$\displaystyle\geqslant\frac{1}{n-1}\sum_{j=1}^{n-1}\left\\{{\frac{1}{\eta}\left[{H\big{(}{\rho^{\prime}_{A_{j}}}\big{)}+H\big{(}{\rho^{\prime}_{B_{j+1}}}\big{)}-2h(\eta)-({1-\eta})\,n\log(2)}\right]-2\log(\eta)}\right\\}$
$\displaystyle=\frac{1}{\eta}\left\\{{\frac{1}{n-1}\sum_{j=1}^{n-1}\left[{\II[\rho^{\prime}]{A_{j}}{B_{j+1}}+H\left({\rho^{\prime}}\right)}\right]-2h(\eta)-({1-\eta})\,n\log(2)}\right\\}-2\log(\eta)$
$\displaystyle=\frac{1}{\eta}\left[{E(\rho^{\prime})+H(\rho^{\prime})-2h(\eta)-({1-\eta})\,n\log(2)}\right]-2\log(\eta)$
$\displaystyle\geqslant\frac{1}{\eta}\left[{E(\rho)-r\mu(\mathcal{G})+H(\rho)-2h(\eta)-({1-\eta})\,n\log(2)}\right]-2\log(\eta)\
.$ (41hs)

The first and second equalities follow from the definitions 41gy and 41gz,
respectively.The last inequality follows because
$E(\rho^{\prime})-E(\rho)\geqslant-r\mu(\mathcal{G})$, by Appendix G, and
because $H(\rho^{\prime})=H(\rho)$, since the von Neumann entropy is unitarily
invariant.Finally, 41hs implies 41hm, since $\zeta$ is arbitrary.∎

##  Appendix H Bounds from short-time Hamiltonian dynamics

Here, we apply the bounds in Appendix G to short-time Hamiltonian dynamics.We
consider a nearest-neighbor Hamiltonian $H$ acting on a 1D chain $S$ of
$n\geqslant 2$ qubits.The chain is initially in a state $\rho$.We show that
the entanglement measure $E(\rho)$ can grow only slowly in time.

######  Corollary 57 (Continuity bound on $E(\rho)$ under local Hamiltonian
evolution).

Let $\rho$ denote any quantum state of $S$.Let $H$ denote a nearest-neighbor,
translationally invariant Hamiltonian governed by local-interaction terms $h$
that can act on only two qubits at a time.It holds that

$\frac{d}{dt}\abs\left\\{{E\left({e^{-iHt}\rho
e^{iHt}}\right)-\rho}\right\\}\leqslant C(n-1)\lVert{h}\rVert\ ,$ (41ht)

for some universal constant $C>0$.

  * Proof. The Hamiltonian has the form

$H=\sum_{j=1}^{n-1}\tau_{j}(h)\ ,$ (41hu)

where $h$ is a fixed interaction term that acts nontrivially only on two
neighboring qubits in $S$ only.For all
$j\in\left\\{{1,2,\ldots,n-1}\right\\}$, $\tau_{j}$ is the shift operator that
places $\tau_{j}(h)$ onto qubits $j$ and $j+1$.By the Trotter formula, it is
manifest that for suitably small times $t>0$, $e^{-iHt}\rho e^{iHt}$ can be
arbitrarily well-approximated by a brickwork circuit involving nearest-
neighbour quantum gates, even for noncommuting $\tau_{j}(h)$ for different
$j$.For each cut specified by $j$, the small incremental entangling bound for
the von-Neumann entropy [142, 27] gives

$\frac{d}{dt}I(S_{1}\ldots S_{j}:S_{j+1}\ldots S_{n})_{\rho}\leqslant
C\lVert{h}\rVert\ .$ (41hv)

Summing this inequality over all cuts yields 41ht.The constant $C$ is proven
in Ref. [142] to be $C=22\log(2)$, where the $2$ originates from the qubits’
local dimension.In numerical studies, it is found to be rather tightly given
by $C=22\log(2)$.∎

The linear growth of $E(\rho)$ in time is tight, in the sense that there exist
local nearest-neighbor Hamiltonians that exhibit such growth for suitably
short times.This claim largely follows from the results of Ref. [143], which
build on those in Ref. [144].Intuitively speaking, the linear growth
originates from a light-cone-like entanglement dynamics governed by Lieb-
Robinson bounds.For free bosons and fermions (called noninteracting bosons and
fermions in this context), such bounds are readily computable.

######  Corollary 58 (Lower bound on $E(\rho)$ under local Hamiltonian
evolution).

Let $\rho$ denote any quantum state of $S$.There exist local Hamiltonians and
product initial conditions $\rho$, with $E(\rho)=0$, such that

$E\left({e^{-iHt}\rho
e^{iHt}}\right)\geqslant(n-1)\left[{\frac{4}{3\pi}-\frac{1}{2}\log(t)-1}\right]\
.$ (41hw)

  * Proof. Let $S$ consist of an odd number $n\geqslant 21$ of qubits.Let $H$ be the Ising Hamiltonian with periodic boundary conditions.Let the initial state be $\rho=\proj{1,1,\ldots,1}$, so that $E(\rho)=0$.The time-evolved state is $\rho(t)=e^{-iHt}\rho e^{iHt}$.For this state, bounds on entanglement entropies for sufficiently large subsystems are known.Since we consider only bipartite cuts of $S$, the subsystem of length $L$ considered in Ref. [143] always satisfies $L=(n-1)/2$.Hence, the constraint $L\geqslant 10$ is always satisfied.It follows from Theorem 1 in Ref. [143], applied to the $n-1$ cuts of $S$, that

$E\bm{(}\rho(t)\bm{)}\geqslant(n-1)\left[{\frac{4}{3\pi}-\frac{1}{2}\log(t)-1}\right]\
.$ (41hx)

∎

##  Appendix J Recovering the hypothesis-testing (relative) entropy at high
complexities

Consider an “unrestricted” agent who can render POVM effects of arbitrarily
high complexities.In this appendix, we formulate conditions under which the
complexity entropy approximates the hypothesis-testing entropy.We first
identify the set of measurements that an unrestricted agent can render with
arbitrary precision.

######  Definition 59 (POVM effects accessible with unbounded complexity).

Let $\\{\mathsf{M}^{r}\\}$ denote any family of POVM-effect-complexity sets
(B.3).The set of _POVM effects accessible with unbounded complexity_ is

$\displaystyle\mathsf{M}^{\infty}\coloneqq\overline{\bigcup_{r\geqslant
0}\mathsf{M}^{r}}\ ,$ (41hy)

where $\overline{A}$ denotes the topological closure of a set $A$.

######  Definition 60 (“Complexity-unrestricted” entropy and relative
entropy).

Let $\mathsf{M}^{\infty}$ denote any set of POVM effects defined by 41hy.Let
$\rho$ denote any subnormalized state; and $\Gamma$, any positive-semidefinite
operator.Let $\eta\in(0,\tr(\rho)]$.We define the “complexity-unrestricted”
relative entropy
$\Dbase{\DSym}_{\mathrm{H}}^{{\infty},\,{\eta}}{\rho}{\Gamma}$ by replacing
$\mathsf{M}^{r}$ with $\mathsf{M}^{\infty}$ in the definition of
$\Dbase{\DSym}_{\mathrm{H}}^{{r},\,{\eta}}{\rho}{\Gamma}$ (D.1).The
“complexity-unrestricted” entropy is
$\Hfnbase{\HSym}{\mathrm{H}}{{\infty},\,{\eta}}\\{{-NoValue-}\\}(\rho)\coloneqq-\Dbase{\DSym}_{\mathrm{H}}^{{\infty},\,{\eta}}{\rho}{\mathds{1}}$.

If $\mathsf{M}^{\infty}$ contains all POVM effects, we recover the hypothesis-
testing (relative) entropy
exactly:$\Hfnbase{\HSym}{\mathrm{H}}{{\infty},\,{\eta}}\\{{-NoValue-}\\}(\rho)=\Hfnbase{\HSym}{\mathrm{H}}{\eta}\\{{-NoValue-}\\}(\rho)$
$[{\Dbase{\DSym}_{\mathrm{H}}^{{\infty},\,{\eta}}{\rho}{\Gamma}=\DHyp[\eta]{\rho}{\Gamma}}]$.Yet,
for many natural families $\\{\mathsf{M}^{r}\\}$ of POVM-effect-complexity
sets, $\mathsf{M}^{\infty}$ contains only a limited portion of all POVM
effects.For instance, let $C$ denote a unitary-complexity measure (B.1), and
let $\mathsf{P}$ denote a set of simple POVM effects (B.4) consisting only of
projectors.Suppose that
$\left\\{{\mathsf{M}^{r}}\right\\}=\left\\{{\mathsf{M}^{r}(\mathsf{P},C)}\right\\}$,
as per 41ch.Since every unitary transformation maps projectors to projectors,
$\mathsf{M}^{\infty}$ is a set of projectors and therefore excludes almost all
POVM effects.However, even when limited in form, $\mathsf{M}^{\infty}$ may
contain enough effects to ensure that
$\Hfnbase{\HSym}{\mathrm{H}}{{r},\,{\eta}}\\{{-NoValue-}\\}(\rho)$ converges
to a value _near_ $\Hfnbase{\HSym}{\mathrm{H}}{\eta}\\{{-NoValue-}\\}(\rho)$
as $r\to\infty$.We now formulate conditions under which
$\Hfnbase{\HSym}{\mathrm{H}}{{\infty},\,{\eta}}\\{{-NoValue-}\\}(\rho)$
$[{\Dbase{\DSym}_{\mathrm{H}}^{{\infty},\,{\eta}}{\rho}{\Gamma}}]$ lies close
to the hypothesis-testing (relative) entropy,
$\Hfnbase{\HSym}{\mathrm{H}}{\eta}\\{{-NoValue-}\\}(\rho)$
$[{\DHyp[\eta]{\rho}{\Gamma}}]$.

######  Definition 61 ($q$–quasi-universal sets of POVM effects).

Let $q\geqslant 0$, and let $\Gamma$ denote any positive-semidefinite
operator.Let $\mathsf{M}^{\infty}$ denote any set of POVM effects defined by
41hy.$\mathsf{M}^{\infty}$ is _$q$ –quasi-universal with respect to $\Gamma$_
if, for all POVM effects $Q$ with $\lVert{Q}\rVert=1$ and for all probability
weights $g\in(0,1)$, there exist accessible effects
$\tilde{Q},\tilde{Q}^{\prime}\in\mathsf{M}^{\infty}$ such that

$\displaystyle g\tilde{Q}\leqslant
Q\leqslant(1-g)\tilde{Q}^{\prime}+g\mathds{1}\$ (41hza)
$\displaystyle\text{and }\hskip
1.42271pt\log\bm{(}\tr({\tilde{Q}^{\prime}\Gamma})\bm{)}-\log\bm{(}\tr({\tilde{Q}\Gamma})\bm{)}\leqslant
q\ .$ (41hzb)

Consider any $Q$ with $\lVert{Q}\rVert=1$.Whenever $\mathsf{M}^{\infty}$ is
quasi-universal (for any $q$), one can “underestimate” $Q$ by forming a convex
mixture from an accessible effect $\tilde{Q}\in\mathsf{M}^{\infty}$ and the
trivial effect $0$.Similarly, one can “overestimate” $Q$ by mixing an
accessible $\tilde{Q}^{\prime}$ and the trivial effect $\mathds{1}$.One can
estimate $Q$ with such mixtures, no matter the mixing probability $g$.The
variable $g$ continuously parametrizes inequalities that approach $0\leqslant
Q\leqslant\tilde{Q}^{\prime}$ as $g\to 0$ and $\tilde{Q}\leqslant
Q\leqslant\mathds{1}$ as $g\to 1$.These boundaries provide intuition about the
parametrization: as one requirement (that $\tilde{Q}$ lower-bound $Q$)
tightens, the other requirement (that $\tilde{Q}^{\prime}$ upper-bound $Q$)
relaxes and vice versa.We exclude the probability weight $g=0$ ($g=1$) to
avoid trivializing the role of $\tilde{Q}$ ($\tilde{Q}^{\prime})$.The
estimations $\tilde{Q}$ and $\tilde{Q}^{\prime}$ lie close to $Q$ because they
lie close to one another, by condition 41hzb:their candidate hypothesis-
testing-entropy values $\log\bm{(}\tr({\tilde{Q}\Gamma})/\eta\bm{)}$ and
$\log\bm{(}\tr({\tilde{Q}^{\prime}\Gamma})/\eta\bm{)}$ differ by $\leqslant q$
for every error intolerance $\eta\in(0,1]$.Without 41hzb, one might trivially
satisfy 41hza with $\tilde{Q}=0$ and $\tilde{Q}^{\prime}=\mathds{1}$.Finally,
as expected, $\mathsf{M}^{\infty}$ is $q$–quasi-universal for every $q$ and
with respect to every $\Gamma$, whenever the set contains all POVM effects.In
this case, one can choose
$\tilde{Q}=\tilde{Q}^{\prime}=Q\in\mathsf{M}^{\infty}$.The hypothesis-testing
(relative) entropy bounds the complexity-unrestricted (relative) entropy from
above and below.

######  Theorem 62 (Recoverability of the hypothesis-testing entropy).

††margin:

Let $\rho$ denote any subnormalized state; and $\Gamma$, any positive-
semidefinite operator.Let $\mathsf{M}^{\infty}$ denote any set of POVM effects
defined by 41hy.Let $q\geqslant 0$, $\eta\in(0,\tr(\rho)]$, and
$g\in(0,1)$.Suppose that $\mathsf{M}^{\infty}$ is $q$–quasi-universal with
respect to $\Gamma$ (Appendix J).Then

$\displaystyle\DHyp[\eta]{\rho}{\Gamma}\geqslant\Dbase{\DSym}_{\mathrm{H}}^{{\infty},\,{\eta}}{\rho}{\Gamma}\geqslant\DHyp[\eta_{g}]{\rho}{\Gamma}-q-\log\left({\frac{\tr({\rho})}{g\eta}}\right)\
,$ (41ia)

wherein $\eta_{g}\coloneqq(1-g)\eta+g\tr({\rho})\geqslant\eta$.If
$\mathsf{M}^{\infty}$ is $q$–quasi-universal with respect to $\mathds{1}$,
then

$\displaystyle\Hfnbase{\HSym}{\mathrm{H}}{\eta}\\{{-NoValue-}\\}(\rho)\leqslant\Hfnbase{\HSym}{\mathrm{H}}{{\infty},\,{\eta}}\\{{-NoValue-}\\}(\rho)\leqslant\Hfnbase{\HSym}{\mathrm{H}}{\eta_{g}}\\{{-NoValue-}\\}(\rho)+q+\log\left({\frac{\tr({\rho})}{g\eta}}\right)\
.$ (41ib)

  * Proof. The $\DHyp[\eta]{\rho}{\Gamma}$ optimization ranges over all POVM effects, while the $\Dbase{\DSym}_{\mathrm{H}}^{{\infty},\,{\eta}}{\rho}{\Gamma}$ optimization ranges over only effects in $\mathsf{M}^{\infty}$.Therefore, $\DHyp[\eta]{\rho}{\Gamma}\geqslant\Dbase{\DSym}_{\mathrm{H}}^{{\infty},\,{\eta}}{\rho}{\Gamma}$.Now, we prove the second inequality in 41ia.Let $Q$ denote an optimal POVM effect for $\DHyp[\eta_{g}]{\rho}{\Gamma}$ in 41cn.Without loss of generality, $\lVert{Q}\rVert=1$, since we could otherwise replace $Q$ by another candidate $Q^{\prime}\coloneqq Q/\lVert{Q}\rVert\geqslant Q$ that achieves the same objective value as $Q$.Moreover, $\tr({Q\rho})\geqslant\eta_{g}$ and $\tr({Q\Gamma})/\tr({Q\rho})=\exp\bm{(}-\DHyp[\eta_{g}]{\rho}{\Gamma}\bm{)}$.Since $\mathsf{M}^{\infty}$ is $q$–quasi-universal, some $\tilde{Q},\tilde{Q}^{\prime}\in\mathsf{M}^{\infty}$ satisfy the conditions 41hz.Condition 41hza implies that

$\displaystyle(1-g)\tr({\tilde{Q}^{\prime}\rho})\geqslant\tr({Q\rho})-g\tr({\rho})\geqslant\eta_{g}-g\tr({\rho})=(1-g)\eta\
.$ (41ic)

Equivalently, $\tr({\tilde{Q}^{\prime}\rho})\geqslant\eta$.Therefore,
$\tilde{Q}^{\prime}$ is a candidate for the
$\Dbase{\DSym}_{\mathrm{H}}^{{\infty},\,{\eta}}{\rho}{\Gamma}$ optimization in
41ct, so

$\displaystyle\Dbase{\DSym}_{\mathrm{H}}^{{\infty},\,{\eta}}{\rho}{\Gamma}$
$\displaystyle\geqslant-\log\left({\frac{\tr\left({\tilde{Q}^{\prime}\Gamma}\right)}{\tr\left({\tilde{Q}^{\prime}\rho}\right)}}\right)\geqslant-\log\left({\frac{\tr\left({\tilde{Q}^{\prime}\Gamma}\right)}{\eta}}\right)\geqslant-\log\left({e^{q}\frac{\tr\left({\tilde{Q}\Gamma}\right)}{\eta}}\right)\geqslant-\log\left({\frac{\tr\left({Q\Gamma}\right)}{g\eta}}\right)-q$
$\displaystyle\geqslant-\log\left({\frac{\tr\left({Q\Gamma}\right)}{\tr\left({Q\rho}\right)}\;\frac{\tr\left({\rho}\right)}{g\eta}}\right)-q=\DHyp[\eta_{g}]{\rho}{\Gamma}-q-\log\left({\frac{\tr\left({\rho}\right)}{g\eta}}\right)\
.$ (41id)

The third inequality holds because $\tr({\tilde{Q}^{\prime}\Gamma})\leqslant
e^{q}\tr({\tilde{Q}\Gamma})$, by condition 41hzb.The fourth inequality follows
from condition 41hza;the fifth from $\tr({Q\rho})\leqslant\tr({\rho})$.∎

We now provide an example of quasi-universal set that arises naturally in
quantum computation.

######  Proposition 63 (Example of $q$–quasi-universal set).

††margin:

Let $\mathcal{G}$ denote any set of unitary gates that is universal for
quantum computation.Let $C_{\mathcal{G}}$ be the unitary-circuit-complexity
measure associated with $\mathcal{G}$ (B.1).Let $\mathsf{P}$ denote a set of
simple POVM effects (B.4) that consists only of projectors and contains at
least one rank-1 projector.Let
$\left\\{{\mathsf{M}^{r}=\mathsf{M}^{r}\left({\mathsf{P},C_{\mathcal{G}}}\right)}\right\\}$
denote the family of POVM-effect-complexity sets defined by 41ch.Let
$\mathsf{M}^{\infty}$ denote the set of POVM effects defined in 41hy, and
assume that the effects in $\mathsf{M}^{\infty}$ act on a Hilbert space of
dimensionality $d$.$\mathsf{M}^{\infty}$ is $q$–quasi-universal with respect
to $\mathds{1}$ (Appendix J), where $q$ is the largest gap within the set
$\mathcal{X}\coloneqq\left\\{{\log\bm{(}\tr(P)\bm{)}:P\in\mathsf{P}}\right\\}\subset[0,\log(d)]$,
i.e.,

$\displaystyle
q\coloneqq\sup\left\\{{\abs{b-a}:(a,b)\subset[0,\log(d)]\setminus\mathcal{X}}\right\\}\
.$ (41ie)

$q$ is the greatest length of any interval $(a,b)\subset[0,\log(d)]$ disjoint
from $\mathcal{X}$.

  * Proof. Let $Q$ denote any POVM effect with $\lVert{Q}\rVert=1$.Let us fix a matrix representation of $Q$ that is diagonal and has decreasing diagonal elements: $Q=\diag({\lambda_{1},\lambda_{2},\ldots,\lambda_{d}})$, with $1=\lambda_{1}\geqslant\lambda_{2}\geqslant\ldots\geqslant\lambda_{d}\geqslant 0$.Consider any $g\in(0,1)$, and let $\ell_{g}\coloneqq\max\\{{i:\lambda_{i}\geqslant g}\\}$ denote the number of $Q$ eigenvalues not less than $g$.$\ell_{g}$ satisfies $1\leqslant\ell_{g}\leqslant d$, since $\lambda_{1}=1\geqslant g$.Let

$\displaystyle\ell_{g}^{-}\coloneqq\max\left\\{{\rank(P)\,:\;P\in\mathsf{P},\
\rank(P)\leqslant\ell_{g}}\right\\}\;\;\;\;\text{and}\;\;\;\;\ell_{g}^{+}\coloneqq\min\left\\{{\rank(P)\,:\;P\in\mathsf{P},\
\rank(P)\geqslant\ell_{g}}\right\\}\ .$ (41if)

As per the definitions of $\ell_{g}^{-}$ and $\ell_{g}^{+}$, there exist
projectors $\tilde{P},\tilde{P}^{\prime}\in\mathsf{P}$ such that
$\rank(\tilde{P})=\ell_{g}^{-}$ and
$\rank(\tilde{P}^{\prime})=\ell_{g}^{+}$.Moreover,
$1\leqslant\ell_{g}^{-}\leqslant\ell_{g}^{+}\leqslant d$, since $\mathsf{P}$
contains both a rank-1 projector and $\mathds{1}$.Let $U$ denote a unitary
such that $\tilde{Q}=U^{\dagger}\,\tilde{P}\,U$ is diagonal:
$\tilde{Q}=\diag(1,\ldots,1,0,\ldots,0)$, with
$\rank(\tilde{Q})=\ell_{g}^{-}$.Likewise, let $U^{\prime}$ denote a unitary
such that $\tilde{Q}=U^{\prime\dagger}\,\tilde{P}^{\prime}\,U^{\prime}$ is
diagonal, with $\rank(\tilde{Q}^{\prime})=\ell_{g}^{+}$.By assumption, $U$ and
$U^{\prime}$ are arbitrarily well-approximated by unitaries of finite
complexity, so $\tilde{Q},\tilde{Q}^{\prime}\in\mathsf{M}^{\infty}$.We now
show that $\tilde{Q}$ and $\tilde{Q}^{\prime}$ satisfy the conditions 41hz in
the definition of a $q$–quasi-universal set.Consider the operator
$g\tilde{Q}$.The first $\ell_{g}\geqslant\ell_{g}^{-}$ diagonal entries of
$g\tilde{Q}$ do not exceed $g$ and, hence, do not exceed the first $\ell_{g}$
diagonal entries of $Q$.Moreover, the last $\ell_{g}$ diagonal entries of
$g\tilde{Q}$ equal 0 and, hence, do not exceed the last $\ell_{g}$ diagonal
entries of $Q$.Thus, $g\tilde{Q}\leqslant Q$.Now, consider the operator
$(1-g)\tilde{Q}^{\prime}+g\mathds{1}$.The first
$\ell_{g}\leqslant\ell_{g}^{+}$ diagonal entries of
$(1-g)\tilde{Q}^{\prime}+g\mathds{1}$ equal 1 and, hence, are not less than
the first $\ell_{g}$ diagonal entries of $Q$.Moreover, the last $\ell_{g}$
diagonal entries of $g\tilde{Q}$ are not less than $g$ and, hence, are not
less than the last $\ell_{g}$ diagonal entries of $Q$.Thus,
$Q\leqslant(1-g)\tilde{Q}^{\prime}+g\mathds{1}$.As such, $\tilde{Q}$ and
$\tilde{Q}^{\prime}$ satisfy condition 41hza.Compare the definitions of
$\ell_{g}^{-}$ and $\ell_{g}^{+}$.There exists no projector $P\in\mathsf{P}$
such that $\ell_{g}^{-}<\rank(P)<\ell_{g}^{+}$.Accordingly, the interval
$\bm{(}\log({\ell_{g}^{-}}),\log({\ell_{g}^{+}})\bm{)}$ is disjoint from
$\mathcal{X}$.$[$If $\ell_{g}^{-}=\ell_{g}^{+}$, then
$\bm{(}\log({\ell_{g}^{-}}),\log({\ell_{g}^{+}})\bm{)}=\emptyset$ by
convention.$]$Thus,
$\log\bm{(}\tr({\tilde{Q}^{\prime}})\bm{)}-\log\bm{(}\tr({\tilde{Q}})\bm{)}=\log({\ell_{g}^{+}})-\log({\ell_{g}^{-}})\leqslant
q$, with $q$ defined as in 41ie, so $\tilde{Q}$ and $\tilde{Q}^{\prime}$
satisfy condition 41hzb.∎

##  Appendix K Data compression under complexity limitations

Here, we establish optimal protocols for data compression under complexity
limitations.We thereby prove the bounds 41ax in V.5.Let $\rho$ denote any
state, and $C$ any unitary-complexity measure, of $n$ qubits.Let $r\geqslant
0$.Let $m\leqslant n$ denote a number of qubits; and $\epsilon\in[0,1)$, an
error parameter.We define an _$(m,r,\epsilon)$ –data-compression protocol_ as
a unitary $U$ that satisfies $C(U)\leqslant r$ and that can compress $\rho$
into $m$ qubits.The compression succeeds if the other $n-m$ qubits end in a
state $\epsilon$-close to $\ket{0^{n-m}}$ in fidelity:there exists a subset
$\mathcal{W}\subset\\{{1,2,\ldots,n}\\}$ of qubits such that
$\abs{\mathcal{W}}=m$ and

$\displaystyle\tr\bm{(}\tr_{\mathcal{W}}(U\rho
U^{\dagger})\,\proj{0^{n-m}}\bm{)}\geqslant 1-\epsilon\ .$ (41ig)

We first prove that some protocol compresses $\rho$ into a number of qubits
proportional to the reduced complexity entropy (D.5.1).

######  Theorem 64 (Achievability of data compression).

††margin:

Let $\rho$ denote any state, and $C$ any unitary-complexity measure (B.1), of
$n$ qubits.Let $\mathsf{P}$ denote the set of simple POVM effects (B.4)
defined in 10.Let $r\geqslant 0$ and $\epsilon\in[0,1)$.Let
$\mathsf{M}^{r}=\mathsf{M}^{r}(\mathsf{P},C)$ denote the POVM-effect-
complexity set defined in 41ch.There exists an $(m,r,\epsilon)$–data-
compression protocol for $\rho$ such that

$\displaystyle
m=\frac{1}{\log(2)}\,\Hfnbase{\HSym}{\mathrm{h}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}(\rho)\
,$ (41ih)

where the reduced complexity entropy
$\Hfnbase{\HSym}{\mathrm{h}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}()$ is
defined with respect to $\mathsf{M}^{r}$.

The above theorem can be reformulated in terms of the complexity entropy
$\Hfnbase{\HSym}{\mathrm{H}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}(\rho)$.D.5.1
shows that the reduced complexity entropy
$\Hfnbase{\HSym}{\mathrm{h}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}(\rho)$
differs from the complexity entropy
$\Hfnbase{\HSym}{\mathrm{H}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}(\rho)$ by at
most $\log(1/[1-\epsilon])$.Combining this fact with Appendix K yields the
bounds 41ax in V.5.If $\epsilon\approx 0$, then the optimal number of qubits
in 41ih is approximately proportional to the complexity entropy:
$m\approx\Hfnbase{\HSym}{\mathrm{H}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}(\rho)/\log(2)$.We
further prove that the protocol in Appendix K is optimal.

######  Theorem 65 (Optimality of data compression).

††margin:

Let $\rho$ denote any state of $n$ qubits.Let $r\geqslant 0$ and
$\epsilon\in[0,1)$.Let $\mathsf{M}^{r}$ denote a POVM-effect-complexity set as
in Appendix K.Every $(m,r,\epsilon)$–data-compression protocol satisfies

$\displaystyle
m\geqslant\frac{1}{\log(2)}\,\Hfnbase{\HSym}{\mathrm{h}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}(\rho)\
,$ (41ii)

where the reduced complexity entropy
$\Hfnbase{\HSym}{\mathrm{h}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}()$ is
defined with respect to $\mathsf{M}^{r}$.

The above data-compression problem maps directly onto the thermodynamic-
erasure problem in III.1.The proofs of Appendices K and K are closely related
to the arguments in III.1 and to the proof of Appendix E.

  * Proof of Appendix K. Consider any $\zeta>0$.There exists a $Q\in\mathsf{M}^{r}$ such that $\tr({Q\rho})\geqslant 1-\epsilon$ and $\log\bm{(}\tr({Q})\bm{)}\leqslant\Hfnbase{\HSym}{\mathrm{h}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}(\rho)+\zeta$.$Q=U^{\dagger}PU$ for some unitary $U$ satisfying $C(U)\leqslant r$ and for some simple POVM effect $P$ acting as $\mathds{1}_{2}$ on every qubit in a subset $\mathcal{W}\subset\\{{1,2,\ldots,n}\\}$ and as $\proj{0}$ on every other qubit.Without loss of generality, assume that $\mathcal{W}$ consists of the first $m\coloneqq\abs{\mathcal{W}}$ of the $n$ qubits.Hence $P=\mathds{1}_{2}^{\otimes m}\otimes\proj{0^{n-m}}$.$U$ is an $(m,r,\epsilon)$–data-compression protocol for $\rho$, since $C(U)\leqslant r$ and condition 41ig holds:

$\displaystyle\tr\bm{(}\tr_{\mathcal{W}}(U\rho
U^{\dagger})\,\proj{0^{n-m}}\bm{)}=\tr([U\rho
U^{\dagger}]\,[\mathds{1}_{2}^{\otimes m}\otimes\proj{0^{n-m}}])=\tr([U\rho
U^{\dagger}]\,P)=\tr({Q\rho})$ $\displaystyle\geqslant 1-\epsilon\ .$ (41ij)

Moreover,

$\displaystyle
m=\log_{2}\bm{(}\tr({P})\bm{)}=\log_{2}\bm{(}\tr({Q})\bm{)}=\frac{1}{\log(2)}\,\log\bm{(}\tr({Q})\bm{)}\leqslant\frac{1}{\log(2)}\left(\Hfnbase{\HSym}{\mathrm{h}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}(\rho)+\zeta\right)\
,$ (41ik)

which implies 41ih, since $\zeta$ is arbitrary.∎

  * Proof of Appendix K. Let $U$ denote any $(m,r,\epsilon)$–data-compression protocol for $\rho$.By definition, there exists a subset $\mathcal{W}\subset\\{{1,2,\ldots,n}\\}$ such that $\abs{\mathcal{W}}=m$ and condition 41ig holds.Without loss of generality, assume that $\mathcal{W}$ consists of the first $m$ of $n$ qubits.Let $P\coloneqq\mathds{1}_{2}^{\otimes m}\otimes\proj{0^{n-m}}$ and $Q\coloneqq U^{\dagger}PU$.By construction, $Q$ is in $\mathsf{M}^{r}$ and satisfies

$\displaystyle\tr({Q\rho})=\tr({P\,U\rho
U^{\dagger}})=\tr\bm{(}\tr_{\mathcal{W}}({U\rho
U^{\dagger}})\,\proj{0^{n-m}}\bm{)}\geqslant 1-\epsilon\ .$ (41il)

$Q$ is therefore a candidate for the
$\Hfnbase{\HSym}{\mathrm{h}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}(\rho)$
optimization, so

$\displaystyle
m=\abs{\mathcal{W}}=\log_{2}\bm{(}\tr({P})\bm{)}=\log_{2}\bm{(}\tr({Q})\bm{)}\geqslant\frac{1}{\log(2)}\Hfnbase{\HSym}{\mathrm{h}}{{r},\,{1-\epsilon}}\\{{-NoValue-}\\}(\rho)\
.$ (41im)

∎

##  Appendix L Decoupling from a reference system

Here, we prove the decoupling bound of V.7.Our proof relies on the following
conjectured property of the complexity conditional entropy.

######  Conjecture 66.

Let $A$, $B$, and $R$ denote distinct quantum systems.Let $\rho_{ABR}$ denote
any state of $ABR$.Let $r\geqslant 0$ and $\eta\in(0,1]$.It holds that

$\displaystyle\Hbase{\HSym}{\mathrm{H}}[\rho][{{r},\,{\eta}}]{B}[R]\leqslant\Hbase{\HSym}{\mathrm{H}}[\rho][{{r},\,{\eta}}]{AB}[R]+\log(d_{A})\
.$ (41in)

Equivalently,

$\displaystyle\Dbase{\DSym}_{\mathrm{H}}^{{r},\,{\eta}}{\rho_{BR}}{\pi_{B}\otimes\rho_{R}}\geqslant\Dbase{\DSym}_{\mathrm{H}}^{{r},\,{\eta}}{\rho_{ABR}}{\pi_{AB}\otimes\rho_{R}}-2\log(d_{A})\
.$ (41io)

The equivalence between 41in and 41io holds because
$\Hbase{\HSym}{\mathrm{H}}[\rho][{{r},\,{\eta}}]{B}[R]=\log(d_{B})-\Dbase{\DSym}_{\mathrm{H}}^{{r},\,{\eta}}{\rho_{BR}}{\pi_{B}\otimes\rho_{R}}$
and because
$\Hbase{\HSym}{\mathrm{H}}[\rho][{{r},\,{\eta}}]{AB}[R]=\log(d_{A}d_{B})-\Dbase{\DSym}_{\mathrm{H}}^{{r},\,{\eta}}{\rho_{ABR}}{\pi_{AB}\otimes\rho_{R}}$,
by the argument in 41fr.41in mirrors a known inequality for the von Neumann
conditional entropy $\HH[\rho]{AB}[R]$.The latter inequality follows from a
chain rule and the lower bound $\HH[\rho]{A}[BR]\geqslant-\log(d_{A})$:

$\displaystyle\HH[\rho]{B}[R]=\HH[\rho]{AB}[R]-\HH[\rho]{A}[BR]\leqslant\HH[\rho]{AB}[R]+\log(d_{A})\
.$ (41ip)

######  Theorem 67 (Upper bound on the number of qubits Alice can decouple
under complexity limitations).

††margin:

Let $A$ denote an $n$-qubit system, and let $R$ denote a quantum system
distinct from $A$.Let $\rho_{AR}$ denote any state of $AR$.Let
$\mathsf{P}=\mathsf{P}_{AR}$ denote any set of simple POVM effects (B.4); and
$C=C_{AR}$, any adjoint-invariant unitary-complexity measure (B.1).Let
$\left\\{{\mathsf{M}^{r}=\mathsf{M}^{r}(\mathsf{P},C)}\right\\}$ denote the
family of POVM-effect-complexity sets defined by 41ch.Let $r_{0}\geqslant 0$
and $r_{1}\geqslant r_{0}$.Let $\eta\in(0,1]$ and $\delta\in(0,1]$.Let $U_{0}$
denote any unitary on $A$ satisfying $C(U_{0})\leqslant r_{0}$.Let $A_{1}$
denote a subsystem of $k\geqslant 0$ qubits in $A$; and $A_{2}$, the subsystem
of the other $n-k$ qubits.Let

$\displaystyle\rho^{\prime}_{AR}$
$\displaystyle\coloneqq(U_{0}\otimes\mathds{1}_{R})\rho_{AR}(U_{0}\otimes\mathds{1}_{R})^{\dagger}\
.$ (41iq)

Suppose that

$\displaystyle\Dbase{\DSym}_{\mathrm{H}}^{{r_{1}},\,{\eta}}{\rho^{\prime}_{A_{2}R}}{\pi_{A_{2}}\otimes\rho_{R}}\leqslant-\log\left(\frac{\delta}{\eta}\right)\
,$ (41ir)

where $\Dbase{\DSym}_{\mathrm{H}}^{{r_{1}},\,{\eta}}{}{}$ is defined with
respect to $\mathsf{M}^{r_{1}}$.Assume Appendix L holds.It holds that

$\displaystyle
k\geqslant\frac{1}{2}\left[n-\frac{1}{\log(2)}\Hbase{\HSym}{\mathrm{H}}[\rho][{{r_{1}-r_{0}},\,{\eta}}]{A}[R]+\log_{2}\left(\frac{\delta}{\eta}\right)\right]\
,$ (41is)

where $\Hbase{\HSym}{\mathrm{H}}[][{{r_{1}-r_{0}},\,{\eta}}]{A}[R]$ is defined
with respect to $\mathsf{M}^{r_{1}-r_{0}}$.

  * Proof. The following chain of inequalities implies 41is:

$\displaystyle-\log\left(\frac{\delta}{\eta}\right)$
$\displaystyle\geqslant\Dbase{\DSym}_{\mathrm{H}}^{{r_{1}},\,{\eta}}{\rho^{\prime}_{A_{2}R}}{\pi_{A_{2}}\otimes\rho_{R}}$
$\displaystyle=\log\left(d_{A_{2}}\right)-\Hbase{\HSym}{\mathrm{H}}[\rho^{\prime}][{{r_{1}},\,{\eta}}]{A_{2}}[R]$
$\displaystyle\geqslant\log\left(d_{A_{2}}\right)-\Hbase{\HSym}{\mathrm{H}}[\rho^{\prime}][{{r_{1}},\,{\eta}}]{A}[R]-\log\left(d_{A_{1}}\right)$
$\displaystyle=(n-2k)\log(2)-\Hbase{\HSym}{\mathrm{H}}[\rho^{\prime}][{{r_{1}},\,{\eta}}]{A}[R]$
$\displaystyle\geqslant(n-2k)\log(2)-\Hbase{\HSym}{\mathrm{H}}[\rho][{{r_{1}-r_{0}},\,{\eta}}]{A}[R]\
.$ (41it)

The first equality follows by the argument in 41fr; the second inequality, by
the assumption that Appendix L holds; and the last inequality, by D.2.∎

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# Rectified motion in an asymmetric channel: the role of hydrodynamic
interactions with walls

Behzad Golshaei  Department of Physics, Institute for Advanced Studies in
Basic Sciences (IASBS), Zanjan 45137-66731, Iran  Ali Najafi  Physics
Department, University of Zanjan, Zanjan 45371-38791, Iran<EMAIL_ADDRESS>
###### Abstract

Dynamics of a Brownian particle in an asymmetric micro-channel that is
subjected to an external oscillating force is numerically analyzed. In
addition to the elastic collisions with the walls that are kind of short range
interactions, the long range hydrodynamic influences of the walls have been
considered. We demonstrate how the geometrical parameters of the channel
change the rectified current of the particle. As a result of numerical
calculations, we show that long range hydrodynamic interactions with walls,
decrease the efficiency of the Brownian ratchet.

###### pacs:

05.40.-a,05.60.-k,47.15.G-

##  I Introduction

For a Brownian particle that is in thermal equilibrium, second low of
thermodynamics does not allow to achieve a rectified motion even when the
particle is moving in an asymmetric periodic potential. In addition to
symmetry breaking with an externally applied asymmetric periodic potential,
the fluctuations need to obey an out of equilibrium statistics to achieve a
rectified motion at the microscopic world. The original idea of the
rectification of random motions at microscopic scales dates back to the work
by M. V. Smoluchovski smolo  where he discussed the issue of extracting useful
work from fluctuations. R. P. Feynman has illustrated this idea by a very
intuitive gedanken experiment composed of ratchet and pawl feynman . Very
recently, a group of experimentalists have shown how the ratchet idea can be
tested at macroscopic world by using a granular gas. In their experiment, a
plate vibrating vertically provides a non equilibrium gas of granular
particles. Collision of these granular particles with four vanes of an
asymmetric rotary part, enforce it to rotate. The overall motion of the vanes
is shown to be a rectified rotation in a direction preferred by the asymmetry
of the vanes granular .

In addition to the above fundamental interests on the physics of Brownian
ratchet, it also provides a basic physical mechanism for describing the
dynamics of most biophysical molecular motors ajdari .

Figure 1: A Brownian particle with radius $a$ moves in a two dimensional
rocked Ratchet micro-channel with period $L_{x}$. The input and out of the
channel have same width given by $L_{z}$. Angle $\theta$, measures the
asymmetry of channel. A typical trajectory of the Brownian particle, moving
near the wall is shown.

Transport of colloidal particles in channels with the size of micrometer is
another related area. Recent technological advances allow researchers to
design and fabricate devices to guide particles on micro and nano-channels
experimantal rocked ratchet ; driftratchet . Flow control and separation of
particles in such channels are the main experimental interests in this field
reimann2002brownian ; particlespration . An important category of micro-
fluidic devices is so-called rocked ratchet, where an applied oscillating
force drives a net drift velocity for particles fluctuating in an asymmetric
potential imposed by the walls of channel . The physical mechanism behind such
systems has been considered in details hanggi ; martis . A very comprehensive
review of the related works is presented in an article by P. Hanggi review .

Hydrodynamic interaction between particles, at intermediate and high volume
fraction of particles is proved to have prominent effects on the efficiency of
Brownian ratchets holger . In addition to interaction between particles, the
long range hydrodynamic interaction with confining walls is proved to have
essential effects on the motion of either passive colloidal particles pralle ;
squires  or active systems roughwall ; suspensionofconfined ; 3S2W . It is the
main goal of this article to address how the interaction with walls will
influence the functionality of a Brownian ratchet. Usually this kind of long
range interactions are neglected in the theoretical and numerical
investigations hnngiprl . In this paper, we focus on hydrodynamic interactions
of particles with the walls of channel. To answer this question, we start by a
perturbation based theory that considers the hydrodynamic effects for a
spherical particle moving near confining walls. Then letting the particle to
fluctuate, we numerically simulate the Brownian dynamics of such a colloidal
particle moving in a medium confined with the walls of an asymmetric channel.

The structure of this article is as follows: In section 2, we define the model
and present the details of approximations for the hydrodynamic interactions.
In section 3, we present the details of numerical scheme that we have used to
simulate the Brownian dynamics. The results are presented in section 4 and
finally we discuss in section 5.

##  II Model and its parameters

A two dimensional rocked ratchet channel is used to study the transport of
Brownian particles. As depicted in fig.1, the channel is characterized by its
periodic length $L_{x}$, input and output opening sizes $L_{z}$ and asymmetry
angle $\theta$. For $\theta=0$, the channel is symmetric and we do not expect
any rectified motion for this case. Moreover we assume that the colloidal
particle has mass $m$ and its radius is given by $a$. In a reference frame
that is shown in the figure, the position vector of the particle is given by:
${\bf r}=(x,z)$, where ${\hat{\bf x}}$ points along the axis of channel. The
following stochastic Langevin differential equation describes the dynamics of
a colloidal particle moving in this channel:

$m{\ddot{\bf r}}={\bf G}\cdot{\dot{\bf r}}+{\bf F}_{hc}+{\bf
F}_{e}(t)+{\bf\Gamma}(t),$ (1)

where ${\bf G}$ stands for the hydrodynamic friction tensor and
${\bf\Gamma}(t)$ shows the random forces due to thermal fluctuations. The
effects of the walls of channel are given by a hard core force ${\bf F}_{hc}$.
As a result of this very short range potential, the dynamics of the particle
obeys the rules of elastic collision at the boundaries. Conservation of energy
and momentum provide relations for the state of the system before and after
each collision. To provide an out of equilibrium condition for the particle,
we apply an external force along the ${\hat{\bf x}}$ direction that is
spatially uniform but period in time:

${\bf F}_{e}(t)=f_{0}\sin(\omega t){\hat{\bf x}},$ (2)

where $f_{0}$ and $\omega$ show the amplitude and frequency of the external
force, respectively. In addition to the above deterministic forces, the
dynamics of the particle is influenced by time dependent random noise
${\bf\Gamma}(t)$ results from thermal fluctuations. The average and
correlation of this stochastic force is given by:

$\langle{\bf\Gamma}(t)\rangle=0,~{}~{}~{}~{}~{}\langle{\bf\Gamma}(t){\bf\Gamma}(t^{\prime})\rangle=2k_{B}T{\bf
G}\delta(t-t^{\prime}),$ (3)

where $k_{B}T$ is the thermal energy.

What we are interested in this article, is the effects of long range
hydrodynamic interaction with the walls of the channel. All of information
related to the hydrodynamic interactions with the walls are encoded in the
friction tensor ${\bf G}$. In addition to the size of colloid, this friction
tensor depends on distance between the particle and the walls. This friction
coefficient, should in principle be derived from solution to Stokes equation,
the equation that governs the dynamics of fluid at the scale of micrometer.
Solution to this equation, provided that the fluid motion obeys the no-slip
boundary conditions on the walls, will reveal the detail form of the friction
tensor. As there is no exact solution to the Stokes equation in an asymmetric
channel, we will use approximate prescriptions for the friction coefficients.

Figure 2: Left: A spherical particle with size $a$ located in a position with
a distance $h$ above a flat wall. Right: Same problem, but seen in a frame of
reference rotated with an angle $\alpha$ with respect to the wall.

Before considering the complicated geometry of our channel, we start by
presenting the results for friction coefficients of a particle moving in a
semi infinite fluid environment that is bounded by a single wall. Instead of
friction tensor, it is more convenient to work with the mobility tensor ${\bf
M}$ that is the inverse of friction tensor when expressed in the matrix
notation: ${\bf M}={\bf G}^{-1}$. Fig. 2(a), shows a spherical particle with
size $a$ immersed in a semi infinite fluid that has a distance $h$ from a
wall. Theoretical analysis based on perturbation theory can give a series
expansion for the mobility tensor in terms of small parameter $(a/h)$. In the
matrix notation, the mobility tensor has the following components:

${\bf M}=\left(\begin{array}[]{cc}M_{xx}&M_{xz}\\\ M_{zx}&M_{zz}\\\
\end{array}\right).$ (4)

Symmetry considerations do not allow non-zero off-diagonal elements for the
mobility tensor: $M_{xz}=M_{zx}=0$ and the other components are given by
swan2007simulation ; HydrodynamicCoupling :

$\displaystyle M_{xx}$ $\displaystyle=$
$\displaystyle\mu\left(1-\frac{9}{16}\left(\frac{a}{h}\right)+\frac{1}{8}\left(\frac{a}{h}\right)^{3}-\frac{1}{16}\left(\frac{a}{h}\right)^{5}+{\cal
O}(\frac{a}{h})^{7}\right),$ $\displaystyle M_{zz}$ $\displaystyle=$
$\displaystyle\mu\left(1-\frac{9}{8}\left(\frac{a}{h}\right)+\frac{1}{2}\left(\frac{a}{h}\right)^{3}-\frac{1}{8}\left(\frac{a}{h}\right)^{5}+{\cal
O}(\frac{a}{h})^{7}\right),$

where $\mu=1/(6\pi\eta a)$, is the self mobility of a spherical particle with
size $a$ moving in a fluid with viscosity $\eta$. Using the above components
for the mobility tensor, one can write a perturbation series for the friction
tensor as:

${\bf G}={\bf G}_{0}+\delta{\bf G},$ (5)

where ${\bf G}_{0}=6\pi\eta a{\bf I}$ is the friction tensor for a sphere
moving in an infinite fluid and ${\bf I}$ is the unit tensor (unit matrix in
matrix representation). Here all of the corrections due to the presence of
wall are collected in $\delta{\bf G}$. Direct calculations can give the
explicit mathematical form of this correction term.

Fig. 2(b), shows the same problem as above, but in a reference frame that is
rotated with an angle $\alpha$ with respect to the wall. It is an straight
forward geometrical calculation, to express the mobility tensor in the rotated
frame. In terms of the mobility tensor in the plate frame (frame in which the
axis are parallel and perpendicular to the wall), the mobility tensor in the
rotated frame can be given by:

${\bf M}={\cal R}^{T}(\alpha)\cdot{\bf M}\cdot{\cal R}(\alpha),$ (6)

where the dot symbol represents the matrix multiplication rule and ${\cal
R}(\alpha)$ is the rotation matrix with an angle $\alpha$ about axis
${\hat{\bf x}}\times{\hat{\bf z}}$.

How a particle moving in medium confined with the walls of a complicated
channel, will response to an external force? Exact solutions to this question
occurs only in a certain type of symmetric geometries. A cylindrical channel
with infinite length and also a rectangular channel with infinite length and
infinite depth are examples with exact solutions exact . Apart from the above
symmetric geometries, there is no analytic solution for the hydrodynamic
friction problem in an asymmetric channel. Among from different approximate
methods, the direct superposition of the corrections from different walls, is
the simplest approximate scheme that can give the effects of walls in a
channel happel ; hydrosuperposition . For a particle moving in a channel with
confining walls, the friction tensor may be written as:

${\bf G}={\bf G}_{0}+\sum_{i}\delta{\bf G}^{i},$ (7)

where $\delta{\bf G}^{i}$ is the correction due to the $i$’th wall of the
channel. Using this approximation we can study the dynamics of a Brownian
particle moving in the channel. Of course the validity of this superposition
method is a challenging issue and it will break near the walls and specially
at the corners. At distances very near to the wall, the short range hard core
interaction dominates the dynamics of the particle and we do not expect to see
any sharp effect from the breakdown of our approximation for hydrodynamic
interactions. Secondly, because of the geometrical constraints, we will see
that the particle does not allow to reach the corners at all. These will
ensure us that the superposition approximation will correctly account the long
range hydrodynamic interactions with the wall of channel and we expect to have
a picture that is at least qualitatively correct.

##  III Numerical simulation

We define the current density as the number of particles per unit length of
the opening that exit from the left side of the channel in a unit time.
Density of current for the particles, is a quantity that reflects how the
motion of particles is rectified. In terms of the average velocity of the
particles, we can write the current density as:

$J=n\langle v_{x}\rangle,$ (8)

where $n$ is the density of particles, number of particles per unit area in
our two dimensional problem. As we are not interested about the hydrodynamic
interactions between the particles, we put a single particle in our channel
and study its dynamics. For this case the density is given by:
$n=(L_{x}L_{z}(1+\frac{1}{2}\Delta\tan\theta))^{-1}$, with
$\Delta=L_{x}/L_{z}$. Averaging over ${\cal N}$ realizations of the system, we
can obtain the average velocity as:

$\langle v_{x}\rangle=\frac{1}{{\cal N}}\sum_{i=1}^{{\cal
N}}\frac{x_{i}(t_{s})-x_{i}(0)}{T},$ (9)

where $t_{s}$ is the time of observation (simulation time).

To numerically solve the Langevin differential equation (Eq. 1), we need to
make it non dimensional. For this purpose we can use $a$ as a scale for length
and $\tau=a^{2}/(\mu k_{B}T)$ as a scale for time and make all variables non
dimensional. After going to the dimensionless system, all of dynamical
equations can be written in terms of dimensionless variables and some
dimensionless numbers. Denoting the non dimensional variables with an over
bar, the non dimensional Langevin equation reads:

${\cal R}e{\bar{\bf r}}^{\prime\prime}={\bar{\bf G}}\cdot{\bar{\bf
r}}^{\prime}+{\bar{f}}_{0}\sin({\bar{\omega}}{\bar{t}}){\hat{x}}+{\bar{\Gamma}},$
(10)

where prime symbol denotes the derivative with respect to non dimensional time
and ${\cal R}e=m\mu/\tau$ is dimensionless Reynolds number and it measures how
the acceleration term in the dynamical equation of the particle is important
with respect to a typical friction force. It is straightforward to see that at
the scale of micrometer, the Reynolds number is very small: ${\cal
R}e=10^{-3}$ . Non dimensional amplitude of the external force is given by
${\bar{f}}_{0}=(\mu\tau/a)f_{0}$ and ${\bar{\bf G}}=\mu{\bf G}$. The noise
satisfies the following correlation function:

$\langle{\bar{\Gamma}}({\bar{t}}){\bar{\Gamma}}({\bar{t}}^{\prime})\rangle=2{\bar{\bf
G}}\delta({\bar{t}}-{\bar{t}}^{\prime}).$ (11)

Non dimensional current density can be written as ${\bar{J}}=J/J_{0}$, where
$J_{0}=(\tau L_{z}(1+\frac{1}{2}\Delta\tan\theta))^{-1}$. To calculate the
current density and average over realizations, we can use the periodic
boundary condition and extract the average value of the current from a long
time dynamics of a single particle. Please note that choosing the above sort
of non dimensional system, allows us to write the average current density as:

${\bar{J}}=\frac{N_{R}-N_{L}}{N},$ (12)

where $N_{R}$ is the number of times that the particle exits from the right
opening and $N_{L}$ is the number of times that the particle enters to the
channel from left side. Here $N=t_{s}/\tau$ and $t_{s}$ stands for the
simulation time.

We use the method of reference ermak  to numerically integrate the equations
of motion. To generate random numbers, we use the Mersenne Twister pseudo-
random number generator.

Numerical parameters that define our system are as follows: we choose a
spherical particle with size $a=1~{}\mu\mathrm{m}$ moving in a channel that is
filled with water and it is at room temperature. The viscosity of water is
$\eta=10^{-3}\mathrm{Pa.sec.}$ and thermal energy is
$k_{B}T=0.02~{}\mathrm{ev}$. These will result a characteristic time scale
that is $\tau=10~{}\mathrm{sec.}$. We choose a time step $\Delta
t=0.005~{}\mathrm{sec.}$ that in dimensionless units is about
$\Delta\bar{t}=0.0005$. Diffusion times along the axis of channel and along a
direction that is perpendicular to its axis are defined by
$\tau_{D}^{x}=(L_{x}/a)^{2}\tau$ and $\tau_{D}^{z}=(L_{z}/a)^{2}\tau$
respectively. In a channel with typical length $L_{x}=15a$, we will see that
$\tau_{D}^{x}=2000~{}\mathrm{sec}$. In the following section we will present
the results of numerical calculations.

Figure 3: Dimensionless current density is plotted as a function of frequency
of external force. Left: simulations have been performed for two cases where
thermal force is present or absent. Simulation time is
$t_{s}=5000~{}\mathrm{sec.}$ and for different simulation time is plotted.
Right: in the absence of thermal noise, the results are shown for different
simulation times. In all graphs $\bar{f}_{0}=10$ and the geometrical variables
are $a=1\mu\mathrm{m}$, $L_{x}=15a$, $L_{z}=5a$.

##  IV Results

Before investigating the hydrodynamic effects of the walls, we first turn off
the hydrodynamic interaction and consider only the elastic collisions with the
walls. To illustrate the functionality of Brownian ratchet, we have plotted
the average current density as a function of frequency of the external force
in fig. 3(left). The results have been shown for two cases where the random
noise is turned on or turned off. The simulation time is
$t_{s}=5000~{}\mathrm{sec}$. One should note that, for this small simulation
time that we have chosen here, only the results at very large frequencies are
acceptable. To obtain true physical results, we should keep in mind that the
simulation time should be large in a way that $\bar{t}_{s}\bar{\omega}\gg 1$.
As we expect, the existence of thermal noise is essential for the
functionality of the Brownian ratchet. In the absence of thermal noise, no
rectification is expected. This is evident in the results, where the current
at very large frequencies (the acceptable results) disappears for the case
where thermal noise is absent. Here the diffusion time along the axis of
channel is about $\tau_{D}^{x}=15^{2}\tau\sim 2000\mathrm{sec}$ and the period
of external force for a special point on the graph with $\bar{\omega}=1$, is
given by $T=2\pi/\omega=(2\pi/1)\times\tau\sim 60\mathrm{sec}$. The value of
current at this frequency is $\bar{J}=2$ which corresponds to a current of
particles that is about $J=2\times J_{0}\sim 10^{5}m^{-1}\mathrm{sec.}^{-1}$.
Here $t_{s}/\tau_{D}^{x}=2.5$, but as a result of external force, the particle
has greater chance to travel many times along the axes of channel. This would
result an overall finite value for both $N_{R}$ and $N_{L}$.

In fig. 3(right), we have studied the system that is subjected to thermal
noise, and investigated its response by changing both the simulation time and
frequency of external force. As one can see, by increasing the time of
simulation, the height of all non zero frequency peaks decreases. Increasing
the simulations time will result a more smoother behavior for the current at
large frequencies. Learning from the results of these graphs, we choose a
proper simulation time and investigate the other physical properties of the
Brownian ratchet in the following part.

Figure 4: Left: dimensionless current density as a function of the amplitude
of external force is plotted for two cases where the hydrodynamic long range
interactions with the walls are absent or present. Right: dimensionless
current density as a function of the asymmetry angle $\theta$ is plotted for
two cases where the hydrodynamic long range interactions with the walls are
absent or present. Simulation time is $t_{s}=10000\mathrm{sec}$,
$\bar{f}_{0}=10$, $\bar{\omega}=100$, $L_{x}=15a$, $L_{z}=5a$ and
$a=1\mu\mathrm{m}$.

Figure 5: Dimensionless current density as a function of the asymmetry angle
$\theta$ for different values of $L_{x}$ and $L_{z}$ is investigated.
Simulation time is $t_{s}=10000\mathrm{sec.}$, $\bar{\omega}=100$ and
$a=1\mu\mathrm{m}$.

In fig. 4, we study how will the hydrodynamic interactions with the walls,
modify the functionality of a Brownian ratchet. We have plotted the current
density in terms of both amplitude of the externally applied force and also
the asymmetry angle $\theta$. As one can see, in both cases the inclusion of
long range hydrodynamic interactions with the walls, will significantly
decrease the average current density. In terms of the amplitude of external
harmonic force, the hydrodynamic interactions with the wall have more
influences at large amplitudes. It is interesting that for a Brownian ratchet
working at its efficient angle, the long range hydrodynamic interactions will
deeply decrease the current.

This current reduction as a result of hydrodynamic interactions, may be
understood in terms of the friction coefficients. As we have pointed out in
section 2, for a particle moving near a single wall, the friction coefficient
will be increased with a factor proportional to $(a/h)$. This means that for
two particles, one moving in an infinite medium and the other moving near a
wall, to achieve a constant velocity for both particles, higher force should
be applied to the particle that is moving near wall. As the fluctuations are
the source of rectified motion here, the increase in the friction coefficient
will result a reduction in the current. This simplified picture that is
inspired by a single wall confinement, can be applied qualitatively for a
particle moving in a complex geometry of a channel.

How do the geometrical parameters of this asymmetric channel influence the
functionality of Brownian ratchet? In fig. 5, we have investigated the
influence of geometrical parameters of the system in current density by
changing both $L_{x}$ and $L_{z}$. The current density is plotted in terms of
the asymmetry angle $\theta$. We see that, for fixed $L_{x}$($L_{z}$), higher
current density can be achieved by choosing smaller $L_{z}$($L_{x}$). In all
cases, the maximum current density corresponds to an angle $\theta\approx
20^{\circ}$. Interplay between the size of the particle and the available area
in the corners, determines the overall behavior of the system. Trapping time
for the particle that is moving in the corners of the channel will change the
efficiency of a Brownian ratchet. Dividing the area of the channel to two
parts, one with a rectangular geometry with an area $L_{x}\times L_{z}$ and
the second part with a triangle area, we see that less area for the
rectangular part, corresponds to more current density.

In conclusion, we have considered the functionality of a Brownian ratchet and
investigate the role of hydrodynamic interactions on the efficiency of this
system. To take into account the interactions with walls of the channel, we
proceeded with an approximate scheme for the mobility tensor of a colloidal
particle moving near boundaries. We discussed about the limitations and
validity of this approximation. As there is no exact solution for the mobility
of a particle inside channel, to achieve more accurate results one can perform
a simulation that includes a coupled equations of motion for colloidal
particle and also the fluid particles. Although the results of such intense
studies will help to obtain more accurate results, but we do not expect to see
a large qualitative deviations with the results that we have obtained here.

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