1\left\\{\text{$\underline{D}$}_{it}=\text{$\underline{d}$}_{t}\right\\}}{\prod_{s=1}^{t}e_{s}\left(d_{s},H_{is}\right)}-\frac{\gamma_{t}Y_{it}\cdot
1\left\\{\text{$\underline{D}$}_{it}=\text{$\underline{d}$}_{t}\right\\}}{\prod_{s=1}^{t}\hat{e}_{s}\left(d_{s},H_{is}\right)}\right)\right.$
$\displaystyle\left.\times\left(\prod_{s=1}^{t}1\left\\{g_{s}\left(H_{is}\right)=d_{s}\right\\}-\prod_{s=1}^{t}1\left\\{\hat{g}_{e,s}^{S}\left(H_{is}\right)=d_{s}\right\\}\right)\right]$
$\displaystyle+W\left(g\right)-W_{n}\left(g\right)+W_{n}\left(\hat{g}_{e}^{S}\right)-W\left(\hat{g}_{e}^{S}\right)$
$\displaystyle\leq\sum_{t=1}^{T}\sum_{\text{$\underline{d}$}_{t}\in\left\\{0,1\right\\}^{t}}\left(\frac{1}{n}\sum_{i=1}^{n}\left|\tau_{t}\left(\underline{d}_{t},H_{it}\right)-\hat{\tau}_{t}\left(\underline{d}_{t},H_{it}\right)\right|\right)+2\sup_{g\in{\cal
G}}\left|W_{n}\left(g\right)-W\left(g\right)\right|$
$\displaystyle\leq\sum_{t=1}^{T}\sum_{\text{$\underline{d}$}_{t}\in\left\\{0,1\right\\}^{t}}\left(\frac{1}{n}\sum_{i=1}^{n}\left|\tau_{t}\left(\underline{d}_{t},H_{it}\right)-\hat{\tau}_{t}\left(\underline{d}_{t},H_{it}\right)\right|\right)$
$\displaystyle+2\sum_{t=1}^{T}\sup_{\underline{g}_{t}\in{\mathcal{G}_{1}\times\cdots\times\mathcal{G}_{t}}}\left|W_{nt}(\text{$\underline{g}$}_{t})-W_{t}(\text{$\underline{g}$}_{t})\right|.$

The first inequality follows from the fact that $\hat{g}_{e}^{S}$ maximizes
$\hat{W}_{n}\left(\cdot\right)$ over ${\cal G}$. The second inequality follows
from the fact that

$\displaystyle\left|\prod_{s=1}^{t}1\left\\{g_{s}\left(H_{is}\right)=d_{s}\right\\}-\prod_{s=1}^{t}1\left\\{\hat{g}_{e,s}^{S}\left(H_{is}\right)=d_{s}\right\\}\right|\leq
1.$

Thus, the average welfare-regret can be bounded from above by

$\displaystyle E_{P^{n}}\left[W_{{\cal
G}}^{\ast}-W\left(\hat{g}_{e}^{S}\right)\right]$
$\displaystyle\leq\sum_{t=1}^{T}\sum_{\text{$\underline{d}$}_{t}\in\left\\{0,1\right\\}^{t}}E_{P^{n}}\left[\frac{1}{n}\sum_{i=1}^{n}\left|\tau_{t}\left(\text{$\underline{d}$}_{t},H_{it}\right)-\hat{\tau}_{t}\left(\text{$\underline{d}$}_{t},H_{it}\right)\right|\right]$
$\displaystyle+2\sum_{t=1}^{T}\sup_{\underline{g}_{t}\in{\mathcal{G}_{1}\times\cdots\times\mathcal{G}_{t}}}E_{P^{n}}\left[\left|W_{nt}(\text{$\underline{g}$}_{t})-W_{t}(\text{$\underline{g}$}_{t})\right|\right].$

Therefore, by the same argument as in the proof of Theorem 3.4 (i) and from
Assumption 5.1 (i), the average welfare-regret is bounded from above as

$\displaystyle E_{P^{n}}\left[W_{{\cal
G}}^{\ast}-W\left(\hat{g}_{e}^{S}\right)\right]$ $\displaystyle\leq
C\sum_{t=1}^{T}\left\\{\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\sqrt{\frac{\sum_{s=1}^{t}v_{s}}{n}}\right\\}+O\left(\phi_{n}^{-1}\right).$

Since this upper bound does not depend on $P\in{\cal P}_{e}\bigcap{\cal
P}\left(M,\kappa,\mathcal{G}\right)$, the upper bound is uniform over ${\cal
P}_{e}\bigcap{\cal P}\left(M,\kappa,\mathcal{G}\right)$. ∎

Before proceeding to the proof of Theorem 5.1 (ii), we define

$\displaystyle\Delta\tilde{Q}_{t,e}$
$\displaystyle\equiv\tilde{Q}_{t}\left(g_{t}^{\ast},\ldots,g_{T}^{\ast}\right)-\tilde{Q}_{t}\left(\hat{g}_{t,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right),$
$\displaystyle\Delta\tilde{Q}_{t,e}^{\dagger}$
$\displaystyle\equiv\tilde{Q}_{t}\left(g_{t}^{\ast},\hat{g}_{t+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)-\tilde{Q}_{t}\left(\hat{g}_{t,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right),$
$\displaystyle\check{Q}_{nt}\left(g_{t},\ldots,g_{T}\right)$
$\displaystyle\equiv
E_{n}\left[\hat{q}_{t}\left(Z,g_{t},\ldots,g_{T}\right)\right]$
$\displaystyle=\sum_{s=t}^{T}E_{n}\left[\frac{(\prod_{\ell=t}^{s}1\left\\{g_{\ell}\left(H_{\ell}\right)=D_{\ell}\right\\})\gamma_{s}Y_{s}}{\prod_{\ell=t}^{s}\hat{e}_{\ell}\left(D_{\ell},H_{\ell}\right)}\right],$

where $\tilde{Q}$ is defined in (14). The following lemmas will be used in the
proof of Theorem 5.1 (ii).

######  Lemma A.7.

Suppose that Assumptions 2.1, 2.2, and 2.4 hold for any
$P\in\mathcal{P}(M,\kappa,\mathcal{G})$, that Assumption 2.3 holds for
$\mathcal{G}$, that Assumption 3.1 holds for a pair $(P,\mathcal{G})$ with any
$P\in\mathcal{P}(M,\kappa,\mathcal{G})$, and that Assumption 5.1 (ii) holds
for any $P\in\mathcal{P}_{e}$. Then, for any
$P\in\mathcal{P}(M,\kappa,\mathcal{G})\cap\mathcal{P}_{e}$, the following
hold:

  * (i)

for $t=1,\ldots,T$,

$\displaystyle
E_{P^{n}}\left[\left|\Delta\tilde{Q}_{t,e}^{\dagger}\right|\right]$
$\displaystyle\leq
C\left(\sum_{s=t}^{T}\frac{\gamma_{s}M_{s}}{\prod_{\ell=t}^{s}\kappa_{\ell}}\right)\sqrt{\frac{\sum_{s=t}^{T}v_{s}}{n}}+O\left(\psi_{n}^{-1}\right),$

where $C$ is the same constant term as introduced in Lemma A.4;

  * (ii)

for $t=1,\ldots,T-1$ and $s=t+1,\ldots,T$,

$\displaystyle\tilde{Q}_{t}\left(g_{t}^{\ast},\ldots,g_{T}^{\ast}\right)-\tilde{Q}_{t}\left(g_{t}^{\ast},\ldots,g_{s}^{\ast},\hat{g}_{s+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)$
$\displaystyle\leq\frac{1}{\prod_{\ell=t}^{s}\kappa_{\ell}}\Delta\tilde{Q}_{s+1,e};$

  * (iii)

$\displaystyle\Delta\tilde{Q}_{1,e}$
$\displaystyle\leq\Delta\tilde{Q}_{1,e}^{\dagger}+\sum_{s=1}^{T-1}\frac{2^{s-1}}{\prod_{t=1}^{s}\kappa_{t}}\Delta\tilde{Q}_{s+1,e}^{\dagger}.$

###### Proof.

(i) It follows for any $\tilde{g}_{t}\in{\cal G}_{t}$ that

$\displaystyle\tilde{Q}_{t}\left(\tilde{g}_{t},\hat{g}_{t+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)-\tilde{Q}_{t}\left(\hat{g}_{t,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)$
$\displaystyle\leq\tilde{Q}_{nt}\left(\tilde{g}_{t},\hat{g}_{t+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)-\check{Q}_{nt}\left(\tilde{g}_{t},\hat{g}_{t+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)-\tilde{Q}_{nt}\left(\hat{g}_{t,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)+\check{Q}_{nt}\left(\hat{g}_{t,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)$
$\displaystyle+\tilde{Q}_{t}\left(\tilde{g}_{t},\hat{g}_{t+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)-\tilde{Q}_{t}\left(\hat{g}_{t,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)+\tilde{Q}_{nt}\left(\hat{g}_{t,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)-\tilde{Q}_{nt}\left(\tilde{g}_{t},\hat{g}_{t+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\sum_{s=t}^{T}\sum_{\text{$\underline{d}$}_{t:s}\in\left\\{0,1\right\\}^{t-s+1}}\left[\left(\frac{1\left\\{\underline{D}_{t:s}=\underline{d}_{t:s}\right\\}\gamma_{s}Y_{s}}{\prod_{\ell=t}^{s}e_{\ell}\left(d_{\ell},H_{\ell}\right)}-\frac{1\left\\{\underline{D}_{t:s}=\underline{d}_{t:s}\right\\}\gamma_{s}Y_{s}}{\prod_{\ell=t}^{s}\hat{e}_{\ell}\left(d_{\ell},H_{\ell}\right)}\right)\right.$
$\displaystyle\left.\times\left(1\left\\{\tilde{g}_{t}(H_{it})=d_{t}\right\\}-1\left\\{\hat{g}_{t,e}^{B}\left(H_{it}\right)=d_{t}\right\\}\right)\right]$
$\displaystyle+\tilde{Q}_{t}\left(\tilde{g}_{t},\hat{g}_{t+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)-\tilde{Q}_{t}\left(\hat{g}_{t,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)+\tilde{Q}_{nt}\left(\tilde{g}_{t},\hat{g}_{t+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)-\tilde{Q}_{nt}\left(\hat{g}_{t,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)$
$\displaystyle\leq\sum_{\text{$\underline{d}$}_{t:T}\in\left\\{0,1\right\\}^{T-t+1}}\left(\frac{1}{n}\sum_{i=1}^{n}\left|\hat{\eta}_{t}\left(\text{$\underline{d}$}_{t:T},H_{iT}\right)-\eta_{t}\left(\text{$\underline{d}$}_{t:T},H_{iT}\right)\right|\right)$
$\displaystyle+2\sup_{\left(g_{t},\ldots,g_{T}\right)\in{\cal
G}_{t}\times\cdots\times{\cal
G}_{T}}\left|\tilde{Q}_{nt}\left(g_{t},\ldots,g_{T}\right)-\tilde{Q}_{t}\left(g_{t},\ldots,g_{T}\right)\right|.$

The first inequality follows from the fact that $\hat{g}_{t,e}^{B}$ maximizes
$\check{Q}_{nt}\left(\cdot,\hat{g}_{t+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)$
over ${\cal G}_{t}$.

Then we have

$\displaystyle
E_{P^{n}}\left[\left|\Delta\tilde{Q}_{t,e}^{\dagger}\right|\right]$
$\displaystyle\leq
2E_{P^{n}}\left[\sup_{\left(g_{t},\ldots,g_{T}\right)\in{\cal
G}_{t}\times\cdots\times{\cal
G}_{T}}\left|\tilde{Q}_{nt}\left(g_{t},\ldots,g_{T}\right)-\tilde{Q}_{t}\left(g_{t},\ldots,g_{T}\right)\right|\right]$
$\displaystyle+\sum_{\text{$\underline{d}$}_{t:T}\in\left\\{0,1\right\\}^{T-t+1}}E_{P^{n}}\left[\frac{1}{n}\sum_{i=1}^{n}\left|\hat{\eta}_{t}\left(\text{$\underline{d}$}_{t:T},H_{iT}\right)-\eta_{t}\left(\text{$\underline{d}$}_{t:T},H_{iT}\right)\right|\right]$

Therefore, applying Lemma A.4 to the first term in the right hand side (as in
the proof of Lemma A.5 (i)) and Assumption 5.1 (ii) to the second term in the
right hand side leads to the result.

(ii) The proof of Lemma A.7 follows from the same argument with the proof of
Lemma A.5 (ii).

(iii) We follow the same strategy as in Lemma A.5 (iii). First, note that

$\displaystyle\Delta\tilde{Q}_{T,e}=\tilde{Q}_{T}\left(g_{T}^{\ast}\right)-\tilde{Q}_{T}\left(\hat{g}_{T,e}^{B}\right)=\Delta\tilde{Q}_{T,e}^{\dagger}.$

Then, for $t=T-1$, we have

$\displaystyle\Delta\tilde{Q}_{T-1,e}$
$\displaystyle=\tilde{Q}_{T-1}\left(g_{T-1}^{\ast},g_{T}^{\ast}\right)-\tilde{Q}_{T-1}\left(\hat{g}_{T-1,e}^{B},\hat{g}_{T,e}^{B}\right)$
$\displaystyle=\tilde{Q}_{T-1}\left(g_{T-1}^{\ast},g_{T}^{\ast}\right)-\tilde{Q}_{T-1}\left(g_{T-1}^{\ast},\hat{g}_{T,e}^{B}\right)+\tilde{Q}_{T-1}\left(g_{T-1}^{\ast},\hat{g}_{T,e}^{B}\right)-\tilde{Q}_{T-1}\left(\hat{g}_{T-1,e}^{B},\hat{g}_{T,e}^{B}\right)$
$\displaystyle\leq\frac{1}{\kappa_{T-1}}\Delta\tilde{Q}_{T,e}^{\dagger}+\Delta\tilde{Q}_{T-1,e}^{\dagger},$

where the inequality follows from Lemma A.7 (ii).

Generally, for any $k=1,\ldots,T-1$, it follows that

$\displaystyle\Delta\tilde{Q}_{T-k,e}$
$\displaystyle=\tilde{Q}_{T-k}\left(g_{T-k}^{\ast},\ldots,g_{T}^{\ast}\right)-\tilde{Q}_{T-k}\left(\hat{g}_{T-k,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)$
$\displaystyle=\sum_{s=T-k}^{T}\left[\tilde{Q}_{T-k}\left(g_{T-k}^{\ast},\ldots,g_{s}^{\ast},\hat{g}_{s+1,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)-\tilde{Q}_{T-k}\left(g_{T-k}^{\ast},\ldots,g_{s-1}^{\ast},\hat{g}_{s,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)\right]$
$\displaystyle\leq\sum_{s=T-k}^{T}\left[\tilde{Q}_{T-k}\left(g_{T-k}^{\ast},\ldots,g_{T}^{\ast}\right)-\tilde{Q}_{T-k}\left(g_{T-k}^{\ast},\ldots,g_{s-1}^{\ast},\hat{g}_{s,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)\right]$
$\displaystyle=\sum_{s=T-k+1}^{T}\left[\tilde{Q}_{T-k}\left(g_{T-k}^{\ast},\ldots,g_{T}^{\ast}\right)-\tilde{Q}_{T-k}\left(g_{T-k}^{\ast},\ldots,g_{s-1}^{\ast},\hat{g}_{s,e}^{B},\ldots,\hat{g}_{T,e}^{B}\right)\right]+\Delta\tilde{Q}_{T-k,e}^{\dagger}$
$\displaystyle\leq\sum_{s=T-k+1}^{T}\frac{1}{\prod_{\ell=T-k}^{s-1}\kappa_{\ell}}\Delta\tilde{Q}_{s,e}+\Delta\tilde{Q}_{T-k,e}^{\dagger},$

where the second line follows by taking a telescope sum; the third line
follows from the fact that $\left(g_{s+1}^{\ast},\ldots,g_{T}^{\ast}\right)$
maximizes
$\tilde{Q}_{T-k}\left(g_{T-k}^{\ast},\ldots,g_{s}^{\ast},\cdot,\ldots,\cdot\right)$
over ${\cal G}_{s+1}\times\cdots\times{\cal G}_{T}$ under Assumption 3.1; the
last line follows from Lemma A.7(ii).

Then, recursively, the following hold:

$\displaystyle\Delta\tilde{Q}_{T-1,e}$
$\displaystyle\leq\frac{1}{\kappa_{T-1}}\Delta\tilde{Q}_{T,e}+\Delta\tilde{Q}_{T-1,e}^{\dagger}=\frac{1}{\kappa_{T-1}}\Delta\tilde{Q}_{T,e}^{\dagger}+\Delta\tilde{Q}_{T-1,e}^{\dagger},$
$\displaystyle\Delta\tilde{Q}_{T-2,e}$
$\displaystyle\leq\frac{1}{\kappa_{T-2}}\Delta\tilde{Q}_{T-1,e}+\frac{1}{\kappa_{T-2}\kappa_{T-1}}\Delta\tilde{Q}_{T,e}+\Delta\tilde{Q}_{T-2,e}^{\dagger}$
$\displaystyle\leq\frac{2}{\kappa_{T-2}\kappa_{T-1}}\Delta\tilde{Q}_{T,e}^{\dagger}+\frac{1}{\kappa_{T-2}}\Delta\tilde{Q}_{T-1,e}^{\dagger}+\Delta\tilde{Q}_{T-2,e}^{\dagger},$
$\displaystyle\vdots$ $\displaystyle\Delta\tilde{Q}_{T-k,e}$
$\displaystyle\leq\sum_{s=1}^{k}\frac{2^{k-s}}{\prod_{t=T-k}^{T-s}\kappa_{t}}\Delta\tilde{Q}_{T-s+1,e}^{\dagger}+\Delta\tilde{Q}_{T-k,e}^{\dagger}.$

Therefore, when $k=T-1$, we have

$\displaystyle\Delta\tilde{Q}_{1,e}$
$\displaystyle\leq\Delta\tilde{Q}_{1,e}^{\dagger}+\sum_{s=1}^{T-1}\frac{2^{T-1-s}}{\prod_{t=1}^{T-s}\kappa_{t}}\Delta\tilde{Q}_{T-s+1,e}^{\dagger}$
$\displaystyle=\Delta\tilde{Q}_{1,e}^{\dagger}+\sum_{s=1}^{T-1}\frac{2^{s-1}}{\prod_{t=1}^{s}\kappa_{t}}\Delta\tilde{Q}_{s+1,e}^{\dagger}.$

∎

###### Proof of Theorem 5.1 (ii):.

Let $P\in\widetilde{\mathcal{P}}_{e}\bigcap{\cal
P}\left(M,\kappa,\mathcal{G}\right)$ be fixed. By the same argument as in the
proof of Theorem 3.4 (ii), it follows for $g^{*}\in\mathop{\rm
arg\leavevmode\nobreak\ max}\limits_{g\in\mathcal{G}}W(g)$ that

$\displaystyle W_{\mathcal{G}}^{\ast}-W(\hat{g}_{e}^{B})$
$\displaystyle=\tilde{Q}_{1}(g^{\ast})-\tilde{Q}_{1}(\hat{g}_{e}^{B})\leq\Delta\tilde{Q}_{1,e}$
$\displaystyle\leq\Delta\tilde{Q}_{1,e}^{\dagger}+\sum_{s=1}^{T-1}\frac{2^{s-1}}{\prod_{t=1}^{s}\kappa_{t}}\Delta\tilde{Q}_{s+1,e}^{\dagger},$

where the second inequality follows from Lemma A.7 (iii). Thus, since
$W_{\mathcal{G}}^{\ast}-W(\hat{g}_{e}^{B})\geq 0$, we have

$\displaystyle E_{P^{n}}\left[\left|W_{{\cal
G}}^{\ast}-W\left(\hat{g}_{e}^{B}\right)\right|\right]\leq
E_{P^{n}}\left[\left|\Delta\tilde{Q}_{1,e}^{\dagger}\right|\right]+\sum_{s=1}^{T-1}\frac{2^{s-1}}{\prod_{t=1}^{s}\kappa_{t}}E_{P^{n}}\left[\left|\Delta\tilde{Q}_{s+1,e}^{\dagger}\right|\right].$

Applying Lemma A.7 (i) to each term in the right hand side gives

$\displaystyle E_{P^{n}}\left[\left|W_{{\cal
G}}^{\ast}-W\left(\hat{g}_{e}^{B}\right)\right|\right]$ $\displaystyle\leq
C\sum_{t=1}^{T}\left\\{\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\sqrt{\frac{\sum_{s=1}^{t}v_{s}}{n}}\right\\}$
$\displaystyle+\sum_{t=2}^{T}\frac{2^{t-2}}{\prod_{s=1}^{t-1}\kappa_{s}}\left(C\sum_{s=t}^{T}\left\\{\frac{\gamma_{s}M_{s}}{\prod_{\ell=t}^{s}\kappa_{\ell}}\sqrt{\frac{\sum_{\ell=t}^{s}v_{\ell}}{n}}\right\\}\right)+O\left(\psi_{n}^{-1}\right).$

Since this upper bound does not depend on
$P\in\widetilde{\mathcal{P}}_{e}\bigcap{\cal
P}\left(M,\kappa,\mathcal{G}\right)$, the upper bound is uniform over
$\widetilde{\mathcal{P}}_{e}\bigcap{\cal P}\left(M,\kappa,\mathcal{G}\right)$.
∎

####  A.2 Proof of Theorem 4.1

This appendix provides the proof of Theorem 4.1. We first introduce lemmas and
a proposition that will be used in the proof of Theorem 4.1.

The following lemma provides a concentration inequality that is frequently
used in the literature on statistical learning theory, the proof of which can
be found, for example, in Mohri et al. (2012).

######  Lemma A.8.

(McDiarmid’s Inequality): Let $S=(Z_{1},\ldots,Z_{n})\in{\cal Z}^{n}$ be a set
of $n$ independent random variables, and $g$ be a mapping from ${\cal Z}^{n}$
to $\mathbb{R}$ such that there exist $c_{1},\ldots,c_{n}>0$ that satisfy the
following conditions:

$\displaystyle\left|g\left(z_{1},\ldots,z_{i},\ldots,z_{n}\right)-g\left(z_{1},\ldots,z_{i}^{\prime},\ldots,z_{n}\right)\right|<c_{i}$

for any $n+1$ points $z_{1},\ldots,z_{n},z_{i}^{\prime}$ in $\mathcal{Z}$ and
all $i\in\left\\{1,\ldots,n\right\\}$. Let $g\left(S\right)$ denote
$g\left(Z_{1},\ldots,Z_{n}\right)$. Then the following inequalities hold for
all $\epsilon>0$:

$\displaystyle\Pr\left[g\left(S\right)-E\left[g\left(S\right)\right]\geq\epsilon\right]$
$\displaystyle\leq\exp\left(\frac{-2\epsilon^{2}}{\sum_{i=1}^{n}c_{i}^{2}}\right),$
$\displaystyle\Pr\left[g\left(S\right)-E\left[g\left(S\right)\right]\leq-\epsilon\right]$
$\displaystyle\leq\exp\left(\frac{-2\epsilon^{2}}{\sum_{i=1}^{n}c_{i}^{2}}\right).$

The following proposition gives a finite-sample upper bound on the exact
maximum welfare-regret of the Simultaneous DEWM method that holds with a high
probability.

######  Proposition A.1.

Suppose that any distribution $P\in{\cal P}\left(M,\kappa,\mathcal{G}\right)$
satisfies Assumptions 2.1, 2.2, and 2.4 and that ${\cal{G}}$ satisfies
Assumption 2.3. Then, for any $\delta\in\left(0,1\right)$, the following holds
for any distribution $P\in\mathcal{P}(M,\kappa,\mathcal{G})$ with probability
at least $1-\delta$:

$\displaystyle\left|W_{{\cal
G}}^{\ast}-W\left(\hat{g}^{S}\right)\right|\leq\frac{1}{\sqrt{n}}\sum_{t=1}^{T}\left[\left(\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)\left\\{2C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{2\log\left(1/\delta\right)}\right\\}\right].$

###### Proof.

The proof follows a similar argument as that of Corollary 3.4 of Mohri et al.
(2012). Let $P\in\mathcal{P}(M,\kappa,\mathcal{G})$ be fixed. From the proof
of Theorem 3.4,

$\displaystyle\left|W_{\mathcal{G}}^{\ast}-W\left(\hat{g}^{S}\right)\right|\leq
2\sup_{g\in{\cal G}}\left|W_{n}\left(g\right)-W\left(g\right)\right|.$ (21)

We will evaluate $\sup_{g\in{\cal
G}}\left|W_{n}\left(g\right)-W\left(g\right)\right|$. Let
$S=\left(Z_{1},\ldots,Z_{n}\right)$ be the sample and define

$\displaystyle A\left(S\right)\equiv\sup_{g\in{\cal
G}}\left\\{W\left(g\right)-W_{S}\left(g\right)\right\\},$

where, for any sample $S$ with size $n$, $W_{S}\left(g\right)$ is defined as
$W_{n}\left(g\right)$ that uses the sample $S$. Introduce
$S^{\prime}=\left(Z_{1},\ldots,Z_{n-1},Z_{n}^{\prime}\right)$, a sample that
is different from $S$ with respect to the final component.

Then, it follows that

$\displaystyle A\left(S\right)-A\left(S^{\prime}\right)$
$\displaystyle=\sup_{g\in{\cal G}}\inf_{g^{\prime}\in{\cal
G}}\left\\{W\left(g\right)-W_{S}\left(g\right)-W\left(g^{\prime}\right)+W_{S^{\prime}}\left(g^{\prime}\right)\right\\}$
$\displaystyle\leq\sup_{g\in{\cal
G}}\left\\{W\left(g\right)-W_{S}\left(g\right)-W\left(g\right)+W_{S^{\prime}}\left(g\right)\right\\}$
$\displaystyle=\frac{1}{n}\sup_{g\in{\cal
G}}\left\\{\sum_{t=1}^{T}w_{t}^{S}\left(Z_{n},\underline{g}_{t}\right)-\sum_{t=1}^{T}w_{t}^{S}\left(Z_{n}^{\prime},\underline{g}_{t}\right)\right\\}$
$\displaystyle\leq\frac{1}{n}\sum_{t=1}^{T}\sup_{g\in\mathcal{G}}\left\\{w_{t}^{S}\left(Z_{n},\underline{g}_{t}\right)-w_{t}^{S}\left(Z_{n}^{\prime},\underline{g}_{t}\right)\right\\}$
$\displaystyle\leq\frac{1}{n}\sum_{t=1}^{T}\left(\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right),$

where the last inequality follows from the fact that under Assumptions 2.2 and
2.4, $w_{t}^{S}\left(Z_{i},\underline{g}_{t}\right)$ is bounded from above by
$\left(\gamma_{t}M_{t}/2\right)/\left(\prod_{s=1}^{t}\kappa_{s}\right)$.

Since we have

$\displaystyle\left|A\left(S\right)-A\left(S^{\prime}\right)\right|\leq\frac{1}{n}\sum_{t=1}^{T}\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}},$

applying Lemma A.8 leads to

$\displaystyle
P\left(\left|A\left(S\right)-E_{P^{n}}\left[A\left(S\right)\right]\right|\geq\epsilon\right)\leq\exp\left(\frac{-2n\epsilon^{2}}{\left(\sum_{t=1}^{T}\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)^{2}}\right)$

for any $\epsilon>0$. This is equivalent to the following inequality: for any
$\delta\in\left(0,1\right)$,

$\displaystyle
P\left(\left|A\left(S\right)-E_{P^{n}}\left[A\left(S\right)\right]\right|\leq\left(\sum_{t=1}^{T}\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)\sqrt{\frac{\log\left(1/\delta\right)}{2n}}\right)\geq
1-\delta.$ (22)

Subsequently, we will evaluate $E_{P^{n}}\left[A\right(S\left)\right]$. Since

$\displaystyle E_{P^{n}}[A(S)]$ $\displaystyle=E_{P^{n}}\left[\sup_{g\in{\cal
G}}\left|\sum_{t=1}^{T}\left(W_{nt}(\underline{g}_{t})-W_{t}(\underline{g}_{t})\right)\right|\right]$
$\displaystyle\leq\sum_{t=1}^{T}E_{P^{n}}\left[\sup_{\text{$\underline{g}$}_{t}\in\mathcal{G}_{1}\times\cdots\times\mathcal{G}_{t}}\left|W_{nt}(\text{$\underline{g}$}_{t})-W_{t}(\text{$\underline{g}$}_{t})\right|\right],$

applying Lemma A.4 combined with Lemma A.3 leads to

$\displaystyle E_{P^{n}}[A(S)]\leq
C\sum_{t=1}^{T}\left\\{\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\sqrt{\frac{\sum_{s=1}^{t}v_{s}}{n}}\right\\},$
(23)

where $C$ is the same constant that appears in Lemma A.4.

Consequently, combining (21), (22), and (23), for any
$\delta\in\left(0,1\right)$, it follows with probability at least $1-\delta$
that

$\displaystyle\left|W_{{\cal G}}^{\ast}-W\left(\hat{g}^{S}\right)\right|$
$\displaystyle\leq
2C\sum_{t=1}^{T}\left[\left(\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)\sqrt{\frac{\sum_{s=1}^{t}v_{s}}{n}}\right]+2\left(\sum_{t=1}^{T}\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)\sqrt{\frac{\log\left(1/\delta\right)}{2n}}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{t=1}^{T}\left[\left(\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)\left\\{2C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{2\log\left(1/\delta\right)}\right\\}\right].$

Since this upper bound does not depend on
$P\in\mathcal{P}(M,\kappa,\mathcal{G})$, this probability inequality is
uniform over $\mathcal{P}(M,\kappa,\mathcal{G})$. ∎

With some abuse of notation, let
${g}^{\ast}=(g_{1}^{\ast},\ldots,g_{T}^{\ast})$ be a solution of the
constrained maximization problem (8). The following lemma shows that a class
of feasible DTRs that satisfy the empirical budget/capacity constraints (10)
contains $g^{\ast}$ with high probability.

######  Lemma A.9.

Suppose that a distribution $P$ satisfies Assumption 2.1 and that
$\sum_{t=1}^{T}K_{tb}=1$ holds for all $b=1,\ldots,B$. Define

$\displaystyle\mathcal{G}_{\alpha_{n}}^{S}\equiv$
$\displaystyle\left\\{g\in\mathcal{G}:\sum_{t=1}^{T}K_{tb}\hat{E}\left[g_{t}(H_{t})\mid
D_{1}=g_{1}(H_{1}),\ldots,D_{t-1}=g_{t-1}(H_{t-1})\right]\leq
C_{b}+\alpha_{n}\right.$ $\displaystyle\left.\mbox{\mbox{ for
}}b=1,\ldots,B\right\\},$

which is a subset of DTRs that satisfy the sample budget constraints (10).
Then, for any $\delta\in\left(0,1\right)$, if
$\alpha_{n}\geq\sqrt{\log\left(B/\delta\right)/\left(2n\right)}$,
$g^{\ast}\in{\cal G}_{\alpha_{n}}^{S}$ holds with probability at least
$1-\delta$ under the distribution $P$.

###### Proof.

It follows that

$\displaystyle P\left(g^{\ast}\notin{\cal G}_{\alpha_{n}}^{S}\right)$
$\displaystyle=$ $\displaystyle
P\left(\max_{b=1,\ldots,B}\left\\{\sum_{t=1}^{T}K_{tb}\hat{E}\left[g_{t}^{\ast}(H_{t})\mid
D_{1}=g_{1}^{\ast}(H_{1}),\ldots,D_{t-1}=g_{t-1}^{\ast}(H_{t-1})\right]-C_{b}\right\\}>\alpha_{n}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{b=1}^{B}P\left(\sum_{t=1}^{T}K_{tb}\hat{E}\left[g_{t}^{\ast}(H_{t})\mid
D_{1}=g_{1}^{\ast}(H_{1}),\ldots,D_{t-1}=g_{t-1}^{\ast}(H_{t-1})\right]-C_{b}>\alpha_{n}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{b=1}^{B}P\left(\sum_{t=1}^{T}K_{tb}\hat{E}\left[g_{t}^{\ast}(H_{t})\mid
D_{1}=g_{1}^{\ast}(H_{1}),\ldots,D_{t-1}=g_{t-1}^{\ast}(H_{t-1})\right]\right.$
$\displaystyle\left.-\sum_{t=1}^{T}K_{tb}E_{P}\left[g_{t}^{\ast}(H_{t})\mid
D_{1}=g_{1}^{\ast}(H_{1}),\ldots,D_{t-1}=g_{t-1}^{\ast}(H_{t-1})\right]>\alpha_{n}\right).$

The second inequality follows from the fact that $g^{\ast}$ satisfies the
population budget/capacity constraints (7).

By Hoeffding’s inequality, it follows that

$\displaystyle P$
$\displaystyle\left(\sum_{t=1}^{T}K_{tb}\hat{E}\left[g_{t}^{\ast}(H_{t})\mid
D_{1}=g_{1}^{\ast}(H_{1}),\ldots,D_{t-1}=g_{t-1}^{\ast}(H_{t-1})\right]\right.$
$\displaystyle\left.-\sum_{t=1}^{T}K_{tb}E_{P}\left[g_{t}^{\ast}(H_{t})\mid
D_{1}=g_{1}^{\ast}(H_{1}),\ldots,D_{t-1}=g_{t-1}^{\ast}(H_{t-1})\right]>\alpha_{n}\right)$
$\displaystyle\leq$
$\displaystyle\exp\left\\{-\frac{2n\alpha_{n}^{2}}{\left(\sum_{t=1}^{T}K_{tb}\right)^{2}}\right\\}=\exp\left(-2n\alpha_{n}^{2}\right)$

for each $b=1,\ldots,B$, where the equality follows from the scale
normalization: $\sum_{t=1}^{T}K_{tb}=1$. Thus, we have

$\displaystyle P\left(g^{\ast}\notin{\cal G}_{\alpha_{n}}^{S}\right)\leq
B\exp\left(-2n\alpha_{n}^{2}\right).$

Therefore, if
$\alpha_{n}\geq\sqrt{\log\left(B/\delta\right)/\left(2n\right)}$,
$g^{\ast}\in{\cal G}_{\alpha_{n}}^{S}$ holds with probability at least
$1-\delta$. ∎

######  Proof of Theorem 4.1 (i):.

Let $P\in\mathcal{P}(M,\kappa,\mathcal{G})$ be fixed. We use the notation
$A\leq_{\delta}B$ to denote that $A\leq B$ holds with probability at least
$1-\delta$. From the proof of Proposition A.1, we have, for any $g\in{\cal
G}$, that

$\displaystyle\left|W\left(g\right)-W_{n}\left(g\right)\right|$
$\displaystyle\leq_{\delta}$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{T}\left[\left(\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)\left\\{2C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{2\log\left(1/\delta\right)}\right\\}\right].$
(24)

Applying the same argument as in the proof of Proposition A.1, it follows for
each $b=1,\ldots,B$ that

$\displaystyle\left|E_{n}\left[\sum_{t=1}^{T}K_{tb}\hat{g}_{t}^{S}\left(H_{t}\right)\right]-E_{P}\left[\sum_{t=1}^{T}K_{tb}\hat{g}_{t}^{S}\left(H_{t}\right)\right]\right|$
$\displaystyle\leq_{\delta}$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{T}\left[K_{tb}\left\\{2C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{2\log\left(1/\delta\right)}\right\\}\right].$
(25)

By Lemma A.9, when
$\alpha_{n}\geq\sqrt{\log\left(6B/\delta\right)/\left(2n\right)}$, we have
$W_{n}\left(g^{\ast}\right)\leq_{\delta/6}W_{n}\left(\hat{g}^{S}\right)$
because $g^{\ast}$ is contained in ${\cal G}_{\alpha_{n}}^{S}$ with
probability at least $1-\delta/6$ and $\hat{g}^{S}$ maximizes
$W_{n}\left(\cdot\right)$ over ${\cal G}_{\alpha_{n}}^{S}$. By combining the
fact that
$W_{n}\left(g^{\ast}\right)\leq_{\delta/6}W_{n}\left(\hat{g}^{S}\right)$ with
(24), it follows that

$\displaystyle W\left(g^{\ast}\right)$
$\displaystyle\leq_{\delta/6}W_{n}\left(g^{\ast}\right)+\frac{1}{\sqrt{n}}\sum_{t=1}^{T}\left[\left(\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)\left\\{2C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{2\log\left(6/\delta\right)}\right\\}\right]$
$\displaystyle\leq_{\delta/6}W_{n}\left(\hat{g}^{S}\right)+\frac{1}{\sqrt{n}}\sum_{t=1}^{T}\left[\left(\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)\left\\{2C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{2\log\left(6/\delta\right)}\right\\}\right]$
$\displaystyle\leq_{\delta/6}W\left(\hat{g}^{S}\right)+\frac{2}{\sqrt{n}}\sum_{t=1}^{T}\left[\left(\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)\left\\{2C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{2\log\left(6/\delta\right)}\right\\}\right].$

The first inequality follows from the inequality in (24); the second
inequality follows from the fact that $\hat{g}^{S}$ maximizes
$W_{n}\left(\cdot\right)$ over ${\cal G}_{\alpha_{n}}^{S}$ and
$g^{\ast}\in{\cal G}_{\alpha_{n}}^{S}$ holds with probability at least
$1-\delta/6$; the third inequality follows from the inequality in (24).
Overall, we have

$\displaystyle
W\left(g^{\ast}\right)\leq_{\delta/2}W\left(\hat{g}^{S}\right)+\frac{1}{\sqrt{n}}\sum_{t=1}^{T}\left[\left(\frac{\gamma_{t}M_{t}}{\prod_{s=1}^{t}\kappa_{s}}\right)\left\\{4C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{8\log\left(6/\delta\right)}\right\\}\right].$
(26)

Furthermore, combining the fact that
$W_{n}\left(g^{\ast}\right)\leq_{\delta/6}W_{n}\left(\hat{g}^{S}\right)$ with
equation (25), it follows for each $b=1,\ldots,B$,

$\displaystyle
E_{P}\left[\sum_{t=1}^{T}K_{tb}\hat{g}_{t}^{S}\left(H_{t}\right)\right]$
$\displaystyle\leq_{\delta/\left(2B\right)}E_{n}\left[\sum_{t=1}^{T}K_{tb}\hat{g}_{t}^{S}\left(H_{t}\right)\right]$
$\displaystyle+\frac{1}{\sqrt{n}}\sum_{t=1}^{T}\left[K_{tb}\left\\{2C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{2\log\left(2B/\delta\right)}\right\\}\right]$
$\displaystyle\leq
C_{b}+\alpha_{n}+\frac{1}{\sqrt{n}}\sum_{t=1}^{T}\left[K_{tb}\left\\{2C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{2\log\left(2B/\delta\right)}\right\\}\right],$

where the first inequality follows from the inequality in (25), and the second
inequality follows from the fact that $\hat{g}^{S}\in{\cal
G}_{\alpha_{n}}^{S}$. Thus, the following holds with probability at least
$1-\delta$: for any $b\in\\{1,\ldots,B\\}$,

$\displaystyle
E_{P}\left[\sum_{t=1}^{T}K_{tb}\hat{g}_{t}^{S}\left(H_{t}\right)-C_{b}\right]$
$\displaystyle\leq_{\delta/(2B)}\alpha_{n}+\frac{1}{\sqrt{n}}\sum_{t=1}^{T}\left[K_{tb}\left\\{2C\sqrt{\sum_{s=1}^{t}v_{s}}+\sqrt{2\log\left(2B/\delta\right)}\right\\}\right].$
(27)

Since the upper bounds in (26) and (27) do not depend on
$P\in\mathcal{P}(M,\kappa,\mathcal{G})$, the probability inequalities (26) and
(27) hold uniformly over $\mathcal{P}(M,\kappa,\mathcal{G})$. Therefore, the
theorem follows from combining the probability inequalities (26) and (27). ∎

###  Appendix B Computation

In this appendix, We explain computation of the Backward and Simultaneous
DEWMs with the linear eligibility score rule introduced in Example 2.2. The
non-convexity of the objective functions make these computations challenging.
However, the optimization problems of both methods can be formulated as Mixed
Integer Linear Programming (MILP) problems, for which some efficient softwares
(e.g., CPLEX; Gurobi) are available. In the following subsections, We
illustrate the MILP formalization in the case of $T=2$. We suppose that the
class of feasible treatment rules at each stage $t=1,2$ takes the form of
$\mathcal{G}_{t}=\left\\{1\left\\{(1,H_{t}^{\prime})\beta_{t}\geq
0\right\\}:\beta_{t}\in\mathbb{R}^{(k+2)t-1}\right\\}$.

####  B.1 Backward DEWM

Using slightly different notation from Section 3.1, the first step of the
Backward DEWM method is

$\displaystyle\max_{g_{2}\in{\cal G}_{2}}$
$\displaystyle\sum_{i=1}^{n}m_{i2}^{B}g_{2},$

where

$\displaystyle m_{i2}^{B}=$
$\displaystyle\left(\frac{D_{i2}}{e_{2}\left(1,H_{i2}\right)}-\frac{1-D_{i2}}{e_{2}\left(0,H_{i2}\right)}\right)\gamma_{2}Y_{i2}.$

Let $\hat{g}_{2}^{B}$ be a maximizer of the above problem. Then, the second
step of the Backward DEWM method is

$\displaystyle\max_{g_{1}\in{\cal G}_{1}}$
$\displaystyle\sum_{i=1}^{n}m_{i1}^{B}g_{1},$

where

$\displaystyle m_{i1}^{B}$
$\displaystyle=\left(\frac{D_{i1}}{e_{1}\left(1,H_{i1}\right)}-\frac{1-D_{i1}}{e_{2}\left(0,H_{i1}\right)}\right)$
$\displaystyle\times\left(\frac{D_{i2}\hat{g}_{2}^{B}\left(H_{i2}\right)}{e_{2}\left(1,H_{i2}\right)}-\frac{\left(1-D_{i2}\right)\left(1-\hat{g}_{2}^{B}\left(H_{i2}\right)\right)}{e_{2}\left(0,H_{i2}\right)}\right)\left(\gamma_{1}Y_{1i}+\gamma_{2}Y_{i2}\right).$

When the class of DTRs is constrained to the class of linear eligibility
rules, each step of the Backward DEWM method described in Section 3.1 can be
formulated as MILP problem. The optimization problem in the first step is
equivalent to the following MILP problem:

(First step)

$\displaystyle\underset{\left(z_{12},\ldots,z_{n2}\right)\in\left\\{0,1\right\\}^{n}}{\max_{\beta_{2}\in\mathbb{R}^{2k+3}}}$
$\displaystyle\sum_{i=1}^{n}m_{i2}^{B}z_{i2}$ s.t.
$\displaystyle\frac{(1,H_{i2}^{\prime})\beta_{2}}{C_{i2}}<z_{i2}\leq
1+\frac{(1,H_{i2}^{\prime})\beta_{2}}{C_{i2}}\ \mbox{for }i=1,\ldots,n,$

where $C_{i2}$ are constants that should satisfy
$C_{i2}>\sup_{\beta_{2}\in\mathbb{R}^{2k+3}}\left|(1,H_{i2}^{\prime})\beta_{2}\right|$.

Subsequently, the optimization problem in the second step is equivalent to the
following MILP problem:

(Second step)

$\displaystyle\underset{\left(z_{11},\ldots,z_{n1}\right)\in\left\\{0,1\right\\}^{n}}{\max_{\beta_{1}\in\mathbb{R}^{k+1}}}$
$\displaystyle\sum_{i=1}^{n}m_{i1}^{B}z_{i1}$ s.t.
$\displaystyle\frac{(1,H_{i1}^{\prime})\beta_{1}}{C_{i1}}<z_{i1}\leq
1+\frac{(1,H_{i1}^{\prime})\beta_{1}}{C_{i1}}\ \mbox{for }i=1,\ldots,n,$

where $C_{i1}$ are constants that should satisfy
$C_{i1}>\sup_{\beta_{1}\in\mathbb{R}^{k+1}}\left|(1,H_{i1}^{\prime})\beta_{1}\right|$.

When we specify the dynamic treatment choice problem as the start (stop) time
decision problem discussed in Section 2.2, the linear constraints $z_{i2}\geq
D_{i1}$ and $D_{i2}\geq z_{i1}$ ($z_{i2}\leq D_{i1}$ and $D_{i2}\leq z_{i1}$)
should be added into the MILP problems for the first and second steps,
respectively. When we specify the problem as the one-shot treatment decision
problem discussed in Section 2.2, the linear constraints $z_{i2}+D_{i1}\leq 1$
and $D_{i2}+z_{i1}\leq 1$ should be added into the MILP problems for the first
and second steps, respectively.

####  B.2 Simultaneous DEWM

In the case of $T=2$, the optimization problem of the Simultaneous DEWM method
is equivalent to

$\displaystyle\max_{\left(g_{1},g_{2}\right)\in{\cal G}}$
$\displaystyle\sum_{i=1}^{n}\left[m_{i1}^{S}g_{1}+m_{i2}^{S}g_{2}+m_{i3}^{S}g_{1}g_{2}\right],$

where $m_{is}^{S}$ for $s=1,2,3$ are defined as

$\displaystyle m_{i1}^{S}$
$\displaystyle=\frac{D_{i1}}{e_{1}\left(1,H_{i1}\right)}\left(\gamma_{1}Y_{i1}+\frac{\left(1-D_{i2}\right)\gamma_{2}Y_{i2}}{e_{2}\left(0,H_{i2}\right)}\right)-\frac{1-D_{i1}}{e_{1}\left(0,H_{i1}\right)}\left(\gamma_{1}Y_{i1}+\frac{\left(1-D_{i2}\right)\gamma_{2}Y_{i2}}{e_{2}\left(0,H_{i2}\right)}\right),$
$\displaystyle m_{i2}^{S}$
$\displaystyle=\left(\frac{\left(1-D_{i1}\right)D_{i2}}{e_{1}\left(0,H_{i1}\right)e_{2}\left(1,H_{i2}\right)}-\frac{\left(1-D_{i1}\right)\left(1-D_{i2}\right)}{e_{1}\left(0,H_{i1}\right)e_{2}\left(0,H_{i2}\right)}\right)\gamma_{2}Y_{i2},$
$\displaystyle m_{i3}^{S}$
$\displaystyle=\sum_{\left(d_{1},d_{2}\right)\in\left\\{0,1\right\\}^{2}}\frac{1\left\\{D_{i1}=d_{1},D_{i2}=d_{2}\right\\}\gamma_{2}Y_{i2}}{e_{1}\left(d_{1},H_{i1}\right)e_{2}\left(d_{2},H_{i2}\right)}.$

When the class of DTRs is constrained to the class of linear eligibility
rules, the above optimization problem is equivalent to the following MILP
problem:

$\displaystyle\underset{\left(z_{1t},\ldots,z_{nt}\right)_{t=1}^{3}\in\\{0,1\\}^{3n}}{\max_{\left(\beta_{1}^{\prime},\beta_{2}^{\prime}\right)^{\prime}\in\mathbb{R}^{3k+4}}}$
$\displaystyle\sum_{i=1}^{n}\left[m_{i1}^{S}z_{i1}+m_{i2}^{S}z_{i2}+m_{i3}^{S}z_{i3}\right]$
s.t. $\displaystyle\frac{(1,H_{it}^{\prime})\beta_{t}}{C_{it}}<z_{it}\leq
1+\frac{(1,H_{it}^{\prime})\beta_{t}}{C_{it}}\ \mbox{for }i=1,\ldots,n\mbox{
and }t=1,2,$ $\displaystyle z_{i3}=z_{i1}z_{i2}\mbox{ for }i=1,\ldots,n,$

where $C_{it}$ are constants that should satisfy
$C_{it}>\sup_{\beta_{t}\in\mathbb{R}^{(k+2)t-1}}\left|(1,H_{it}^{\prime})\beta_{t}\right|$.

When we specify the dynamic treatment choice problem as the start (stop) time
decision problem discussed in section 2.2, the linear constraints $z_{i2}\geq
z_{i1}$ ($z_{i2}\leq z_{i1}$) should be added into the MILP problem. When we
specify the problem as the one-shot treatment decision problem discussed in
section 2.2, the linear constraint $z_{i1}+z_{i2}\leq 1$ should be added into
the MILP problem.

####  B.3 Budget/Capacity Constraint

The budget/capacity constraints studied in Section 4 can be incorporated into
the MILP problem for the Simultaneous DEWM. The optimization problem (9) with
the class of linear eligibility score rules is formulated as the following
MILP problem:

$\displaystyle\underset{\left(z_{1t},\ldots,z_{nt}\right)_{t=1}^{3}\in\\{0,1\\}^{3n}}{\max_{\left(\beta_{1}^{\prime},\beta_{2}^{\prime}\right)^{\prime}\in\mathbb{R}^{3k+4}}}$
$\displaystyle\sum_{i=1}^{n}\left[m_{i1}^{S}z_{i1}+m_{i2}^{S}z_{i2}+m_{i3}^{S}z_{i3}\right]$
s.t. $\displaystyle\frac{(1,H_{it}^{\prime})\beta_{t}}{C_{it}}<z_{it}\leq
1+\frac{(1,H_{it}^{\prime})\beta_{t}}{C_{it}}\ \mbox{for }i=1,\ldots,n\mbox{
and }t=1,2,$ $\displaystyle z_{i3}=z_{i1}z_{i2}\mbox{ for }i=1,\ldots,n,$
$\displaystyle\frac{1}{n}\sum_{t=1}^{2}\sum_{i=1}^{n}K_{tb}z_{it}\leq
C_{b}+\alpha_{n}\ \mbox{for }b=1,\ldots,B\mbox{ and }t=1,2,$

where $C_{it}$ are constants that should satisfy
$C_{it}>\sup_{\beta_{t}\in\mathbb{R}^{(k+2)t-1}}\left|(1,H_{it}^{\prime})\beta_{t}\right|$.
The linear constraints in the last line correspond to the budget/capacity
constraints.

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# Symmetry Resolved Entanglement Entropy in Hyperbolic de Sitter Space

Himanshu Gaur<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Bombay, Powai, Mumbai, Maharashtra 400076 India Urjit A. Yajnik
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Bombay, Powai, Mumbai, Maharashtra 400076 India

###### Abstract

In this paper, we study the relation between entanglement and global internal
symmetries on de Sitter space. We consider two symmetric causally disconnected
regions in the hyperbolic chart on de Sitter space. Since entanglement
measures characterises correlations, the study of entanglement between the two
causally disconnected regions gives information about the long range
correlations in de Sitter space. When a theory possesses an additive global
internal symmetry, the entanglement measures for a state with fixed global
charge may be decomposed into local charge sectors in either subsystem and
thus providing a finer resolution of entanglement. Here we will consider two
theories: free complex scalar field, and free Dirac field on de Sitter space.
Both theories possess global internal $U(1)$ symmetry. We study the symmetry
resolved entanglement entropy for both theories in the Bunch-Davies vacuum
state. We find that the symmetry resolved entanglement entropy has
equipartition into local charge sectors upto the terms that scale as
$V_{H_{3}}^{0}$ in the limit of large $V_{H_{3}}$, where $V_{H_{3}}$ is the
volume of either region. This equipartition however is only broken by the
terms of order $O(1/V_{H_{3}})$. Consequently, we have equipartition of
symmetry resolved entanglement entropy in the limit of infinite volume.

###### Contents

  1. 1 Introduction
  2. 2 Hyperbolic de Sitter space
  3. 3 Symmetry resolution of entanglement
  4. 4 Complex scalar field
    1. 4.1 Symmetry resolved entanglement
  5. 5 Dirac field
    1. 5.1 Symmetry resolved entanglement
  6. 6 Conclusion

##  1 Introduction

The phenomenon of entanglement is one of the most fascinating feature of
quantum theories. Study of entanglement has provided key insights in quantum
many body systems [1, 2, 3], black hole thermodynamics [4] and gauge gravity
duality [5, 6].

The study of long range correlations in de Sitter space is particularly
enlightening. Due to the expansion in de Sitter space, the mutually disjoint
observers eventually become causally disconnected, but however still remain
correlated. The long range correlations thus contain information about the
causally disconnected degrees of freedom. To this end study of entanglement
between two causally disconnected regions of de Sitter space is of
considerable interest, since entanglement characterizes correlations. In the
seminal work of Ref. [7], the entanglement entropy between the two symmetric
causally disconnected regions in hyperbolic chart on de Sitter space was
studied. They considered a scalar field with minimal coupling in Bunch-Davies
vacuum and showed that the entanglement entropy is non vanishing. Since then
there have been several investigations of entanglement entropy [8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19], entanglement negativity [20, 21], quantum
discord [22, 23, 24], and bell type inequalities [25, 26, 27, 28, 24, 29] on
de Sitter space in various contexts.

To study entanglement we partition the system into two parts, namely $A$ and
its compliment $B$, such that the Hilbert space has the decomposition
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. If the system is in state
$|\Psi\rangle$, the entanglement entropy $S$ is given by

$S=-\mathrm{Tr}\left[\rho_{A}\ln\rho_{A}\right],$ (1)

where the reduced density matrix $\rho_{A}$ is given by
$\mathrm{Tr}_{B}|\Psi\rangle\langle\Psi|$. We now consider a theory that
possesses an internal global symmetry such that the corresponding conserved
charge is additive i.e. $Q=Q_{A}+Q_{B}$, where $Q$ is conserved global charge
and $Q_{A}$, and $Q_{B}$ are local charge in region $A$, and $B$ respectively.
In such theories the entanglement entropy may be decomposed into local charge
sectors [30]. The symmetry decomposition of entanglement has been shown to
have equipartition in lower dimensional critical systems with global $U(1)$
symmetry [31, 32, 33]. Symmetry decomposition of entanglement has also been
studied in the context of $\text{AdS}_{3}/\text{CFT}_{2}$ correspondence [34,
35, 36, 37, 38].

In the present work, we study the symmetry decomposition of entanglement
entropy between two causally disconnected regions in de Sitter space into the
local charge sectors for theories with global $U(1)$ symmetry. We will
consider two theories, namely minimally coupled complex scalar field, and
minimally coupled Dirac field. Both the theories possess an internal global
$U(1)$ symmetry and are considered to be in the Bunch-Davies vacuum state. We
will work in the hyperbolic chart on de Sitter space. In this chart we
naturally obtain two symmetrical causally disconnected regions namely $R$, and
$L$. The study of symmetry resolved entanglement provides a finer
characterisation of long range correlations in de Sitter space.

This paper is organised as follows. In Section 2 we briefly discuss the
hyperbolic chart on de Sitter space. In Section 3 we discuss the symmetry
decomposition of entanglement entropy in theories with global $U(1)$ symmetry.
In Section 4 we find the symmetry resolved entanglement entropy for complex
scalar field. In Section 5 we find the symmetry resolved entanglement entropy
for Dirac field. Finally, in Section 6 we discuss our results and conclude
this work.

##  2 Hyperbolic de Sitter space

In this section we will discuss the hyperbolic chart on de Sitter space. We
consider the Euclidean de Sitter space and prepare hyperbolic chart on the
Lorentzian de Sitter space by analytically continuing a global chart on
Euclidean de Sitter space. This is done in order to find the Bunch Davies
vacuum or Euclidean vacuum as discussed in Ref. [39]. The hyperbolic de Sitter
consists of three open regions $R$, $L$, and $C$.

The Euclidean de Sitter may be considered as a hypersphere of radius $H^{-1}$
(Hubble constant of de Sitter) in a 5-dimensional Euclidean space. The
hypersphere is given by parametrising the Euclidean co-ordinates
$(\tilde{x}^{0},x^{1},x^{2},x^{3},x^{4})$

$\tilde{x}^{0}=\cos\tau\cos
r,\;x^{1}=\sin\tau,\;\left(\begin{array}[]{c}x^{2}\\\ x^{3}\\\
x^{4}\end{array}\right)=\cos\tau\sin r\left(\begin{array}[]{c}\cos\theta\\\
\sin\theta\cos\varphi\\\ \sin\theta\sin\varphi\end{array}\right),$ (2)

where $(-\pi/2\leq\tau\leq\pi/2)$ and $(0\leq r\leq\pi)$. The metric on the
hypersphere is given by

$\mathrm{d}s_{E}^{2}=H^{-2}\left(\mathrm{d}\tau^{2}+\cos^{2}\tau\left(\mathrm{d}r^{2}+\sin^{2}r\mathrm{d}\Omega^{2}\right)\right).$
(3)

By analytically continuing $\tilde{x}^{0}\to x^{0}$, we obtain a chart on the
Lorentzian de Sitter space. This chart on de Sitter may be divided into three
patches $R$, $L$, and $C$ and they are described by the co-ordinates

$\displaystyle\left\\{\begin{array}[]{ll}t_{R}=i(\tau-\pi/2),&(t_{R}\geq 0)\\\
r_{R}=ir,&(r_{R}\geq 0)\end{array}\right.$ (6)
$\displaystyle\left\\{\begin{array}[]{ll}t_{C}=\tau,&(\pi/2\geq
t_{C}\geq-\pi/2)\\\ r_{C}=i(r-\pi/2),&(\infty\geq
r_{C}\geq-\infty)\end{array}\right.$ (9)
$\displaystyle\left\\{\begin{array}[]{ll}t_{L}=i(-\tau-\pi/2),&(t_{L}\geq
0)\\\ r_{L}=ir,&(r_{L}\geq 0)\end{array}\right.,$ (12)

with their respective metrics

$\displaystyle\mathrm{d}s_{R}^{2}$
$\displaystyle=H^{-2}\left(-\mathrm{d}t_{R}^{2}+\sinh^{2}t_{R}\left(\mathrm{d}r_{R}^{2}+\sinh^{2}r_{R}\mathrm{d}\Omega^{2}\right)\right)$
(13) $\displaystyle\mathrm{d}s_{C}^{2}$
$\displaystyle=H^{-2}\left(\mathrm{d}t_{C}^{2}+\cos^{2}t_{R}\left(-\mathrm{d}r_{C}^{2}+\cos^{2}r_{C}\mathrm{d}\Omega^{2}\right)\right)$
(14) $\displaystyle\mathrm{d}s_{L}^{2}$
$\displaystyle=H^{-2}\left(-\mathrm{d}t_{L}^{2}+\sinh^{2}t_{L}\left(\mathrm{d}r_{L}^{2}+\sinh^{2}r_{L}\mathrm{d}\Omega^{2}\right)\right).$
(15)

Figure 1:  Penrose diagram for hyperbolic chart on de Sitter space. The three
patches $R$, $L$, and $C$ are denoted in the figure. Thinner red line on
either $R$ and $L$ patch are constant $t_{R}$ and $t_{L}$ surfaces
respectively. The thick black line is the chosen Cauchy surface.

The Penrose diagram for the hyperbolic chart on de Sitter is shown in Figure
1. Now in a field theory to find the global euclidean modes a Cauchy surface
must be chosen. This is done by choosing an large hyperbolic radius $r_{R}$
and $r_{L}$ on either patch at $t_{R}=\text{constant}$ and
$t_{L}=\text{constant}$ respectively. Finally, these two parts are connected
via a bridge through $C$. This construction is shown in Figure 1. When the
radius on either patch is appropriately large, we may ignore the bridge while
normalising the modes on the Cauchy surface.

##  3 Symmetry resolution of entanglement

In this section we will discuss the symmetry decomposition of entanglement
into the local charge sectors for a system with global $U(1)$ symmetry [30].
We will consider that our system is partitioned into two parts, namely $A$ and
its compliment $B$. The partition is such that the Hilbert space has the
decomposition $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$.

We further assume that the charge $\hat{Q}$ corresponding to the global $U(1)$
symmetry is additive in $A$ and $B$ i.e. $\hat{Q}=\hat{Q}_{A}+\hat{Q}_{B}$.
When the system is in a state $|\Psi\rangle$ of fixed global charge we have
the relation $[\rho,\hat{Q}]=0$, where $\rho=|\Psi\rangle\langle\Psi|$. Taking
the trace over $B$ degrees of freedom gives $[\rho_{A},\hat{Q}_{A}]=0$. This
implies that the RDM $\rho_{A}$ is block diagonal in local charge sectors of
$\hat{Q}_{A}$. We may project $\rho_{A}$ into the local charge sector $q$
using the projection operators $\Pi_{q}$

$p_{q}\rho_{A,q}=\Pi_{q}\rho_{A}\Pi_{q},$ (16)

where $q$ is an eigenvalue of $\hat{Q}_{A}$, $\rho_{A,q}$ is the density
matrix in charge sector $q$ and $p_{q}$ is the probability of getting $q$ upon
measurement of $\hat{Q}_{A}$. The study of entanglement in the charge sectors
$q$ has been termed symmetry resolved entanglement. The Rényi entropy and
entanglement entropy in the local charge sector $q$ is given by

$\displaystyle S_{n,q}$
$\displaystyle=\frac{1}{1-n}\mathrm{Tr}\ln\left[\rho_{A,q}^{n}\right],$ (17)
$\displaystyle S_{q}$
$\displaystyle=-\partial_{n}\mathrm{Tr}\ln\left.\left[\rho_{A,q}^{n}\right]\right|_{n\to
1}$ (18)

respectively. This block structure of density matrix however may not manifest
easily and for this purpose charged moments $Z_{n}(\alpha)$ are introduced.
For the present case the charged moments are given by

$Z_{n}(\alpha)=\mathrm{Tr}\left[\rho_{A}^{n}e^{i\alpha\hat{Q}_{A}}\right].$
(19)

Taking their fourier transform gives the trace of
$\left(p_{q}\rho_{A,q}\right)^{n}$. For brevity we denote this via
$\mathcal{Z}_{n}(q)$ and it is given by

$\mathcal{Z}_{n}(q)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}\alpha\,Z_{n}(\alpha)e^{-iq\alpha}.$
(20)

The symmetry resolved Rényi entropy $S_{n,q}$, and Entanglement entropy
$S_{q}$ in charge sector $q$ are then given by using eq.(20) in eq.(17) and
eq.(18) respectively,

$\displaystyle S_{n,q}$
$\displaystyle=\frac{1}{1-n}\ln\left[\frac{\mathcal{Z}_{n}(q)}{\mathcal{Z}_{1}^{n}(q)}\right],$
(21) $\displaystyle S_{q}$
$\displaystyle=-\partial_{n}\ln\left.\left[\frac{\mathcal{Z}_{n}(q)}{\mathcal{Z}_{1}^{n}(q)}\right]\right|_{n\to
1}.$ (22)

##  4 Complex scalar field

In this section we find the symmetry resolved entanglement entropy of complex
scalar field on hyperbolic de Sitter space.

We consider a complex scalar field with a minimal coupling on the de Sitter
space, whose action reads

$S_{S}=-\int\mathrm{d}^{4}x\,\sqrt{-g}\,\left(g^{\mu\nu}\nabla_{\mu}\phi^{*}\nabla_{\nu}\phi+m^{2}\phi^{*}\phi\right).$
(23)

Complex boson field has a global $U(1)$ symmetry under the transformation
$\phi\to\phi\,e^{i\alpha}$ and $\phi^{*}\to\phi^{*}e^{-i\alpha}$, with the
corresponding conserved charge $Q$

$Q=i\int\mathrm{d}S^{\mu}\sqrt{-g}(\phi\,\partial_{\mu}\phi^{*}-\phi^{*}\partial_{\mu}\phi)$
(24)

where $\mathrm{d}S^{\mu}$ is the differential surface element on the Cauchy
surface and
$i\sqrt{-g}(\phi\,\partial_{\mu}\phi^{*}-\phi^{*}\partial_{\mu}\phi)$ is the
conserved current $J_{\mu}$. As discussed in Section 2 in hyperbolic slicing
of de Sitter space we have three patches. Here we are interested in the
correlations between the disconnected patches $R$ and $L$. Since, for the
Cauchy surface chosen in Section 2 we may ignore the $C$ patch, what we then
have is effectively a theory on $R\cup L$. The mode equation on either patch
is given by

$\left[\frac{1}{\sinh^{3}t}\frac{\partial}{\partial
t}\sinh^{3}t\frac{\partial}{\partial
t}-\frac{1}{\sinh^{2}t}\boldsymbol{L}^{2}+\frac{9}{4}-\nu^{2}\right]u(t,r,\Omega)=0,$
(25)

where $\boldsymbol{L}^{2}$ is the Laplacian on the unit 3-hyperboloid and the
parameter $\nu$ introduced above is

$\nu=\sqrt{\frac{9}{4}-\frac{m^{2}}{H^{2}}}.$ (26)

The case of coupling $\xi$ to the scalar curvature may be similarly considered
after replacing $m^{2}$ by $m_{eff}^{2}=m^{2}+12\xi H^{2}$ in eq.(26). We have
the case of conformally coupled massless scalar at $\nu=\frac{1}{2}$ and
minimally coupled massless scalar at $\nu=\frac{3}{2}$. The corresponding
local charge $Q_{A}$ on the $R$ patch is given by

$Q_{A}=\int_{R}\mathrm{d}^{3}\,x\,J_{0},$ (27)

where the integral is over the constant $t_{R}$ surface.

###  4.1 Symmetry resolved entanglement

The global Bunch Davies vacuum however does not co-inside with the $R$, and
$L$ vacua. This is easily seen by taking the mode expansion of the field
$\phi$ in terms of globally well defined modes and locally well defined modes
on either patch [39]. The global vacuum state for boson field when written in
$R$ and $L$ modes is a Gaussian state. The exact expression for scalar field
has been obtained in the Ref. [7]. The case of complex boson can be handled
similarly and the Bunch Davies vacuum in terms of the positive frequency mode
operator $a_{i,p,l,m}$, and negative frequency mode operator $b_{i,p,l,m}$ on
the $R$ and $L$ patch is given by

$|{\Omega}\rangle_{p,l,m}\propto
e^{m_{ij}a_{i}^{\dagger}b_{j}^{\dagger}}|{R}\rangle|{L}\rangle,$ (28)

where $a_{R}|R\rangle=b_{R}|R\rangle=0$, $a_{L}|L\rangle=b_{L}|L\rangle=0$,
$i\in\left\\{R,L\right\\}$ and

$m_{ij}=\frac{\sqrt{2}e^{-p\pi}}{\sqrt{\cosh 2\pi p+\cos
2\pi\nu}}\left(\begin{array}[]{cc}\cos\pi\nu&i\sinh p\pi\\\ i\sinh
p\pi&\cos\pi\nu\end{array}\right).$ (29)

The mode operators satisfy the commutation relations
$[a_{i,p,l,m},a_{i^{\prime},p^{\prime},l^{\prime},m^{\prime}}^{\dagger}]$
$=[b_{i,p,l,m},b_{i^{\prime},p^{\prime},l^{\prime},m^{\prime}}^{\dagger}]$
$=\delta_{i,i^{\prime}}\delta_{p,p^{\prime}}\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}$,
and rest are zero. In eq.(28) we suppressed the quantum numbers $(p,l,m)$ on
operators $a_{i}$, and $b_{i}$ and will continue to do so in what follows for
brevity. The quantum numbers $(p,l,m)$ corresponds to the Laplacian on the
hyperbola appearing in eq.(25). The local charge operator in terms of bosonic
mode operators is

$Q_{A}=\sum_{p,l,m}a^{\dagger}_{R}a_{R}-b^{\dagger}_{R}b_{R}.$ (30)

We now introduce the Schmidt basis so that the vacuum state may be written as

$|\Omega\rangle_{p,l,m}\propto
e^{\gamma_{p}\left(\tilde{a}_{R}^{\dagger}\tilde{b}_{L}^{\dagger}+\tilde{b}_{R}^{\dagger}\tilde{a}_{L}^{\dagger}\right)}|\tilde{R}\rangle|\tilde{L}\rangle,$
(31)

with the relation
$\tilde{a}_{R}|\tilde{R}\rangle=\tilde{b}_{R}|\tilde{R}\rangle=0$ and
$\tilde{a}_{L}|\tilde{L}\rangle=\tilde{b}_{L}|\tilde{L}\rangle=0$. The two
sets of Bosonic operators are related via the Bogoliubov transformation

$\displaystyle\tilde{a}_{R}$
$\displaystyle=\alpha{a}_{R}+\beta{b}_{R}^{\dagger},\qquad\tilde{b}_{R}=\alpha{b}_{R}+\beta{a}_{R}^{\dagger}$
(32) $\displaystyle\tilde{a}_{L}$
$\displaystyle={\alpha}^{*}{a}_{L}+{\beta}^{*}{b}_{L}^{\dagger},\qquad\tilde{b}_{L}={\alpha}^{*}{b}_{L}+{\beta}^{*}{a}_{L}^{\dagger}.$
(33)

The mode operators satisfy the commutation relation
$[\tilde{a}_{i},\tilde{a}_{i^{\prime}}^{\dagger}]=[\tilde{b}_{i},\tilde{b}_{i^{\prime}}^{\dagger}]=\delta_{i,i^{\prime}}$
and consequently the Bogoliubov coefficients satisfy the relation
$|\alpha|^{2}-|\beta|^{2}=1$. From eq.(28), eq.(29), and eq.(31)-eq.(33) we
obtain $\gamma_{p}$ (in eq.(31)) to be

$\gamma_{p}=i\frac{\sqrt{2}}{\sqrt{\cosh 2\pi p+\cos 2\pi\nu}+\sqrt{\cosh 2\pi
p+\cos 2\pi\nu+2}}.$ (34)

The reduced density matrix in the Schmidt basis is then obtained by taking the
trace over $L$ degrees of freedom. The reduced density matrix is given by

$\left(\rho_{A}\right)_{p,l,m}=\frac{1}{\left(1-|\gamma_{p}|^{2}\right)^{2}}\sum_{j,k=0}^{\infty}|\gamma_{p}|^{2j+2k}|j,k,p,l,m\rangle\langle
j,k,p,l,m|.$ (35)

The local charge operator $\hat{Q}$ in terms of the new bosonic operators is
easily obtained from eq.(32) and eq.(33) to be

$\hat{Q}_{A}=\sum_{p,l,m}\tilde{a}_{{R}}^{\dagger}\tilde{a}_{{R}}-\tilde{b}_{{R}}^{\dagger}\tilde{b}_{{R}}.$
(36)

The charged moments in sector $(p,l,m)$ are obtained using the eq.(35) and
eq.(36) in eq.(19) and are given by

$\left(Z_{n}\left(\alpha\right)\right)_{p,l,m}=\frac{\left(1-|\gamma_{p}|^{2}\right)^{2n}}{1-2|\gamma_{p}|^{2n}\cos\left(\alpha\right)+|\gamma_{p}|^{4n}}.$
(37)

The summation over all modes $(p,l,m)$ is equivalent to the volume of the
hyperbola and an integration in $p$ over the density of states [7]. The
charged moments are given by

$Z_{n}(\alpha)=\exp\left\\{V_{H_{3}}\int_{0}^{\infty}\mathrm{d}p\,\frac{p^{2}}{2\pi^{2}}\ln\left(\frac{\left(1-|\gamma_{p}|^{2}\right)^{2n}}{1-2|\gamma_{p}|^{2n}\cos\left(\alpha\right)+|\gamma_{p}|^{4n}}\right)\right\\},$
(38)

where $V_{H_{3}}$ is the volume of the $R$ patch on the Cauchy surface and
$\frac{p^{2}}{2\pi^{2}}$ is the density of modes for bosons on the hyperbola
[40]. To evaluate $\mathcal{Z}_{n}(q)$ using eq.(20), we first note that the
volume $V_{H_{3}}\gg 1$ and take the taylor expansion of the exponent in
eq.(38) upto $O(\alpha^{2})$. We have

$\displaystyle\int_{0}^{\infty}\mathrm{d}p\frac{p^{2}}{2\pi^{2}}\ln\left(\frac{\left(1-|\gamma_{p}|^{2}\right)^{2n}}{1-2|\gamma_{p}|^{2n}\cos\left(\alpha\right)+|\gamma_{p}|^{4n}}\right)$
$\displaystyle\approx\int_{0}^{\infty}\mathrm{d}p\frac{p^{2}}{2\pi^{2}}\ln\left(\frac{\left(1-|\gamma_{p}|^{2}\right)^{2n}}{1-2|\gamma_{p}|^{2n}+|\gamma_{p}|^{4n}}\right)-\alpha^{2}\frac{\Lambda_{n}^{s}}{2},$
(39) $\displaystyle\text{where,}\qquad\Lambda_{n}^{s}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}p\frac{p^{2}}{2\pi^{2}}\frac{2|\gamma_{p}|^{2n}}{1-2|\gamma|^{2n}+|\gamma|^{4n}}.$
(40)

This is a good approximation for $Z_{n}(\alpha)$ in the range
$\alpha\in(-\pi,\pi)$. The fourier transform of $Z_{n}(\alpha)$ is evaluated
by taking the Gaussian approximation of the integral to be

$\mathcal{Z}_{n}(q)\approx\frac{Z_{n}(0)}{\sqrt{2\pi\Lambda_{n}^{s}V_{H_{3}}}}\exp\left\\{-\frac{q^{2}}{2\Lambda_{n}^{s}V_{H_{3}}}\right\\}.$
(41)

The quantity $\mathcal{Z}_{1}(q)$ is the charge distribution probability of
the local charge $\hat{Q}_{A}$. We observe that the charge $q$ has a Gaussian
distribution with the standard deviation being proportional to the volume
$V_{H_{3}}$. The symmetry resolved entanglement entropy is obtained by using
eq.(41) in eq.(20)

$S_{q}\approx
S-\frac{1}{2}\ln\left[V_{H_{3}}\right]+\left\\{\frac{1}{2}\left.\frac{\partial_{n}\Lambda_{n}^{s}}{\Lambda_{n}^{s}}\right|_{n\to
1}-\frac{1}{2}\ln\left[2\pi\Lambda_{1}^{s}\right]\right\\}-\frac{q^{2}}{2V_{H_{3}}}\left(\left.\frac{\partial_{n}\Lambda_{n}^{s}}{\left(\Lambda_{n}^{s}\right)^{2}}\right|_{n\to
1}+\frac{1}{\Lambda_{1}^{s}}\right),$ (42)

where $S$ is the entanglement entropy of the complex scalar field. We notice
that the symmetry resolved entanglement entropy in the large $V_{H_{3}}$ limit
has an equipartition into the charge sectors $q$ upto the leading order terms
in the volume $V_{H_{3}}$. This equipartition is broken only by the terms of
order $O\left(1/V_{H_{3}}\right)$. Hence, in the limit $V_{H_{3}}\to\infty$ we
have equipartition to all orders.

Figure 2:  On the left is plotted $\Lambda_{1}$ given by eq.(40). The standard
deviation for the charge distribution is proportional to $\Lambda_{1}$. It is
shown to have a local maxima at $\nu=\frac{1}{2}$ and $\nu=\frac{3}{2}$. On
the right is plotted "curly bracket term" in eq.(42) and it is denoted by
$O\left(V_{H_{3}}^{0}\right)$ above. This term is also shown to have a local
maxima at $\nu=\frac{1}{2}$ and $\nu=\frac{3}{2}$.

We see from Figure 2 that $\Lambda_{1}^{s}$ has a local maxima at conformally
coupled massless scalar and minimally coupled massless scalar cases,
reflecting higher probability of non-zero local charge at these points than
other values of $\nu^{2}$. The term that scale as $V_{H_{3}}^{0}$ (i.e "curly
bracket term") in eq.(42) is plotted in Figure 2. This contribution to
symmetry resolved entanglement entropy is again maximised at conformally
coupled massless scalar and minimally coupled massless scalar cases, similar
to the entanglement entropy behaviour for the scalar field. We also notice the
strange behaviour for both the plots in Figure 2 in the low mass regime
$\left(\nu\geq\frac{1}{2}\right)$, a similar behaviour is also present in
entanglement entropy of the scalar field [7], since we would expect
correlations to become stronger as mass decreases. We mention here that there
is a supercurvature mode $\left(p=i\left(\nu-\frac{1}{2}\right)\right)$ in
this regime [39]. The entanglement contribution, if any, of this mode has not
been calculated so far and can possibly alter the behaviour of entanglement in
this region. This mode is not considered in this work as well. However it is
true that this supercurvature mode is not entangled with any other mode since
it has a different value of $p$.

##  5 Dirac field

In this section we find the symmetry resolved entanglement entropy for Dirac
field on hyperbolic de Sitter space.

We consider the Dirac field on the de Sitter space with minimal coupling,
whose action reads

$S_{D}=\int\mathrm{d}^{4}\,x\sqrt{-g}\bar{\Psi}\left(\gamma^{\mu}\mathrm{D}_{\mu}-m\right)\Psi.$
(43)

Dirac field has a global internal $U(1)$ symmetry under $\Psi\to
e^{i\alpha}\Psi$, and $\Psi^{\dagger}\to e^{-i\alpha}\Psi^{\dagger}$ and the
corresponding conserved charge is

$Q=-\int\mathrm{d}\,S_{\mu}\bar{\Psi}\gamma^{\mu}\Psi,$ (44)

where $\mathrm{d}\,S_{\mu}$ is the differential surface element on the Cauchy
surface. We consider the hyperbolic chart on de Sitter space just as we did in
the last section. The problem of computing the Bunch Davis vacuum and
entanglement entropy for Dirac field in the same setting has been considered
in Ref.[8]. The local charge $Q_{A}$ on the $R$ patch is

$Q_{A}=\int_{R}\mathrm{d}^{3}\,x\sqrt{-g}\Psi^{\dagger}\Psi,$ (45)

where the integral is on the constant $t_{R}$ surface.

###  5.1 Symmetry resolved entanglement

In the case of fermions we may write the Bunch Davies vacuum $|\Omega\rangle$
as a direct product $|0\rangle^{+}|0\rangle^{-}$. The state $|0\rangle^{+}$
$\left(|0\rangle^{-}\right)$ is annihilated by spin down (spin up) global
positive frequency mode, and spin up (spin down) global negative frequency
mode annihilation operators. The state $|0\rangle^{+}$ in terms of mode
operators on the $R$ patch $c^{R}_{(\downarrow,p,l,m)}$,
$d^{R}_{(\uparrow,p,l,m)}$ and on the $L$ patch $c^{L}_{(\uparrow,p,l,m)}$,
$d^{L}_{(\downarrow,p,l,m)}$ is given by

$|0\rangle_{(p,l,m)}^{+}\propto
e^{m_{ij}c^{i\dagger}d^{j\dagger}}|0\rangle_{R}^{+}|0\rangle_{L}^{-},$ (46)

where we have the relations
$c^{R}_{\downarrow}|0\rangle_{R}^{+}=d^{R}_{\uparrow}|0\rangle_{R}^{+}=0$ and
$c^{L}_{\uparrow}|0\rangle_{L}^{-}=d^{L}_{\downarrow}|0\rangle_{L}^{-}=0$. The
state $|0\rangle^{-}$ is written similarly by using the complimentary spin
mode operators on both patches. The mode operators satisfy the anti-
commutation relation
$\\{c_{s,p,l,m}^{i},c_{s^{\prime},p^{\prime},l^{\prime},m^{\prime}}^{i^{\prime}\dagger}\\}=\\{d_{s,p,l,m}^{i},d_{s^{\prime},p^{\prime},l^{\prime},m^{\prime}}^{i^{\prime}\dagger}\\}=\delta_{i,i^{\prime}}\delta_{s,s^{\prime}}\delta_{p,p^{\prime}}\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}$,
and rest are zero. From here on we will suppress the spins on mode operators
as well. The matrix $m_{ij}$ in this case is given by

$\displaystyle m_{ij}=$
$\displaystyle-\frac{B^{*}}{A^{2}+1}\left(\begin{array}[]{cc}A&-1\\\
1&A\end{array}\right),\qquad\text{where}$ (49) $\displaystyle
A=\frac{\sinh\frac{\pi m}{H}}{\cosh\pi p},$ $\displaystyle\qquad
B=\frac{e^{-\pi
p}\Gamma\left(\frac{1}{2}-ip\right)^{2}}{\Gamma\left(\frac{1}{2}-ip-i\frac{m}{H}\right)\Gamma\left(\frac{1}{2}-ip+i\frac{m}{H}\right)}.$
(50)

The local charge operator $Q_{A}$ in terms of mode operators on the $R$ patch
is given by

$Q_{A}=\sum_{s,p,l,m}c^{R\dagger}c^{R}-d^{R\dagger}d^{R},$ (51)

where $s$ in the summation stands for spin. The Schmidt basis were introduced
for this problem through Bogoliubov transformations

$\displaystyle\tilde{c}_{R}$
$\displaystyle=u{c}^{R}+v{d}^{R{\dagger}},\qquad\tilde{d}_{R}=u{d}^{R}-v{c}^{R\dagger}$
(52) $\displaystyle\tilde{c}_{L}$
$\displaystyle={u}^{*}{c}^{L}-{v}^{*}{d}^{L\dagger},\qquad\tilde{d}_{L}={u}^{*}{d}^{L}+{v}^{*}{c}^{L\dagger}.$
(53)

in the Ref.[8]. The mode operators satisfy the anti-commutation relations
$\\{\tilde{c}_{i},\tilde{c}_{i^{\prime}}^{\dagger}\\}=\\{\tilde{d}_{i},\tilde{d}_{i^{\prime}}^{\dagger}\\}=\delta_{i,i^{\prime}}$
and rest are zero. The Bogoliubov coefficients satisfy the relation
$|u|^{2}+|v|^{2}=1$. The state $|0\rangle^{+}$ in the Schmidt basis is given
by

$|0\rangle^{+}_{p,l,m}\propto
e^{\gamma_{p}\left(\tilde{c}_{R}^{\dagger}\tilde{d}_{L}^{\dagger}+\tilde{d}_{R}^{\dagger}\tilde{c}_{L}^{\dagger}\right)}|0\rangle_{R^{\prime}}^{+}|0\rangle_{L^{\prime}}^{-},$
(54)

with the relations
$\tilde{c}_{R}|0\rangle_{R^{\prime}}^{+}=\tilde{d}_{R}|0\rangle_{R^{\prime}}^{+}=0$
and
$\tilde{c}_{L}|0\rangle_{L^{\prime}}^{+}=\tilde{d}_{L}|0\rangle_{L^{\prime}}^{+}=0$.
The parameter $\gamma_{p}$ is given by

$\gamma_{p}=\frac{1}{2m_{12}}\left(m_{11}^{2}+m_{12}^{2}-1+\sqrt{\left(m_{11}^{2}+m_{12}^{2}-1\right)^{2}+4m_{12}^{2}}\right).$
(55)

The reduced density matrix is then obtained by taking the trace over the $L$
degrees of freedom. It is given by

$\begin{split}&{\rho_{A}}_{s,p,l,m}=\\\ &\hskip
14.45377pt\frac{1}{\left(1+|\gamma_{p}|^{2}\right)^{2}}\left(|0\rangle^{+}_{R^{\prime}}{}_{R^{\prime}}^{+}\langle
0|+|\gamma_{p}|^{2}|1,0\rangle^{+}_{R^{\prime}}{}_{R^{\prime}}^{+}\langle
1,0|+|\gamma_{p}|^{2}|0,1\rangle^{+}_{R^{\prime}}{}_{R^{\prime}}^{+}\langle
0,1|+|\gamma_{p}|^{4}|1,1\rangle^{+}_{R^{\prime}}{}_{R^{\prime}}^{+}\langle
1,1|\right)\end{split}$ (56)

The local charge operator $Q_{A}$ on the $R$ patch is obtained by using
eq.(52), and eq.(53) in eq.(51)

$Q_{A}=\sum_{s,p,l,m}\tilde{c}_{R^{\prime}}^{\dagger}\tilde{c}_{R^{\prime}}-\tilde{d}_{R^{\prime}}^{\dagger}\tilde{d}_{R^{\prime}}.$
(57)

Figure 3:  On the left is plotted $\Lambda_{1}^{f}$ given by eq.(59). The
standard deviation of the charge distribution is proportional to
$\Lambda_{1}^{f}$ and it shown to monotonically increase with $\nu^{2}$. On
the right is plotted the "curly bracket term" in eq.(61). This term is denoted
by $O(V_{H_{3}}^{0})$ and is shown to monotonically decrease with
$\frac{m^{2}}{H^{2}}$.

The charged moments for fermions are obtained using the eq.(56), and eq.(57)
in eq.(20). We have

$\displaystyle Z_{n}(\alpha)$ $\displaystyle\approx
Z_{n}(0)\exp\left\\{-\frac{V_{H_{3}}\Lambda_{n}^{f}\alpha^{2}}{2}\right\\},\qquad\text{where}$
(58) $\displaystyle\Lambda_{n}^{f}$
$\displaystyle=2\int_{0}^{\infty}\mathrm{d}p\left(\frac{1/4+p^{2}}{2\pi^{2}}\right)|\gamma|^{2n}$
(59)

here we have only taken terms upto $O(\alpha^{2})$ in the exponent and used
the density of states for fermions on hyperbola to be
$({1/4+p^{2}})/{2\pi^{2}}$ [40, 41]. $\mathcal{Z}_{n}(q)$ is obtained by
taking the fourier transform of eq.(58). Taking the Gaussian approximation in
the integral we obtain

$\mathcal{Z}_{n}(q)\approx\frac{Z_{n}(0)}{\sqrt{2\pi\Lambda_{n}^{f}V_{H_{3}}}}\exp\left\\{-\frac{q^{2}}{2\Lambda_{n}^{f}V_{H_{3}}}\right\\}.$
(60)

We note that similar to the case of complex scalar field the local charge has
a Gaussian distribution with standard deviation proportional to the volume
$V_{H_{3}}$. Using eq.(60), the symmetry resolved entanglement entropy is
obtained to be

$S_{q}=S-\frac{1}{2}\ln\left[V_{H_{3}}\right]+\left\\{\frac{1}{2}\left.\frac{\partial_{n}\Lambda_{n}^{f}}{\Lambda_{n}^{f}}\right|_{n\to
1}-\frac{1}{2}\ln\left[2\pi\Lambda_{1}^{f}\right]\right\\}-\frac{q^{2}}{2V_{H_{3}}}\left(\left.\frac{\partial_{n}\Lambda_{n}^{f}}{\left(\Lambda_{n}^{f}\right)^{2}}\right|_{n\to
1}+\frac{1}{\Lambda_{1}^{f}}\right),$ (61)

here $S$ is the entanglement entropy for the Dirac field. Similar to the
bosonic case we see that the symmetry resolved entanglement entropy has an
equipartition into the local charge sectors upto the leading order terms in
$V_{H_{3}}$. The plot for the term that scale as $V_{H_{3}}^{0}$ (i.e. "curly
bracket term") is shown in Figure 3, where it is shown to monotonically
increase with $\nu^{2}=\frac{9}{4}-\frac{m^{2}}{H^{2}}$ and hence the
contribution to symmetry resolved entanglement entropy at this order
monotonically decreases with $\frac{m^{2}}{H^{2}}$. The equipartition of
entanglement entropy is broken by an $O\left(1/V_{H_{3}}\right)$ term in this
case as well.

In the case of Dirac fermions there is no strange behaviour in the low mass
limit. It is also true that there are no supercurvature modes present for
Dirac fermions. We also observe that the $O\left(V_{H_{3}}^{0}\right)$
contribution to symmetry resolved entanglement entropy is higher in Dirac
fermion case than the scalar case in the limit $m\to 0$, similar to the
behaviour of entanglement entropy [8].

##  6 Conclusion

We have studied the symmetry decomposition of entanglement entropy for
theories in their ground state with global $U(1)$ symmetry into the local
charge sectors on de Sitter background. We considered the entanglement between
two causally disconnected regions, namely $R$ and $L$ patch, in the hyperbolic
chart on de Sitter space. We studied two cases, free complex scalar field, and
free Dirac field. We first showed that the local charge in either region has a
Gaussian distribution for both the fields. We then found that for both of
these cases the symmetry resolved entanglement entropy has an equipartition
into the local charge sectors $q$ upto terms that scales as $V_{H_{3}}^{0}$,
where $V_{H_{3}}$ is the volume of the constant time slice of the subregion
(i.e. the $R$ patch). We also found that the leading term is just the
entanglement entropy for the field in the expression for symmetry resolved
entanglement entropy. The equipartition of symmetry resolved entanglement
entropy is however broken by a term proportional to the inverse of the same
volume. This terms vanishes in the limit of infinite volume and hence we have
the equipartition of symmetry resolved entanglement entropy.

In the low mass regime $\frac{1}{2}\leq\nu\leq\frac{3}{2}$, we saw a strange
oscillatory behaviour for the $O\left(V_{H_{3}}^{0}\right)$ contribution to
the symmetry resolved entanglement entropy in the complex scalar case, similar
to the behaviour of the entanglement entropy [7]. In this regime there is a
supercurvature mode present whose contribution, if any, was not studied in
this work. The contribution from supercurvature mode to entanglement entropy,
if any, has not been calculated yet. The $O\left(V_{H_{3}}^{0}\right)$
contribution term is also shown to have a local maxima at $\nu=\frac{1}{2}$
(massless conformally coupled field) and $\nu=\frac{3}{2}$ (massless minimally
coupled field). In the Dirac field case there are no supercurvature modes and
the $O\left(V_{H_{3}}\right)$ contribution is shown to monotonically decrease
with $\frac{m^{2}}{H^{2}}$. We also found that this contribution to symmetry
resolved entanglement entropy is higher in Dirac field than the scalar field
in the limit $m\to 0$.

## Acknowledgments

HG is supported by the Prime Minister’s Research fellowship provided by
Ministry of Education, Government of India. UAY is supported by an Institute
Chair Professorship.

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