. 6), which is possible using the inverse of the
channel that achieves Eq. K.158. For this to succeed—and for
$\sigma^{\nu}_{f_{n}}(T)$ to realize the Pauli algebra on site $i_{n}$—we must
have

$\displaystyle d(f^{\vphantom{\prime}}_{n},j^{\vphantom{\prime}}_{m,n})$
$\displaystyle\leq 2vT-v$ (K.159a) $\displaystyle
d(j^{\vphantom{\prime}}_{s+1,n},j^{\vphantom{\prime}}_{s,n})$
$\displaystyle\leq 2v(T-1)$ (K.159b) $\displaystyle
d(j^{\vphantom{\prime}}_{1,n},i^{\vphantom{\prime}}_{n})$ $\displaystyle\leq
v(T-1)\,,~{}~{}$ (K.159c)

where the middle line holds for $1\leq s<m$. These relations follow from the
requirement that all light cones come within $v$ of one another prior to the
final channel in the Heisenberg picture (with depth $\Delta T\geq 1$), except
that of the leftmost $\sigma^{\nu}$ operator generated in Eq. K.158, which can
propagate an extra $v$ sites to $i_{n}$. Putting this all together, a protocol
with $m$ measurement regions teleports the $n$th logical qubit a distance $L$
(13) satisfying

$\displaystyle L\leq vT+2mv(T-1)\,,~{}~{}$ (K.160)

and now, we consider the remaining $k-1$ logical qubits.

To proceed, we first recall several previously established facts. First, Prop.
9 proves that (_i_) the _same_ measured observables cannot be attached to
distinct logical basis operators and remain useful to teleportation and (_ii_)
a given region requires _two_ measurements per logical qubit. This implies
that each of the $m$ measurement “regions” must involve $2k$ single-qubit
measurements; moreover, pairs of measurements corresponding to the same
logical operator should neighbor one another (i.e., be no further than $v$
from each other). The optimal strategy is to have the sites $i_{n}$ (and
$f_{n}$) for different $n$ correspond to alternating sites (in order), and for
each measurement region to comprise $2k$ consecutive sites, of which the pair
of sites $2n-1,2n$ correspond to the observables attached to the logical basis
operators for the $n$th logical qubit. As a result, the total number of
measurements required is $M=2mk$; by Prop. 9 and Def. 2, only an additional
$2k$ measurements can increase $L$, and thus we replace $m$ with $\lfloor
M/2k\rfloor$ in Eq. K.160 to recover Eq .47.

Finally, we prove Eq. 48. As always, the maximum distance between equivalent
sites $j^{\,}_{s,n}$ and $j^{\,}_{s+1,n}$ is $\ell\leq 2v(T-1)$. However, we
note that at least $2k$ sites must be measured in each region; hence, we must
have $\ell\geq 2k$. Combining these inequalities, we find that $2v(T-1)\geq
2k$, which simplifies to Eq. 48. In the case $k=1$, we find $T\geq 2$, as
noted in Lemmas 11 and 12. ∎

##  Appendix L Proof of Corollary 14

###### Proof.

The first important result of Lemma 8 is

$\left[A^{\vphantom{\prime}}_{j},\,\Gamma(t^{\vphantom{\prime}}_{j})\right]=0\,,~{}~{}$
(39)

where $\Gamma(t^{\,}_{j})$ is some final-state logical operator $\Gamma$
evolved to the (Heisenberg) time $t^{\,}_{j}$ immediately prior to the
measurement of $A^{\,}_{j}$. The other important result is

$\Gamma(t^{\vphantom{\prime}}_{j-1})=\operatorname*{tr}_{{\rm
ss}~{}j}\left[\,\mathcal{M}^{\dagger}_{j}\,\Gamma(t^{\vphantom{\prime}}_{j})\,\mathcal{M}^{\vphantom{\dagger}}_{j}\,\right]=\bar{A}^{\lambda^{\,}_{j,\Gamma}}_{j}\,\Gamma(t^{\vphantom{\prime}}_{j})\,,~{}$
(40)

where $\lambda^{\,}_{j,\Gamma}\in\\{0,1\\}$ is determined by the recovery
channels conditioned on the state of the $j$th Stinespring qubit, and
$\bar{A}^{\,}_{j}$ is the involutory part (33) of the observable $A^{\,}_{j}$.

The final measured observable $A^{\,}_{M}$ therefore satisfies

$\displaystyle\left[A^{\vphantom{\prime}}_{M},\,\sigma^{\nu}_{f}\right]=0\,,~{}~{}$
(L.161)

for all final logical sites $f\in F$ and all $\nu\in\\{x,y,z\\}$, which
requires that $A^{\,}_{M}$ has no support in $F$ (since only $\mathds{1}_{f}$
commutes with $\sigma^{\nu}_{f}$ for all nontrivial values of $\nu$).

The penultimate observable $A^{\,}_{M-1}$ also has no support on any logical
site $f\in F$ to which $A^{\,}_{M}$ is not attached; for any logical operator
$\Gamma$ to which $A^{\,}_{M}$ _is_ attached,

$\displaystyle 0$
$\displaystyle=\left[A^{\vphantom{\prime}}_{M-1},\,\bar{A}^{\vphantom{\prime}}_{M}\,\Gamma\right]$
$\displaystyle=\left[A^{\vphantom{\prime}}_{M-1},\,\bar{A}^{\vphantom{\prime}}_{M}\right]\,\Gamma+\bar{A}^{\vphantom{\prime}}_{M}\,\left[A^{\vphantom{\prime}}_{M-1},\,\Gamma\right]\,,~{}$
(L.162)

where at least one of the two commutators in the sum above must vanish, as
$A^{\,}_{M-1}$ (32) remains a single-qubit operator in the canonical-form
protocol $\mathcal{W}$ (31) by the requirement of Def. 6 that no unitaries be
applied to previously measured sites. Hence, both commutators must vanish,
meaning that

$\left[A^{\,}_{M-1},\,A^{\,}_{M}\right]=0\,,~{}~{}$ (L.163)

and also that $A^{\,}_{M-1}$ has no support in $F$.

Repeating this analysis for all subsequent observables $A^{\,}_{j}$, we again
find constraints of the same form as Eq. L.162, with $\bar{A}^{\,}_{M}$
replaced by some product of observables $\bar{A}^{\vphantom{\prime}}_{j+1}$
through $\bar{A}^{\vphantom{\prime}}_{M}$. For each $j<M$, we find that none
of the measured observables previously encountered in the Heisenberg evolution
of $\Gamma$ have support in $F$. As a result, Eq. 39 can be split into a
vanishing sum of two commutators à la Eq. L.162, where at least one of the
commutators must be zero. Hence, both commutators must vanish, meaning that
$A^{\,}_{j}$ has no support in $F$ and

$\left[A^{\vphantom{\prime}}_{j},\,\bar{A}^{\lambda_{j+1,\Gamma}}_{j+1}\cdots\bar{A}^{\lambda_{M,\Gamma}}_{M}\right]=0\,,~{}~{}$
(L.164)

meaning that $A^{\,}_{j}$ commutes with _every_ product of observables
$A^{\,}_{j+1}$ through $A^{\,}_{M}$ attached to _any_ logical operator
$\Gamma=\sigma^{\nu}_{f}$ via Eq. 40 of Lemma 8.

Importantly, all of these observables are single-qubit operators, and
trivially commute unless they act on the same qubit. By Eq. L.164, either
_all_ observables correspond to different sites $j\notin F$ _or_ , following
the measurement of $A^{\,}_{j}$ (in the Schrödinger picture), two observables
that anticommute with $A^{\,}_{j}$ are subsequently measured _and_ both are
attached to the same logical operators. However, these observables must
commute with each other, and hence must be the same, so the product of their
involutory parts (33) is the identity. Hence, this pair of measurements can be
omitted entirely. Any measurements intermediate to this pair act on other
sites, and hence, trivially commute.

This also agrees with the basic intuition that the subsequent anticommuting
measurements necessarily result in random outcomes, yet the final logical
state must be the same along _every_ outcome trajectory, meaning that no
nontrivial operations can be conditioned on the outcomes of the measurements
that anticommute with $A^{\,}_{j}$. More important, however, is the fact that
this anticommuting operator is effectively removed from _all_ logical
operators prior to encountering $A^{\,}_{j}$ (in the Heisenberg picture), and
hence, these measurements may be omitted entirely without affecting any
commutation condition (L.164) on the observables or the evolution of any
logical operator. ∎

##  Appendix M Proof of Lemma 15

###### Proof.

The resource state is given by

$\lvert\Psi^{\vphantom{\prime}}_{t}\rangle=\mathcal{U}\,\lvert\Psi^{\vphantom{\prime}}_{0}\rangle\,,~{}~{}$
(36)

where
$\lvert\Psi^{\,}_{0}\rangle=\lvert\psi\rangle\otimes\lvert\Phi^{\,}_{0}\rangle$
(6) is the initial state, with $\lvert\Phi^{\,}_{0}\rangle$ a product state by
Def. 3. Importantly, the teleportation conditions (8) of Prop. 1 guarantee
that

$\sigma^{\nu}_{i}\,\mathsf{S}^{\vphantom{\prime}}_{\nu}=\operatorname*{tr}_{\rm
ss}\left[\,\mathcal{W}^{\dagger}\,\sigma^{\nu}_{f}\,\mathcal{W}\,\widetilde{P}^{(\boldsymbol{0})}_{\rm
ss}\,\right]\,,~{}~{}$ (8)

where
$\mathsf{S}^{\,}_{\nu}\,\lvert\Psi^{\,}_{0}\rangle=\lvert\Psi^{\,}_{0}\rangle$
is a stabilizer
$\mathsf{S}^{\,}_{\nu}\in\operatorname*{stab}\left(\lvert\Psi^{\,}_{0}\rangle\right)$.

The logical operator that acts on the resource state is

$\displaystyle\sigma^{\nu}_{f}(T-t)$ $\displaystyle=\operatorname*{tr}_{\rm
ss}\left[\,\mathcal{M}^{\dagger}\,\mathcal{R}^{\dagger}\,\sigma^{\nu}_{f}\,\mathcal{R}\,\mathcal{M}\,\widetilde{P}^{(\boldsymbol{0})}_{\rm
ss}\,\right]$
$\displaystyle=\bar{A}_{1}^{\lambda^{\,}_{1,\nu}}\cdots\bar{A}_{M}^{\lambda^{\,}_{M,\nu}}\,\sigma^{\nu}_{f}=\bar{A}^{\vphantom{\prime}}_{\nu}\,\sigma^{\nu}_{f}$
$\displaystyle=\left(\mathcal{U}\,\sigma^{\nu}_{i}\,\mathcal{U}^{\dagger}\right)\,\left(\mathcal{U}\,\mathsf{S}^{\vphantom{\prime}}_{\nu}\,\mathcal{U}^{\dagger}\right)\,,~{}~{}$
(M.165)

by Eqs. 8 and 40, and we define

$\displaystyle\mathsf{S}^{\vphantom{\prime}}_{\nu,t}$
$\displaystyle\equiv\mathcal{U}\,\mathsf{S}^{\,}_{\nu}\,\mathcal{U}^{\dagger}$
(M.166)

which is an element of the (unitary) stabilizer group
$\operatorname*{stab}\left(\lvert\Psi^{\,}_{t}\rangle\right)$ for the
_resource_ state, since

$\displaystyle\mathsf{S}^{\vphantom{\prime}}_{\nu,t}\,\lvert\Psi^{\vphantom{\prime}}_{t}\rangle$
$\displaystyle=\left(\mathcal{U}\,\mathsf{S}^{\vphantom{\prime}}_{\nu}\,\mathcal{U}^{\dagger}\right)\,\mathcal{U}\,\lvert\Psi^{\vphantom{\prime}}_{0}\rangle$
$\displaystyle=\mathcal{U}\,\mathsf{S}^{\vphantom{\prime}}_{\nu}\,\lvert\Psi^{\vphantom{\prime}}_{0}\rangle=\mathcal{U}\,\lvert\Psi^{\vphantom{\prime}}_{0}\rangle=\lvert\Psi^{\vphantom{\prime}}_{t}\rangle\,,~{}~{}$
(M.167)

as expected. Using Eqs. M.165 and M.166, we have

$\displaystyle\mathsf{S}^{\vphantom{\prime}}_{\nu,t}$
$\displaystyle=\mathcal{U}\,\sigma^{\nu}_{i}\,\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu}\,\sigma^{\nu}_{f}\,,~{}~{}$
(M.168)

and thus, by Eq. M.167, we have that

$\displaystyle 1$
$\displaystyle=\left\langle\Psi^{\vphantom{\prime}}_{t}\middle|\,\mathsf{S}^{\vphantom{\prime}}_{\nu,t}\,\middle|\Psi^{\vphantom{\prime}}_{t}\right\rangle$
$\displaystyle=\left\langle\Psi^{\vphantom{\prime}}_{t}\middle|\,\mathcal{U}\,\sigma^{\nu}_{i}\,\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu}\,\sigma^{\nu}_{f}\,\middle|\Psi^{\vphantom{\prime}}_{t}\right\rangle\,,~{}~{}$
(54)

as stated in the Lemma. The commutation of the “bulk” operator content of the
string order parameter $\mathsf{S}^{\,}_{\nu,t}$ (M.168) is guaranteed by Cor.
14 for all _necessary_ measurements, where we simply omit unnecessary
measurements. The anticommutation of the endpoint operators follows
automatically from the Pauli algebra. ∎

##  Appendix N Proof of Lemma 16

###### Proof.

We first show that there exist subregions of the interval $[i,f]$ with
different _left_ endpoints such that a string order parameter analogous to Eq.
54 has unit expectation value in the _same_ resource state
$\lvert\Psi^{\,}_{t}\rangle$ (36). This requires that multiple observables
$\bar{A}^{\,}_{j}$ are attached to both logical operators, which is guaranteed
when $L\gg vT$. We show the same for different _right_ endpoints subsequently.

The combination of Eqs. 8 and 40 imply that

$\displaystyle\sigma^{\nu}_{i}\,\mathsf{S}^{\vphantom{\prime}}_{\nu}$
$\displaystyle=\left(\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu}\,\mathcal{U}\right)\left(\mathcal{U}^{\dagger}\,\sigma^{\nu}_{f}\,\mathcal{U}\right)\,,~{}~{}$
(N.169)

and importantly, the fact that $L>vT$ (as required for physical teleportation
via Def. 3) guarantees that
$\mathcal{U}^{\dagger}\,\sigma^{\nu}_{f}\,\mathcal{U}$ only contributes to the
stabilizer part $\mathsf{S}^{\,}_{\nu}$ of Eq. N.169. Hence, the logical part
$\sigma^{\nu}_{i}$ must recover from evolving the leftmost measured
observables in $\bar{A}^{\,}_{\nu}$ (55) under $\mathcal{U}$.

This product of operators can be rewritten as

$\bar{A}^{\vphantom{\prime}}_{\nu}=\bar{A}^{\vphantom{\prime}}_{\nu,1}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,m}\,,~{}~{}$
(N.170)

where each operator $\bar{A}^{\,}_{\nu,s}$ corresponds to a particular
“measurement region,” and may represent a product of the involutory parts (33)
of two or more measured observables attached to $\sigma^{\nu}_{f}$ via Eq. 40
of Lemma 8.

Using the above, we rewrite Eq. N.169 as

$\displaystyle\sigma^{\nu}_{i}\,\mathsf{S}^{\vphantom{\prime}}_{\nu}$
$\displaystyle=\left(\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,1}\,\mathcal{U}\right)\left(\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,2}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,m}\,\sigma^{\nu}_{f}\,\mathcal{U}\right)\,,~{}$
(N.171)

and since only the leftmost term on the right-hand side is causally connected
to $\sigma^{\nu}_{i}$ via $\mathcal{U}$, we define

$\sigma^{\nu}_{i}\,\Sigma^{\nu}_{1}=\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,1}\,\mathcal{U}\,,~{}~{}$
(N.172)

where $\Sigma^{\nu}_{1}$ acts nontrivially only to the right of site $i$, when
the above is applied to the initial state $\lvert\Psi^{\,}_{0}\rangle$ (6).
Since the right-hand sides of Eq. N.172 for $\nu=x,z$ commute with one
another, the left-hand sides must as well. This implies that, e.g.,
$\Sigma^{x}_{1}$ and $\Sigma^{z}_{1}$ reproduce the Pauli algebra to the right
of $i$, up stabilizer elements of
$\operatorname*{stab}\left(\lvert\Psi^{\,}_{0}\rangle\right)$.

The string order parameter (54) from Lemma 15 can be written as the following
element of $\operatorname*{stab}\left(\lvert\Psi^{\,}_{t}\rangle\right)$,

$\mathsf{S}^{\vphantom{\prime}}_{\nu,t}=\mathcal{U}\,\sigma^{\nu}_{i}\,\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu}\,\sigma^{\nu}_{f}\,,~{}~{}$
(M.168)

which we next massage using Eq. N.172 to find

$\displaystyle\mathsf{S}^{\vphantom{\prime}}_{\nu,t}$
$\displaystyle=\mathcal{U}\,\sigma^{\nu}_{i}\,\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,1}\,\left(\mathcal{U}\,\mathcal{U}^{\dagger}\right)\,\bar{A}^{\vphantom{\prime}}_{\nu,2}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,m}\,\sigma^{\nu}_{f}$
$\displaystyle=\mathcal{U}\,\sigma^{\nu}_{i}\,\left(\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,1}\,\mathcal{U}\right)\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,2}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,m}\,\sigma^{\nu}_{f}$
$\displaystyle=\mathcal{U}\,\sigma^{\nu}_{i}\,\left(\sigma^{\nu}_{i}\,\Sigma^{\nu}_{1}\right)\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,2}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,m}\,\sigma^{\nu}_{f}$
$\displaystyle=\left(\mathcal{U}\,\Sigma^{\nu}_{1}\,\mathcal{U}^{\dagger}\right)\,\bar{A}^{\vphantom{\prime}}_{\nu,2}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,m}\,\sigma^{\nu}_{f}\,,~{}~{}$
(N.173)

which has the same right endpoint operator $\sigma^{\nu}_{f}$ as in Eq. 54,
but the left endpoint operator has changed from

$\mathcal{U}\,\sigma^{\nu}_{i}\,\mathcal{U}^{\dagger}\to\mathcal{U}\,\Sigma^{\nu}_{1}\,\mathcal{U}^{\dagger}\,,~{}~{}$
(N.174)

where the support of $\Sigma^{\nu}_{1}$ is entirely to the right of
$\sigma^{\nu}_{i}$, and thus constitutes an equivalent string order parameter
$\mathsf{S}^{\,}_{\nu,t}$ (N.173) of the same form as Eq. 54 with a _new_ left
endpoint, where the left-endpoint operators for $\nu=x,z$ are guaranteed to
anticommute by Eq. N.172.

Importantly, we observe that the new string-order parameter in Eq. N.173 has
the same important properties as the naïve counterpart (54). Both objects have
(_i_) unit expectation value in the resource state
$\lvert\Psi^{\,}_{t}\rangle$ (36); (_ii_) a right-endpoint operator given by
$\sigma^{\nu}_{f}$; (_iii_) a left-endpoint operator of the form
$\mathcal{U}\,\tau^{\nu}\,\mathcal{U}^{\dagger}$, where the objects
$\tau^{\nu}$ satisfy the Pauli algebra; and (_iv_) all of the operators
between the two endpoint commute for different values of $\nu$.

There are two key features that facilitate iterating this procedure to recover
new endpoints. The first is that the left-endpoint operators (N.174) are of
the same form as the original Pauli operator in Eq. 54. The second is that
there exists a “leftmost” choice of $\Sigma^{\nu}_{1}$ (N.172) such that the
clustered observables $\bar{A}^{\,}_{\nu,3}$ through $\bar{A}^{\,}_{\nu,m}$
are not connected to $\Sigma^{\nu}_{1}$ under $\mathcal{U}$. Essentially, a
set of equivalent operators recovers upon multiplying $\Sigma^{\nu}_{1}$
(N.172) by stabilizer elements $\mathsf{S}^{\,}_{0}$ of
$\operatorname*{stab}\left(\lvert\Psi^{\,}_{0}\rangle\right)$; the “leftmost”
left-endpoint operator (N.174) is the operator
$\Sigma^{\nu}_{1}\,\mathsf{S}^{\,}_{0}$ for
$\mathsf{S}^{\,}_{0}\in\operatorname*{stab}\left(\lvert\Psi^{\,}_{0}\rangle\right)$
whose support is closest to—but disjoint from—the initial site $i$ (i.e.,
whose right weight is closest to $i$).

We find the next left endpoint by defining

$\Sigma^{\nu}_{1}\,\Sigma^{\nu}_{2}=\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,2}\,\mathcal{U}\,,~{}~{}$
(N.175)

in analogy to Eq. N.172. The procedure then repeats as before, where in the
general case we define

$\sigma^{\nu}_{i}\,\Sigma^{\nu}_{s}=\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,1}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,s}\,\mathcal{U}\,,~{}~{}$
(N.176)

which is equivalent to the familiar form

$\Sigma^{\nu}_{s-1}\,\Sigma^{\nu}_{s}=\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,s}\,\mathcal{U}\,,~{}$
(N.177)

where the above two definitions reproduce Eq. 59.

One can iterate this procedure until all bulk operators except
$\bar{A}^{\,}_{\nu,m}$ have been exhausted. Choosing the leftmost form of each
$\Sigma^{\nu}_{s}$ guarantees that each $\Sigma^{\nu}_{s}$ only has support to
the right of $\Sigma^{\nu}_{s-1}$, and thus constitutes a genuinely new
endpoint. This choice only affects the total number of measurement “regions”
$m$, which delineate endpoints.

We then find a set of string order parameters

$\displaystyle\mathsf{S}^{\vphantom{\prime}}_{\nu,t,s}$
$\displaystyle=\left(\mathcal{U}\,\Sigma^{\nu}_{s}\,\mathcal{U}^{\dagger}\right)\,\bar{A}^{\vphantom{\prime}}_{\nu,s+1}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,m}\,\left(\sigma^{\nu}_{f}\right)$
(N.178)

for all $s\in[1,m]$, such that
$\left\langle\Psi^{\,}_{t}\middle|\mathsf{S}^{\,}_{\nu,t,s}\middle|\Psi^{\,}_{t}\right\rangle=1$
(60) indicates perfect string order of $\lvert\Psi^{\,}_{t}\rangle$ (36).

We next identify new _right_ endpoints. Returning to Eq. N.171, using the same
operators $\Sigma^{\nu}_{s}$ defined in Eqs. N.176 and N.177. Importantly, we
take $\Sigma^{\nu}_{s}$ to be the operator
$\Sigma^{\nu}_{s}\,\mathsf{S}^{\,}_{0}$ for
$\mathsf{S}^{\,}_{0}\in\operatorname*{stab}\left(\lvert\Psi^{\,}_{0}\rangle\right)$
with the leftmost right weight (i.e., whose support is farthest to the left).
All such operators are equivalent, since these quantities are (effectively)
evaluated in the initial state $\lvert\Psi^{\,}_{0}\rangle$ (6), so
$\mathsf{S}^{\,}_{0}\cong\mathds{1}$.

For any $s\in[1,m]$, we define

$\displaystyle\mathsf{S}^{\prime}_{\nu,t,s}$
$\displaystyle\equiv\mathsf{S}^{\vphantom{\prime}}_{\nu,t,0}\,\mathsf{S}^{\dagger}_{\nu,t,s}$
$\displaystyle=\left(\mathcal{U}\,\sigma^{\nu}_{i}\,\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,1}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,m}\,\sigma^{\nu}_{f}\right)$
$\displaystyle\times\left(\sigma^{\nu}_{f}\,\bar{A}^{\vphantom{\prime}}_{\nu,m}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,s+1}\,\mathcal{U}\,\Sigma^{\nu}_{s}\,\mathcal{U}^{\dagger}\right)$
$\displaystyle=\left(\mathcal{U}\,\sigma^{\nu}_{i}\,\mathcal{U}^{\dagger}\right)\,\bar{A}^{\vphantom{\prime}}_{\nu,1}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,s}\,\left(\mathcal{U}\,\Sigma^{\nu}_{s}\,\mathcal{U}^{\dagger}\right)\,\,,~{}~{}$
(N.179)

which has the same left-endpoint operator
$\mathcal{U}\,\sigma^{\nu}_{i}\,\mathcal{U}^{\dagger}$ as
$\mathsf{S}^{\,}_{\nu,t,0}$ (54) but distinct right-endpoint operators
$\Sigma^{\nu}_{s}$, which are the same endpoint operators in Eq. 57.

So far, we have shown that new left and right endpoints can independently be
chosen in Eqs. N.178 and N.179, respectively. Finally, we recover the generic
result stated in the Lemma with arbitrary left endpoints via

$\displaystyle\mathcal{S}^{\vphantom{\prime}}_{\nu,a,b}$
$\displaystyle\equiv\mathsf{S}^{\vphantom{\prime}}_{\nu,t,a}\,\mathsf{S}^{\dagger}_{\nu,t,b}$
(N.180)
$\displaystyle=\left(\mathcal{U}\,\Sigma^{\nu}_{a}\,\mathcal{U}^{\dagger}\,\bar{A}^{\vphantom{\prime}}_{\nu,a+1}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,m}\,\sigma^{\nu}_{f}\right)$
(N.181)
$\displaystyle\times\left(\sigma^{\nu}_{f}\,\bar{A}^{\vphantom{\prime}}_{\nu,m}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,b+1}\,\mathcal{U}\,\Sigma^{\nu}_{b}\,\mathcal{U}^{\dagger}\right)$
(N.182)
$\displaystyle=\left(\mathcal{U}\,\Sigma^{\nu}_{a}\,\mathcal{U}^{\dagger}\right)\,\bar{A}^{\vphantom{\prime}}_{\nu,a+1}\cdots\bar{A}^{\vphantom{\prime}}_{\nu,b}\,\left(\mathcal{U}\,\Sigma^{\nu}_{b}\,\mathcal{U}^{\dagger}\right)\,\,,~{}~{}$
(57)

and since the support of $\Sigma^{\nu}_{s+1}$ is entirely to the right of
$\Sigma^{\nu}_{s}$, the support of $\Sigma^{\nu}_{s}$ is to the left of
$\Sigma^{\nu}_{s+1}$. Hence, the operators $\Sigma^{\nu}_{s}$ constitute
distinct left _and_ right endpoint operators for each $s\in[1,m]$, where
$\left\langle*\middle|\Psi^{\,}_{t}\middle|\mathcal{S}^{\,}_{\nu,a,b}\right\rangle{\Psi^{\,}_{t}}=1$
(53) indicates nontrivial string order of $\lvert\Psi^{\,}_{t}\rangle$ (36). ∎

## References

  * Bennett  _et al._ (1993) Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
  * Briegel  _et al._ (1998) H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
  * Gottesman and Chuang (1999) Daniel Gottesman and Isaac L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations,” Nature 402, 390–393 (1999).
  * Raussendorf and Briegel (2001) Robert Raussendorf and Hans J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188–5191 (2001).
  * Popp  _et al._ (2005) M. Popp, F. Verstraete, M. A. Martí n-Delgado, and J. I. Cirac, “Localizable entanglement,” Phys. Rev. A 71 (2005).
  * Verstraete  _et al._ (2004a) F. Verstraete, M. Popp, and J. I. Cirac, “Entanglement versus correlations in spin systems,” Phys. Rev. Lett. 92, 027901 (2004a).
  * Verstraete  _et al._ (2004b) F. Verstraete, M. A. Martín-Delgado, and J. I. Cirac, “Diverging entanglement length in gapped quantum spin systems,” Phys. Rev. Lett. 92, 087201 (2004b).
  * Jin and Korepin (2004) B.-Q. Jin and V. E. Korepin, “Localizable entanglement in antiferromagnetic spin chains,” Phys. Rev. A 69, 062314 (2004).
  * Pachos and Plenio (2004) Jiannis K. Pachos and Martin B. Plenio, “Three-spin interactions in optical lattices and criticality in cluster hamiltonians,” Phys. Rev. Lett. 93, 056402 (2004).
  * Venuti and Roncaglia (2005) L. Campos Venuti and M. Roncaglia, “Analytic relations between localizable entanglement and string correlations in spin systems,” Phys. Rev. Lett. 94, 207207 (2005).
  * Skrøvseth and Bartlett (2009) Stein Olav Skrøvseth and Stephen D. Bartlett, “Phase transitions and localizable entanglement in cluster-state spin chains with Ising couplings and local fields,” Phys. Rev. A 80, 022316 (2009).
  * Raussendorf  _et al._ (2005) Robert Raussendorf, Sergey Bravyi, and Jim Harrington, “Long-range quantum entanglement in noisy cluster states,” Phys. Rev. A 71, 062313 (2005).
  * Van den Nest  _et al._ (2006) Maarten Van den Nest, Akimasa Miyake, Wolfgang Dür, and Hans J. Briegel, “Universal resources for measurement-based quantum computation,” Phys. Rev. Lett. 97, 150504 (2006).
  * Gross and Eisert (2007) D. Gross and J. Eisert, “Novel schemes for measurement-based quantum computation,” Phys. Rev. Lett. 98, 220503 (2007).
  * Gross  _et al._ (2009) D. Gross, S. T. Flammia, and J. Eisert, “Most quantum states are too entangled to be useful as computational resources,” Phys. Rev. Lett. 102, 190501 (2009).
  * Bremner  _et al._ (2009) Michael J. Bremner, Caterina Mora, and Andreas Winter, “Are random pure states useful for quantum computation?” Phys. Rev. Lett. 102, 190502 (2009).
  * Doherty and Bartlett (2009) Andrew C. Doherty and Stephen D. Bartlett, “Identifying phases of quantum many-body systems that are universal for quantum computation,” Phys. Rev. Lett. 103, 020506 (2009).
  * Gross and Eisert (2010) D. Gross and J. Eisert, “Quantum computational webs,” Phys. Rev. A 82, 040303 (2010).
  * Else  _et al._ (2012) Dominic V. Else, Ilai Schwarz, Stephen D. Bartlett, and Andrew C. Doherty, “Symmetry-protected phases for measurement-based quantum computation,” Phys. Rev. Lett. 108 (2012).
  * Prakash and Wei (2015) Abhishodh Prakash and Tzu-Chieh Wei, “Ground states of one-dimensional symmetry-protected topological phases and their utility as resource states for quantum computation,” Phys. Rev. A 92, 022310 (2015).
  * Miller and Miyake (2016) Jacob Miller and Akimasa Miyake, “Hierarchy of universal entanglement in 2D measurement-based quantum computation,” NPJ Quant. Inf. 2, 16036 (2016).
  * Stephen  _et al._ (2017) David T. Stephen, Dong-Sheng Wang, Abhishodh Prakash, Tzu-Chieh Wei, and Robert Raussendorf, “Computational power of symmetry-protected topological phases,” Phys. Rev. Lett. 119, 010504 (2017).
  * Marvian (2017) Iman Marvian, “Symmetry-protected topological entanglement,” Phys. Rev. B 95, 045111 (2017).
  * Wei (2018) Tzu-Chieh Wei, “Quantum spin models for measurement-based quantum computation,” Advances in Physics: X 3, 1461026 (2018), https://doi.org/10.1080/23746149.2018.1461026 .
  * Chen  _et al._ (2011) Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen, “Classification of gapped symmetric phases in one-dimensional spin systems,” Phys. Rev. B 83, 035107 (2011).
  * Pollmann  _et al._ (2010) Frank Pollmann, Ari M. Turner, Erez Berg, and Masaki Oshikawa, “Entanglement spectrum of a topological phase in one dimension,” Phys. Rev. B 81 (2010), 10.1103/PhysRevB.81.064439.
  * Schuch  _et al._ (2011) Norbert Schuch, David Pé rez-García, and Ignacio Cirac, “Classifying quantum phases using matrix product states and projected entangled pair states,” Phys. Rev. B 84 (2011), 10.1103/physrevb.84.165139.
  * Chen  _et al._ (2013) Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen, “Symmetry protected topological orders and the group cohomology of their symmetry group,” Phys. Rev. B 87, 155114 (2013).
  * Miyake (2010) Akimasa Miyake, “Quantum computation on the edge of a symmetry-protected topological order,” Phys. Rev. Lett. 105, 040501 (2010).
  * den Nijs and Rommelse (1989) Marcel den Nijs and Koos Rommelse, “Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains,” Phys. Rev. B 40, 4709–4734 (1989).
  * Kennedy and Tasaki (1992) Tom Kennedy and Hal Tasaki, “Hidden ${\mathrm{z}}_{2}$×${\mathrm{z}}_{2}$ symmetry breaking in haldane-gap antiferromagnets,” Phys. Rev. B 45, 304–307 (1992).
  * Pollmann and Turner (2012) Frank Pollmann and Ari M. Turner, “Detection of symmetry-protected topological phases in one dimension,” Phys. Rev. B 86, 125441 (2012).
  * Raussendorf  _et al._ (2022) Robert Raussendorf, Wang Yang, and Arnab Adhikary, “Measurement-based quantum computation in finite one-dimensional systems: String order implies computational power,” (2022), arXiv:2210.05089 [quant-ph] .
  * Lieb and Robinson (1972) Elliott H. Lieb and Derek W. Robinson, “The finite group velocity of quantum spin systems,” Commun. Math. Phys. 28, 251–257 (1972).
  * Poulin (2010) David Poulin, “Lieb-Robinson bound and locality for general Markovian quantum dynamics,” Phys. Rev. Lett. 104, 190401 (2010).
  * Friedman  _et al._ (2022a) Aaron J. Friedman, Chao Yin, Yifan Hong, and Andrew Lucas, “Locality and error correction in quantum dynamics with measurement,” arXiv (2022a), arXiv:2206.09929 [quant-ph] .
  * Stinespring (1955) W. Forrest Stinespring, “Positive functions on ${C}^{*}$-algebras,” Proc. Amer. Math. Soc. 6, 211 (1955).
  * Choi (1975) Man-Duen Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. 10, 285–290 (1975).
  * Takesaki (2012) Masamichi Takesaki,  _Theory of Operator Algebras I_ (Springer New York, NY, 2012).
  * Friedman  _et al._ (2022b) Aaron J. Friedman, Oliver Hart, and Rahul Nandkishore, “Measurement-induced phases of matter require adaptive dynamics,” arXiv (2022b), arXiv:2210.11111 [quant-ph] .
  * Barberena and Friedman (2023) Diego Barberena and Aaron J. Friedman, “Ideal quantum measurements,” arXiv , to appear (2023).
  * Briegel and Raussendorf (2001) Hans J. Briegel and Robert Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910–913 (2001).
  * Gottesman (1997) Daniel Gottesman, “Stabilizer codes and quantum error correction,” (1997), arXiv:quant-ph/9705052 [quant-ph] .
  * Rossi  _et al._ (2013) M Rossi, M Huber, D Bruß, and C Macchiavello, “Quantum hypergraph states,” New J. Phys. 15, 113022 (2013).
  * Gachechiladze  _et al._ (2019) Mariami Gachechiladze, Otfried Gühne, and Akimasa Miyake, “Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states,” Phys. Rev. A 99, 052304 (2019).
  * Hong  _et al._ (2023) Yifan Hong, David T. Stephen, and Aaron. J Friedman, “Virtual teleportation and symmetry-protected topological order,” arXiv , to appear (2023).
  * Hastings (2010) M. B. Hastings, “Locality in quantum systems,” arXiv (2010), 10.48550/ARXIV.1008.5137.
  * Wang and Hazzard (2020) Zhiyuan Wang and Kaden R.A. Hazzard, “Tightening the Lieb-Robinson bound in locally interacting systems,” PRX Quantum 1, 010303 (2020).
  * Foss-Feig  _et al._ (2015) Michael Foss-Feig, Zhe-Xuan Gong, Charles W. Clark, and Alexey V. Gorshkov, “Nearly linear light cones in long-range interacting quantum systems,” Phys. Rev. Lett. 114, 157201 (2015).
  * Chen and Lucas (2019) Chi-Fang Chen and Andrew Lucas, “Finite speed of quantum scrambling with long-range interactions,” Phys. Rev. Lett. 123, 250605 (2019), arXiv:1907.07637 [quant-ph] .
  * Kuwahara and Saito (2020) Tomotaka Kuwahara and Keiji Saito, “Strictly linear light cones in long-range interacting systems of arbitrary dimensions,” Phys. Rev. X 10, 031010 (2020), 1910.14477 .
  * Tran  _et al._ (2020) Minh C. Tran, Chi-Fang Chen, Adam Ehrenberg, Andrew Y. Guo, Abhinav Deshpande, Yifan Hong, Zhe-Xuan Gong, Alexey V. Gorshkov, and Andrew Lucas, “Hierarchy of linear light cones with long-range interactions,” Phys. Rev. X 10, 031009 (2020).
  * Chen  _et al._ (2023) Chi-Fang Chen, Andrew Lucas, and Chao Yin, “Speed limits and locality in many-body quantum dynamics,” (2023), arXiv:2303.07386 [quant-ph] .
  * Greenberger  _et al._ (1989) Daniel M Greenberger, Michael A Horne, and Anton Zeilinger, “Going beyond Bell’s theorem,” in  _Bell’s theorem, quantum theory and conceptions of the universe_ (Springer, 1989) pp. 69–72.
  * Piroli  _et al._ (2021) Lorenzo Piroli, Georgios Styliaris, and J. Ignacio Cirac, “Quantum circuits assisted by local operations and classical communication: Transformations and phases of matter,” Phys. Rev. Lett. 127, 220503 (2021).
  * Nielsen (2003) Michael A. Nielsen, “Quantum computation by measurement and quantum memory,” Phys. Lett. A 308, 96–100 (2003).
  * Leung (2002) D. W. Leung, “Two-qubit projective measurements are universal for quantum computation,” (2002), arXiv:quant-ph/0111122 [quant-ph] .
  * Raussendorf  _et al._ (2017) Robert Raussendorf, Dong-Sheng Wang, Abhishodh Prakash, Tzu-Chieh Wei, and David T. Stephen, “Symmetry-protected topological phases with uniform computational power in one dimension,” Phys. Rev. A 96, 012302 (2017).
  * Pérez-García  _et al._ (2008) D. Pérez-García, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac, “String order and symmetries in quantum spin lattices,” Phys. Rev. Lett. 100 (2008), 10.1103/PhysRevLett.100.167202.
  * Berkovich and Zhmud (1998) I_A G Berkovich and EM Zhmud,  _Characters of finite groups_ , Vol. 1 (American Mathematical Soc., 1998).
  * Huang and Chen (2015) Yichen Huang and Xie Chen, “Quantum circuit complexity of one-dimensional topological phases,” Phys. Rev. B 91, 195143 (2015).
  * Zeng and Wen (2015) Bei Zeng and Xiao-Gang Wen, “Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity,” Phys. Rev. B 91, 125121 (2015).
  * Stephen  _et al._ (2022) David T. Stephen, Wen Wei Ho, Tzu-Chieh Wei, Robert Raussendorf, and Ruben Verresen, “Universal measurement-based quantum computation in a one-dimensional architecture enabled by dual-unitary circuits,” arXiv (2022), arXiv:2209.06191 [quant-ph] .
  * Affleck  _et al._ (1987) Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki, “Rigorous results on valence-bond ground states in antiferromagnets,” Phys. Rev. Lett. 59, 799–802 (1987).
  * Verresen  _et al._ (2017) Ruben Verresen, Roderich Moessner, and Frank Pollmann, “One-dimensional symmetry protected topological phases and their transitions,” Phys. Rev. B 96 (2017).
  * Stephen (2023) David T. Stephen, “Non-onsite symmetries and localizable entanglement in split-index matrix product states,” , to appear (2023).
  * Jozsa (1994) Richard Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).