Communication, Dynamical Resource Theory, and Thermodynamics

Resource is a concept widely used in the study of physics: It can be an effect or a phenomenon, helping us achieve advantages that can never occur in its absence. A quantitative understanding of different resources is thus vital for further applications. For this reason, an approach called resource theory comes, aiming to provide a general strategy to depict different resources RT-RMP .

A resource theory can be interpreted as a triplet $(R,\mathcal{F}_{R},\mathcal{O}_{R})$, consisting of the resource itself $R$ (e.g., entanglement Ent-RMP ), the set of quantities without the resource $\mathcal{F}_{R}$ (called free quantities; e.g., separable states), and the set of physical processes that will not generate the resource $\mathcal{O}_{R}$ (called free operations of $R$; e.g., local operation and classical communication channels QCI-book ). Every $\mathcal{E}\in\mathcal{O}_{R}$ must satisfy $\mathcal{E}(\eta)\in\mathcal{F}_{R}\;\forall\eta\in\mathcal{F}_{R}$, which is sometimes called the golden rule of resource theories RT-RMP . Operations satisfying this condition are called resource non-generating for $R$, which form the largest possible set of free operations of $R$. A resource theory allows ones to quantify the resource via a resource monotone, $Q_{R}$, which is a non-negative-valued function satisfying two conditions: (i) $Q_{R}(q)=0$ if $q\in\mathcal{F}_{R}$; and (ii) $Q_{R}[\mathcal{E}(q)]\leq Q_{R}(q)\;\forall q\;\&\;\forall\mathcal{E}\in\mathcal{O}_{R}$. This is a “ruler” attributing numbers to different resource contents.

Adopting this general approach, one can study specific resources such as (but not limited to) entanglement Ent-RMP ; Vedral1997 ; Vidal2002 , coherence  Coherence-RMP ; Baumgratz2014 , nonlocality Bell ; Bell-RMP ; Wolfe2019 , steering Wiseman2007 ; Jones2007 ; steering-review ; Skrzypczyk2014 ; Piani2015 ; Gallego2015 ; RMP-steering , asymmetry Gour2008 ; Marvian2016 ; Takagi2019-4 , and athermality Brandao2013 ; Brandao2015 ; Horodecki2013 ; Lostaglio2018 ; Serafini2019 ; Narasimhachar2019 . Together with various features of general resource theories Horodecki2013-2 ; Brandao2015-2 ; del_Rio2015 ; Coecke2016 ; Gour2017 ; Anshu2018 ; Regula2018 ; Bu2018 ; Liu2017 ; Lami2018 ; RT-RMP ; Takagi2019 ; Takagi2019-2 ; Liu2019 ; Fang2019 ; Korzekwa2019 ; Regula2020 , one is able to concretely picture the originally vague notion of resources for states – while the unique roles of dynamical resources have not been noticed until recently. Resource theories of channels footnote0 have therefore drawn much attention lately and been studied intensively Hsieh2017 ; Kuo2018 ; Pirandola2017 ; Dana2017 ; Bu2018 ; Wilde2018 ; Diaz2018 ; Zhuang2018 ; Gour2019-3 ; Theurer2019 ; Seddon2019 ; Rosset2018 ; LiuWinter2019 ; LiuYuan2019 ; Gour2019 ; Gour2019-2 ; Bauml2019 ; Wang2019 ; Berk2019 ; Takagi2019-3 ; Hsieh2020-1 ; Saxena2019 ; Zhang2020 . Unlike the state resources, which are static, channel resources are dynamical properties, thereby providing links to dynamical problems such as communication Takagi2019-3 and resource preservation Hsieh2020-1 ; Saxena2019 .

Very recently, the interplay between resource theories and classical communication has been investigated Korzekwa2019 ; Takagi2019-3 (see also Ref. Kristjansson2020 ), successfully providing new insights and widening our understanding. For instance, a neat proof of the strong converse property of non-signaling assisted classical capacity has been established Takagi2019-3 . Also, amounts of classical messages encodable into the resource content of states has been estimated, and different physical meanings can be concluded by considering specific resources Korzekwa2019 . Hence, the interplay between resource theory and classical communication is a potential platform for interdisciplinary studies. To continue this research line, it is thus necessary to understand communication setups constrained by different static resources. A general treatment on this can clarify the role of static resources in communication and provide potential applications in different physical settings. This motivates us to ask:

How do resource constraints affect classical communications?

In this work, we consider classical communication scenarios where the information processing channel is forbidden to supply additional resources, thereby being a free operation (there are some subtleties about this setting, and we refer the reader to Appendix A for a detailed discussion). The basic setup will be given in Sec. II. In Sec. III, we show that the corresponding one-shot classical capacity is upper bounded by the ability to preserve the resource, which is called resource preservability Hsieh2020-1 , plus a resourceless contribution term. Furthermore, when the underlying resource is asymmetry, a lower bound can be obtained. As an application, we use our approach to bridge classical communication and thermodynamics in Sec. IV: Under the thermalization model introduced in Ref. Sparaciari2019 , the one-shot classical capacity of a Gibbs-preserving coherence non-generating channel is upper bounded by its coherence preservability plus the smallest bath size needed to thermalize all outputs of its incoherent version Sparaciari2019 ; Hsieh2020-1 . A direct physical message is that, under this setting, transmitting classical information through a coherence-annihilating channel necessarily needs a large enough bath size as the thermodynamic cost. This provides a dynamical analogue of the famous Landauer’s principle Landauer1961 and illustrates how dynamical resource theory can connect seemingly different fields, inspiring us to further investigate the interplay of classical communication and thermalization. To this end, in Sec. V we first study a tool related to the ability to simultaneously maintain orthogonality and maximal entanglement. Using this concept, in Sec. VI we found that it is possible for locally performed thermalization channels to globally transmit classical information when the local baths are correlated classically through pre-shared randomness. This result reveals the huge difference between local and global dynamics, providing the recent discovery in Ref. Hsieh2020-2 a generalization and application to communication. Finally, we conclude in Sec. VII.

To process classical information depicted by a finite sequence of integers $\{m\}_{m=0}^{M-1}$ through a quantum channel $\mathcal{N}$, one needs to encode them into a set of quantum states $\{\rho_{m}\}_{m=0}^{M-1}$; likewise, a decoding is needed to extract the information from outputs of $\mathcal{N}$, which can be done by a positive operator-valued measurement (POVM) $\{E_{m}\}_{m=0}^{M-1}$ QCI-book . They can be written jointly as $\Theta_{M}=(\{\rho\}_{m=0}^{M-1},\{E_{m}\}_{m=0}^{M-1})$, called an $M$-code, which depicts the transformation $\rho_{m}\mapsto{\rm tr}\left[E_{m}\mathcal{N}(\rho_{m})\right]$ for each $m$. To see how faithfully one can extract the input messages $\{m\}_{m=0}^{M-1}$, the one-shot classical capacity with error $\epsilon$ Wang2012 ; Datta2013 of $\mathcal{N}$ can be defined as a measure:

$\displaystyle C_{\rm(1)}^{\epsilon}(\mathcal{N})\coloneqq\max\left\{\log_{2}{M}\;|\;\exists\Theta_{M},\;p_{s}(\Theta_{M},\mathcal{N})\geq 1-\epsilon\right\},$

(1)

where the average success probability is given by

$\displaystyle p_{s}(\Theta_{M},\mathcal{N})\coloneqq\frac{1}{M}\sum_{m=0}^{M-1}{\rm tr}\left[E_{m}\mathcal{N}(\rho_{m})\right].$

(2)

Before introducing the main results, we briefly review relevant ingredients of resource preservability Hsieh2020-1 (or simply $R$-preservability when the state resource $R$ is given), which is a dynamical resource depicting the ability to preserve $R$. To start with, we impose basic assumptions on a given state resource theory $(R,\mathcal{F}_{R},\mathcal{O}_{R})$:

1. 1.

   Identity and partial trace are both free operations.

2. 2.

   $\mathcal{O}_{R}$ is closed under tensor products, convex sums, and compositions  footnote:Markovianity .

These assumptions are strict enough for an analytically feasible study and still general enough to be shared by many known resource theories (see Appendix B for a further discussion). For a given state resource theory $(R,\mathcal{F}_{R},\mathcal{O}_{R})$, the induced $R$-preservability theory is a channel resource theory written as ($R$-preservability, $\mathcal{O}_{R}^{N}$, $\mathbb{F}_{R}$). It is defined on all channels in $\mathcal{O}_{R}$. In this channel resource theory, the free quantities are members of the set $\mathcal{O}_{R}^{N}$ called resource annihilating channels Hsieh2020-1 ; footnote:R-DestroyingMap :

$\displaystyle\mathcal{O}_{R}^{N}\coloneqq\{\Lambda\in\mathcal{O}_{R}\;|\;\Lambda(\rho)\in\mathcal{F}_{R}\;\forall\rho\}.$

(3)

They are channels in $\mathcal{O}_{R}$ which can only output free states. A special class of resource annihilating channels are those who cannot output any resourceful state even assisted by ancillary resource annihilating channels; specifically, no $R$-preservability can be activated Palazuelos2012 ; Cavalcanti2013 ; Hsieh2016 ; Quintino2016 ; Hsieh2020-1 ; Zhang2020 . Such channels are called absolutely resource annihilating channels Hsieh2020-1 , which are elements of the set $\widetilde{\mathcal{O}}_{R}^{N}\coloneqq\{\widetilde{\Lambda}\in\mathcal{O}_{R}^{N}\;|\;\widetilde{\Lambda}\otimes\Lambda\in\mathcal{O}_{R}^{N}\;\forall\Lambda\in\mathcal{O}_{R}^{N}\}.$ Free operations of $R$-preservability, which are collectively denoted by the set $\mathbb{F}_{R}$, are super-channels Chiribella2008 ; Chiribella2008-2 given by Hsieh2020-1 $\mathcal{E}\mapsto\Lambda_{+}\circ(\mathcal{E}\otimes\widetilde{\Lambda})\circ\Lambda_{-}$ with $\Lambda_{+},\Lambda_{-}\in\mathcal{O}_{R}$ and $\widetilde{\Lambda}\in\widetilde{\mathcal{O}}_{R}^{N}$. Finally, to quantify $R$-preservability, consider a contractive generalized distance measure $D$ on quantum states; that is, it is a real-valued function satisfying (i) $D(\rho,\sigma)\geq 0$ and equality holds if and only if $\rho=\sigma$, and (ii) (data-processing inequality) $D[\mathcal{E}(\rho),\mathcal{E}(\sigma)]\leq D(\rho,\sigma)$ for all $\rho,\sigma$ and channels $\mathcal{E}$. The following $R$-preservability monotone induced by $D$ Hsieh2020-1 will be used in this work:

$\displaystyle P_{D}(\mathcal{E})\coloneqq\inf_{\Lambda\in\mathcal{O}_{R}^{N}}D^{R}(\mathcal{E},\Lambda),$

(4)

where $D^{R}(\mathcal{E}_{\rm S},\Lambda_{\rm S})\coloneqq\sup_{\widetilde{\Lambda}_{\rm A},\rho_{\rm SA}}D[(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA}),(\Lambda_{\rm S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA})]$ maximizes over every possible ancillary system ${\rm A}$, joint input $\rho_{\rm SA}$, and absolutely resource annihilating channel $\widetilde{\Lambda}_{\rm SA}$. Geometrically, Eq. (4) can be understood as a distance between $\mathcal{E}$ and the set $\mathcal{O}_{R}^{N}$ that is adjusted by absolutely resource annihilating channels. With Assumptions 1 and 2, in Appendix B we show that $P_{D}$ is indeed a monotone Hsieh2020-1 , in the sense that it is a non-negative-valued function such that $P_{D}(\mathcal{E})=0$ if $\mathcal{E}\in\mathcal{O}_{R}^{N}$ footnote:FaithfulnessRemark , and $P_{D}[\mathfrak{F}(\mathcal{E})]\leq P_{D}(\mathcal{E})$ for every channel $\mathcal{E}$ and $\mathfrak{F}\in\mathbb{F}_{R}$. Note that, in this work, we only ask $R$-preservability monotones to satisfy these two conditions, which are the core features of a monotone. Further properties, such as Eq. (7) in Ref. Hsieh2020-1 , will need additional assumptions, and we leave the details in Appendix B. Finally, the distance measure that will be mainly used in this work is the max-relative entropy Datta2009 for states defined as (and, conventionally, we adopt $\inf\emptyset=\infty$)

$\displaystyle D_{\rm max}(\rho\|\sigma)\coloneqq\log_{2}\inf\{\lambda\geq 0\,|\,\rho\leq\lambda\sigma\}.$

(5)

Hence, $P_{D_{\rm max}}(\mathcal{N})$ is the minimal amount of noise one needs to add to turn $\mathcal{N}$ into resource annihilating [see also Eq. (C)]. For a given error $\kappa>0$, we define $\mathcal{O}_{R}^{N}(\kappa;\mathcal{N})\coloneqq\{\Lambda\in\mathcal{O}_{R}^{N}\,|\,|D_{\rm max}^{R}(\mathcal{N}\|\Lambda)-P_{D_{\rm max}}(\mathcal{N})|\leq\kappa\}$. It contains resource annihilating channels achievable by adding the smallest amount of noise to $\mathcal{N}$, up to the error $\kappa$. We call them resourceless versions of $\mathcal{N}$ up to the error $\kappa$, and use the notation $\Lambda^{\mathcal{N}}\in\mathcal{O}_{R}^{N}(\kappa;\mathcal{N})$ to emphasize their dependence on $\mathcal{N}$.

To introduce the first result, define

$\displaystyle\Gamma_{\kappa}(\mathcal{N})\coloneqq\log_{2}\inf_{\Lambda^{\mathcal{N}}\in\mathcal{O}_{R}^{N}(\kappa;\mathcal{N})}\sup_{\Theta_{M}}\sum_{m=0}^{M-1}{\rm tr}\left[E_{m}\Lambda^{\mathcal{N}}(\rho_{m})\right],$

(6)

where $\sup_{\Theta_{M}}$ maximizes over every $M\in\mathbb{N}$ and $M$-code $\Theta_{M}$. $2^{\Gamma_{\kappa}(\mathcal{N})}$ tells us the highest number of discriminable states through every resourceless version of $\mathcal{N}$, up to the error $\kappa$. We also consider $P_{D}^{\delta}(\mathcal{E})\coloneqq\inf_{\left\|\mathcal{E}-\mathcal{E}^{\prime}\right\|_{\diamond}\leq 2\delta}P_{D}(\mathcal{E}^{\prime})$ and $\Gamma_{\kappa}^{\delta}(\mathcal{E})\coloneqq\sup_{\left\|\mathcal{E}-\mathcal{E}^{\prime}\right\|_{\diamond}\leq 2\delta}\Gamma_{\kappa}^{\delta}(\mathcal{E}^{\prime})$, which smooth the optimizations over all channels $\mathcal{E}^{\prime}$ closed to $\mathcal{E}$. Also, $\left\|\mathcal{E}\right\|_{\diamond}\coloneqq\sup_{{\rm A},\rho_{\rm SA}}\left\|(\mathcal{E}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SA})\right\|_{1}$ is the diamond norm. Combining Refs. Takagi2019-3 ; Hsieh2020-1 , in Appendix C we prove the following upper bound:

This upper bound Eq. (7) can be interpreted as follows: It is the highest amount of carriable classical information by every resourceless version of $\mathcal{N}$, i.e., ${\Gamma_{\kappa}^{\delta}}(\mathcal{N})$, plus the contribution from the ability of $\mathcal{N}$ to preserve $R$, i.e., $P_{D_{\rm max}}^{\delta}(\mathcal{N})$. This also suggests that $C_{\rm(1)}^{\epsilon}(\mathcal{N})-{\Gamma_{\kappa}^{\delta}(\mathcal{N})}$ characterizes the resource advantage, since it estimates the amount of transmissible classical information via the ability of $\mathcal{N}$ to preserve $R$. To see the tightness of Eq. (7) (up to error terms containing $\epsilon,\delta$), the inequality is saturated by every state preparation channel outputting a fixed free state $(\cdot)\mapsto\eta$ with $\eta\in\mathcal{F}_{R}$. There also exist examples attending the inequality with non-zero classical capacity. For instance, in a $d$-dimensional system, when $R$ is coherence and $\mathcal{N}$ is the dephasing channel, i.e., $(\cdot)\mapsto\sum_{i=1}^{d}|i\rangle\langle i|\cdot|i\rangle\langle i|$, we have both side as $\log_{2}d$.

As the last remark on Eq. (7), when the optimal amount of classical information can be encoded into free states, the optimal capacity should intuitively be achievable by channels with zero $R$-preservability, and any general result should respect this fact. Hence, resource preservability monotone cannot upper bound classical capacity solely, and Theorem 1 is consistent with this expectation due to the term $\Gamma_{\kappa}^{\delta}(\mathcal{N})$. However, such resourceless advantages no longer exist when more constraints are made for specific purposes (e.g., Sec. V).

Theorem 1 can link different physical properties to classical communication, which is illustrated by the following result. Let $P_{D|R}$ denote Eq. (4) with the state resource $R$ and write $R=\gamma$ when the state resource is the athermality induced by the thermal state $\gamma$ (we postpone its formal definition to Sec. IV). Then in Appendix C we show that:

Corollary 2 implies that the ability of a Gibbs-preserving $\mathcal{N}\in\mathcal{O}_{R}$ to transmit classical information is limited by its ability to preserve $R$, plus the ability of its resourceless version (to $R$) to preserve athermality. This observation will allow us to prove one of the main results of this work in Sec. IV.

It is worth mentioning that our result bridges two seemingly different physical concepts: Classical capacity Takagi2019-3 and heat bath size needed for thermalization Sparaciari2019 ; Hsieh2020-1 . To introduce the result, we give a quick review of the resource theory of athermality and related ingredients for thermalization bath sizes Sparaciari2019 . Athermality is the status out of thermal equilibrium. With a fixed system dimension $d$, the unique free state is the thermal equilibrium state, or the thermal state. With a given system Hamiltonian $H_{\rm S}$ and temperature $T$, the thermal state is uniquely given by $\gamma=\frac{e^{-\beta H_{\rm S}}}{{\rm tr}(e^{-\beta H_{\rm S}})},$ where $\beta=\frac{1}{k_{B}T}$ is the inverse temperature and $k_{B}$ is the Boltzmann constant. For multiple systems with tensor product, all free states in this resource theory are $\gamma^{\otimes k}$ for some positive integer $k$ (i.e., all allowed dimensions are $d^{k}$ with some $k$). In this work, we adopt Gibbs-preserving channels as the free operations. They are channels $\mathcal{E}$ keeping thermal states invariant: $\mathcal{E}(\gamma^{\otimes k})=\gamma^{\otimes l}$, where $d^{k}$ and $d^{l}$ are the input and output dimensions, respectively. Physically, these are dynamics that will not drive thermal equilibrium away from equilibrium.

To formally study thermalization, we follow Ref. Sparaciari2019 and define a channel (jointly acting on system ${\rm S}$ plus bath ${\rm B}$) $\mathcal{E}_{\rm SB}:{\rm SB}\to{\rm SB}$ to $\epsilon$-thermalize a system state $\rho_{\rm S}$ if

$\displaystyle\left\|\mathcal{E}_{\rm SB}\left(\rho_{\rm S}\otimes\gamma^{\otimes(n-1)}\right)-\gamma^{\otimes n}\right\|_{1}\leq\epsilon.$

(10)

That is, $\mathcal{E}_{\rm SB}$ needs to globally thermalize ${\rm SB}$, where the thermal state $\gamma$ is determined by the Hamiltonian and the temperature of ${\rm S}$, and the initial state of ${\rm B}$ is the $n-1$ copies of $\gamma$. To depict such thermalization processes dynamically, we consider the collision model introduced in Ref. Sparaciari2019 . To avoid complexity, we refer the reader to Appendix E for a brief introduction of this model, and here we let $\mathcal{C}_{n}$ be the set of all channels on ${\rm SB}$ that can be realized by this model. Then the quantity $n^{\epsilon}_{\gamma}(\rho_{\rm S})\coloneqq\inf\{n\in\mathbb{N}\,|\,\exists\,\mathcal{E}_{\rm SB}\in\mathcal{C}_{n}\;{\rm s.t.\;Eq.~{}\eqref{Eq:Thermalize-Def}\;holds}\}$ can be understood as the smallest bath size needed to $\epsilon$-thermalize $\rho_{\rm S}$ under this model Sparaciari2019 . This concept can be generalized to any channel $\mathcal{N}$ by defining Hsieh2020-1

$\displaystyle\mathcal{B}^{\epsilon}_{\gamma}(\mathcal{N})\coloneqq\sup_{\rho}n^{\epsilon}_{\gamma}[\mathcal{N}(\rho)]-1,$

(11)

which maximizes over all the smallest bath sizes among all outputs of $\mathcal{N}$. This is thus the smallest bath size needed to $\epsilon$-thermalize all outputs of $\mathcal{N}$ under the given collision model.

Now we mention a core assumption made in Ref. Sparaciari2019 used to regularize the analysis. A given Hamiltonian $H$ with energy levels $\{E_{i}\}_{i=1}^{d}$ is said to satisfy the energy subspace condition if for every positive integer $M$ and every pair of different vectors $\{{\bf m}\neq{\bf m}^{\prime}\}\subset\left(\mathbb{N}\cup\{0\}\right)^{d}$ satisfying $\sum_{i=1}^{d}m_{i}=\sum_{i=1}^{d}m^{\prime}_{i}=M$, we have $\sum_{i=1}^{d}m_{i}E_{i}\neq\sum_{i=1}^{d}m^{\prime}_{i}E_{i}$. Hence, energy levels cannot be integer multiples of each other, and energy degeneracy is also forbidden. As an application of Theorem 1 (and Corollary 2), in Appendix F we show the following bound [we implicitly assume the system Hamiltonian is the one realizing the given thermal state $\gamma$ with some temperature, and its energy eigenbasis defines the coherence, $R={\rm Coh}$; also, we use $p_{\rm min}(\gamma)$ to denote the smallest eigenvalue of $\gamma$]:

Theorem 4 illustrates how a dynamical resource theory can bridge a pure thermodynamic quantity to a pure communication quantity. To illustrate this, let us first consider the special case when $\mathcal{N}$ is coherence-annihilating footnote:Coherence-Annihilating . In this case, one is able to choose $\mathcal{N}=\Lambda^{\mathcal{N}}$ and $\kappa=0$. Within this setup, if $\mathcal{N}$ can communicate a high amount of classical information, it necessarily requires a large bath to thermalize all its outputs. On the other hand, if this channel has a small thermalization bath size, it unavoidably has a weak ability to communicate classical information. Importantly, Theorem 4 provides a physical message that is in spirit similar to the Landauer’s principle Landauer1961 . Landauer’s principle says that preparing a pre-defined pure state, e.g., $|0\rangle$, from an initial state $\rho$ requires at least $S(\rho)k_{B}T\ln 2$ amount of energy del_Lio2011 , where $S(\rho)\coloneqq-{\rm tr}(\rho\log_{2}\rho)$ is the von Neumann entropy of $\rho$. In this sense, energy can be regarded as the thermodynamic cost needed to create classical information carried by an orthonormal basis $\{|m\rangle\}_{m=0}^{M-1}$. Theorem 4 provides a different, dynamical perspective: Under the given setting, transmitting $n$ bits of classical information, i.e., $C_{\rm(1)}^{\epsilon}(\mathcal{N})=n$, necessarily requires the bath size $\mathcal{B}^{\epsilon}_{\gamma}(\mathcal{N})$ to be at least $2^{n}-1$, up to some small terms. In this case, the bath size can be interpreted as the thermodynamic cost needed to transmit classical information. When the ability of the channel $\mathcal{N}$ to preserve coherence is turned on, interestingly, the cost to transmit classical information becomes a hybrid term: It is the bath size of $\mathcal{N}$’s incoherent version [i.e., $\Lambda^{\mathcal{N}}\in\mathcal{O}_{\rm Coh}^{N}(\kappa,\mathcal{N})$] plus the ability of $\mathcal{N}$ to preserve coherence. In other words, this is the sum of the thermodynamic cost of $\mathcal{N}$’s classical counterpart, and the quantum effect maintained by $\mathcal{N}$. By treating Landauer’s principle as a bridge between thermodynamics and a static property of classical information, Theorem 4 brings a connection between thermodynamics and a dynamical feature of classical information.

Note that, as expected, a full thermalization process (i.e., a state preparation channel of the given thermal state) cannot transmit any amount of classical information, since the bath size needed for thermalization is precisely zero. This also implies that the inequality in Theorem 4 is tight. However, locally-performed thermalization processes can actually allow global transmission of classical information, which only needs shared randomness as a resource. In this sense, thermalization together with a classical resource can still achieve nontrivial classical communication. We will introduce this result more in-dept in Sec. VI, and the following section is for a tool needed for its proof.

Once a question can be formulated into a classical communication problem, our approach can be used to study connections between the given question and different resource constraints. To illustrate this, we study the following question: How robust is the structure of orthogonal maximal entanglement under dynamics? As the motivation, a maximally entangled basis is a well-known tool in quantum information science, promising applications such as quantum teleportation Bennett1993 and superdense coding QCI-book . The key is the simultaneous existence of maximal entanglement and orthogonality, and maintaining both of them through a physical evolution is vital for applications afterward. To model this question, we impose two restrictions in the classical communication scenarios used in this work: (i) The encoding $\{\rho_{m}\}_{m=0}^{M-1}$ are mutually orthonormal maximally entangled states $\{|\Phi_{m}\rangle\}_{m=0}^{M-1}$. (ii) The decoding $\{E_{m}\}_{m=0}^{M-1}$ are projective measurements done by a (sub-)basis of orthogonal maximally entangled states $\{|\Phi^{\prime}_{m}\rangle\langle\Phi^{\prime}_{m}|\}_{m=0}^{M-1}$. The corresponding one-shot classical capacity characterizes the ability of a given channel $\mathcal{N}$ footnote:Local-Dimension to simultaneously maintain orthogonality and maximal entanglement:

$\displaystyle C_{{\rm ME},(1)}^{\epsilon}(\mathcal{N})\coloneqq\log_{2}\max\{M\,|\,p_{s|{\rm ME}}(M,\mathcal{N})\geq 1-\epsilon\},$

(13)

where the success probability reads $p_{s|{\rm ME}}(M,\mathcal{N})\coloneqq\sup_{\{|\Phi_{m}\rangle\},\{|\Phi^{\prime}_{m}\rangle\}}\frac{1}{M}\sum_{m=0}^{M-1}\langle\Phi^{\prime}_{m}|\mathcal{N}(|\Phi_{m}\rangle\langle\Phi_{m}|)|\Phi^{\prime}_{m}\rangle,$ and the maximization is taken over all sets of orthogonal maximally entangled states of size $M$, denoted by $\{|\Phi_{m}\rangle\},\{|\Phi^{\prime}_{m}\rangle\}$. Thus, $C_{{\rm ME},(1)}^{\epsilon}(\mathcal{N})$ is the highest maintainable pairs of mutually orthonormal maximally entangled states under the dynamics $\mathcal{N}$, up to an error smaller than $\epsilon$. To introduce the result, we say a state $\rho$ is multi-copy nonlocal/steerable Palazuelos2012 ; Cavalcanti2013 ; Hsieh2016 ; Quintino2016 if there exists an integer $k$ such that $\rho^{\otimes k}$ is nonlocal/steerable. Also, recall that $\mathbb{F}_{R}$ is the set of free operation of $R$-preservability defined in Sec. II. Then in Appendix G we show that

Theorem 5 provides upper bounds on $C_{\rm ME,(1)}(\mathcal{N})$ when $\mathcal{N}$ is a free operation of specific resources, which holds even when the channel is assisted by additional structures constrained by the resource (which is given by $\mathbb{F}_{R}$). Theorem 5 also brings an alternative operational interpretation of $R$-preservability: For the resources $R$ mentioned above, $R$-preservability bounds the channel’s simultaneous maintainability of orthogonality and maximal entanglement in resource-constrained scenarios.

We remark that one can also interpret Eq. (13) as a measure of the ability to admit superdense coding, and Theorem 5 therefore serves as an upper bound on this ability. Furthermore, Theorem 5 brings a connection between fully entangled fraction Horodecki1999-2 ; Albeverio2002 and $R$-preservability. We leave the details in Appendix H.

The connection between Communication and Thermodynamics given by Theorem 4 implies that it is impossible for a full thermalization, i.e., $\Phi_{\gamma}:(\cdot)\mapsto\gamma{\rm tr}(\cdot)$ for a given thermal state $\gamma$, to transmit any amount of classical information. This is consistent with our physical intuition since a full thermalization means a complete destroy of the input information, leaving no freedom for the output. However, this impossibility can be flipped with the help of pre-shared randomness, which is a pure classical resource. More precisely, when local baths in a bipartite setting are allowed to be correlated through shared randomness, it is possible for locally performed full thermalizations to globally transmit classical information.

To make the statement precise, let us recall the following notions from Ref. Hsieh2020-2 . A bipartite channel $\mathcal{E}_{\rm AB}$ on ${\rm AB}$ is called a local thermalization to a given pair of thermal states $(\gamma_{\rm A},\gamma_{\rm B})$ if (1) there exist an ancillary system ${\rm A^{\prime}B^{\prime}}$, unitary channels $\mathcal{U}_{\rm AA^{\prime}},\mathcal{U}_{\rm BB^{\prime}}$, and a thermal state $\widetilde{\gamma}_{\rm A^{\prime}B^{\prime}}$ separable in ${\rm AB}$ bipartition such that $\mathcal{E}_{\rm AB}(\rho)={\rm tr}_{\rm A^{\prime}B^{\prime}}\left[\mathcal{U}_{\rm AA^{\prime}}\otimes\mathcal{U}_{\rm BB^{\prime}}(\rho\otimes\widetilde{\gamma}_{\rm A^{\prime}B^{\prime}})\right],$ and (2) ${\rm tr}_{\rm B}\circ\mathcal{E}_{\rm AB}(\rho)=\gamma_{\rm A}$ and ${\rm tr}_{\rm A}\circ\mathcal{E}_{\rm AB}(\rho)=\gamma_{\rm B}$ for all $\rho$. As mentioned in Ref. Hsieh2020-2 , local thermalization is local in two senses: It is a local operation plus pre-shared randomness channel, and it is locally equivalent to a full thermalization process. The main message from Ref. Hsieh2020-2 is that entanglement preserving local thermalization exists; that is, for every non-pure full-rank $\gamma_{\rm A},\gamma_{\rm B}$, there exists a local thermalization $\mathcal{E}_{\rm AB}$ such that $\mathcal{E}_{\rm AB}(\rho)$ is entangled for some $\rho$. It turns out that such channels can also transmit classical information. To state the result, consider a tripartite system ${\rm ABC}$ with local dimension $d,d,d^{2}+1$, respectively. We focus on the bipartition ${\rm A|BC}$; namely, the subsystem ${\rm C}$ can be understood as an ancillary system possessed by the agent in ${\rm B}$. Let $\gamma_{{\rm X}}$ be the thermal state of the subsystem ${\rm X=A,B,C}$ with temperature $T_{\rm X}$ and Hamiltonian $H_{\rm X}$. Following Ref. Hsieh2020-2 , we assume each $H_{\rm X}$ is non-degenerate and finite-energy. Then, Combining Ref. Hsieh2020-2 and results in Sec. V, in Appendix I we prove that locally performed thermalization is able to transmit classical information with the help of pre-shared randomness [let $p_{\rm min|AB}\coloneqq\min\{p_{\rm min}(\gamma_{{\rm A}});p_{\rm min}(\gamma_{{\rm B}})\}$ and $\gamma_{\rm BC}\coloneqq\gamma_{\rm B}\otimes\gamma_{\rm C}$; see also Fig. 1]:

Under the global dynamics given by Theorem 6, although the local agents will observe a full thermalization process, transmitting classical information is still possible via the joint bipartite dynamics. This is because classical information can be locally encoded in the global quantum correlation. Note that the amount of transmissible classical information largely depends on local temperatures: When the local systems are too cold (which corresponds to the limit $p_{\rm min|AB}\to 0$), a low temperature will force the local thermal state be closed to a pure state (here we assume no energy degeneracy), resulting in no possibility to maintain global correlation. Also note that although $\mathcal{E}_{\rm A|BC}$ is locally identical to a full thermalization process, its global behavior is far from a thermalization. This gap between local and global dynamics leads to the possibility for a classical communication through local thermalizations when shared randomness is accessed to. Notably, this illustrate that, although being a classical resource, shared randomness provides advantages to maintain quantum correlation and assist classical communication through thermalization processes.

We study classical communication scenarios with free operations of a given resource as the information processor. The one-shot classical capacity is upper bounded by resource preservability Hsieh2020-1 plus a term of resourceless contribution. This upper bound provides an alternative interpretation of resource preservability. Furthermore, when asymmetry is the resource, a lower bound can also be obtained.

As an application, we use our result to bridge two seemingly different concepts: Under the thermalization model given by Ref. Sparaciari2019 , for every Gibbs-preserving coherence non-generating channel, its smallest channel bath size (i.e., a bath size needed to thermalize all outputs of a given channel) plus its coherence preservability will upper bound its one-shot classical capacity. Thus, under this setting, transmitting $n$ bits of classical information through a coherence-annihilating channel requires the channel bath size to be at least $2^{n}-1$, up to some small terms. This reveals an implicit thermodynamic cost of transmitting classical information, providing a dynamical analogue of the Landauer’s principle Landauer1961 and illustrating how a dynamical resource theory allows applications to connect different dynamical phenomena.

We further apply our approach to study channel’s simultaneous maintainability of orthogonality and maximal entanglements. Formulating the question into a communication form, a capacity-like measure can be introduced and upper bounded by resource preservability.

Finally, applying the thermalization channel introduced in Ref. Hsieh2020-2 to a bipartite setting, we show that classically correlated local baths allow a decent amount of one-shot classical capacity even when both local systems are completely thermalized. Hence, classical information processing and thermalization processes can coexist, which only requires shared randomness as a resource. This result also means that when a many-body system is in contact with a global bath having classical correlations within, it is possible to maintain classical information even after it has been thermalized locally. Share randomness as a resource is enough to guarantee that certain amounts of classical information can be extracted after local thermalizations, provided that local temperatures are not too low.

Several open questions remain. First, currently all the results related to resource preservability (e.g., Ref. Hsieh2020-1 ) are focusing on the one-shot regime, and a future direction is to understand its asymptotic behavior. Furthermore, whether one can derive a lower bound similar to Theorem 3 in terms of resource preservability and even extend the result to other state resources are still unknown. These questions could be difficult and largely depend on the choice of resources, since, e.g., the lower bound on the capacity used in Ref. Korzekwa2019 is given by the information spectrum relative entropy, which is not a contractive generalized distance measure Leditzky-PhD and hence cannot induce legal resource preservability monotone. Also, whether one can obtain any result similar to Theorem 5 in the context of coherence is still unknown. Finally, as a direct consequence of Theorem 4, we have the following conjecture: Transmitting $n$ bits of classical information through a Gibbs-preserving coherence-annihilating channel requires the corresponding channel bath size to be at least $2^{n}-1$. This conjecture could largely depend on the underlying communication setup and the thermalization model.

We hope the physical messages provided by this work can offer alternative interpretations in the interplay of dynamical resource theory, classical communication, thermodynamics, and different forms of inseparability.

We thank Antonio Acín, Stefan B$\ddot{\rm a}$uml, Dario De Santis, Yeong-Cherng Liang, Matteo Lostaglio, Mohammad Mehboudi, Gabriel Senno, and Ryuji Takagi for fruitful discussions and comments. This project is part of the ICFOstepstone - PhD Programme for Early-Stage Researchers in Photonics, funded by the Marie Skłodowska-Curie Co-funding of regional, national and international programmes (GA665884) of the European Commission, as well as by the ‘Severo Ochoa 2016-2019’ program at ICFO (SEV-2015-0522), funded by the Spanish Ministry of Economy, Industry, and Competitiveness (MINECO). We also acknowledge support from the Spanish MINECO (Severo Ochoa SEV-2015-0522), Fundació Cellex and Mir-Puig, Generalitat de Catalunya (SGR1381 and CERCA Programme).