Resource Theory of Coherence - Beyond States

TL;DR

This paper extends the resource theory of quantum coherence from states to quantum operations, introducing coherence capacity and cost for channels, and revealing phenomena like bound coherence in operations.

Contribution

It generalizes coherence resource theory to quantum channels, providing formulas for coherence capacity and cost, and discovering bound coherence in operations.

Findings

Single-letter formula for coherence capacity of unitaries

Maximally coherent states suffice for channel simulation with incoherent operations

Existence of bound coherence in quantum operations

Abstract

We generalize the recently proposed resource theory of coherence (or superposition) [Baumgratz, Cramer & Plenio, Phys. Rev. Lett. 113:140401; Winter & Yang, Phys. Rev. Lett. 116:120404] to the setting where not only the free ("incoherent") resources, but also the objects manipulated are quantum operations, rather than states. In particular, we discuss an information theoretic notion of coherence capacity of a quantum channel, and prove a single-letter formula for it in the case of unitaries. Then we move to the coherence cost of simulating a channel, and prove achievability results for unitaries and general channels acting on a $d$-dimensional system; we show that a maximally coherent state of rank $d$ is always sufficient as a resource if incoherent operations are allowed. We also show lower bounds on the simulation cost of channels that allow us to conclude that there exists bound coherence in operations, i.e. maps with non-zero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence.

Figures

Equations

$$|\psi ⟩=\sum _i{\psi }_i|i⟩,$$

$$\tilde{T}(\rho )=\sum _{\alpha }{K}_{\alpha }\rho K_{\alpha }^{†}\otimes |\alpha ⟩⟨\alpha |.$$

$$C(\varphi)\mathrel{\mathop{\mathchar 58\relax}}=S\bigl{(}\Delta(\varphi)\bigr{)},$$

$$C_f(\rho )\text{mathchar}58=\min \sum _i{p}_{i}C({\psi }_i)\text{ s.t. }\rho =\sum _i{p}_i{\psi }_i.$$

$$C_{r}(\rho)\mathrel{\mathop{\mathchar 58\relax}}=\min_{\sigma\in\Delta}D(\rho\|\sigma)=S\bigl{(}\Delta(\rho)\bigr{)}-S(\rho),$$

$$\rho_{1}=\mathcal{I}_{1}\bigl{(}(T\otimes{\operatorname{id}})\rho_{0}\bigr{)}.$$

$$\rho_{t+1}=\mathcal{I}_{t+1}\bigl{(}(T\otimes{\operatorname{id}})\rho_{t}\bigr{)}.$$

$$⟨{\Psi }_{2^{nR}}|{\rho }_n^{A_n}|{\Psi }_{2^{nR}}⟩\ge 1-ϵ.$$

$$C_{\text{gen}}(T)\geq\sup_{|\varphi\rangle\in A\otimes C}C_{r}\bigl{(}(T\otimes{\operatorname{id}})\varphi\bigr{)}-C(\varphi),$$

$$C_{\text{gen}}(T)\leq\sup_{\rho\text{ on }A\otimes C}C_{r}\bigl{(}(T\otimes{\operatorname{id}})\rho\bigr{)}-C_{r}(\rho),$$

$$C_{\text{gen}}(V\cdot V^{†})$$

$$\displaystyle=\max_{|\varphi\rangle\in A}C\bigl{(}V\varphi V^{\dagger}\bigr{)}-C(\varphi).$$

$$P_{r}(T)=\max_{\rho\text{ on }A}C_{r}\bigl{(}T(\rho)\bigr{)}-C_{r}(\rho),$$

$$\widetilde{P}_{r}(T)=\max_{|\varphi\rangle\in A}C_{r}\bigl{(}T(\varphi)\bigr{)}-C(\varphi).$$

$$P_r^{(k)}(T)\text{mathchar}58=P_r(T\otimes {\mathrm{id}}_k),{\tilde{P}}_r^{(k)}(T)\text{mathchar}58={\tilde{P}}_r(T\otimes {\mathrm{id}}_k),$$

$$C_{\text{gen}}(V\cdot V^{†})=\underset{k}{\sup }{\tilde{P}}_r^{(k)}(V\cdot V^{†})={\tilde{P}}_r(V\cdot V^{†}),$$

$$\underset{k}{\sup }{\tilde{P}}_r^{(k)}(T)\le C_{\text{gen}}(T)\le \underset{k}{\sup }{P}_r^{(k)}(T).$$

$${\Psi }_2^{\otimes \lfloor nC(\phi )+n\delta \rfloor }$$

$${\rho }^{\otimes n}$$

$${\sigma }^{\otimes \lfloor n{R}_0\rfloor }\stackrel{\text{IO}}{⟼}\approx {\Psi }_2^{\otimes \lfloor nC(\phi )+n\delta \rfloor }.$$

$$\begin{split}nR-2\delta n-2&\leq C_{r}(\rho_{n})\\ &=\sum_{t=0}^{n-1}C_{r}(\rho_{t+1})-C_{r}(\rho_{t})\\ &\leq\sum_{t=1}^{n-1}C_{r}\bigl{(}(T\otimes{\operatorname{id}})\rho_{t}\bigr{)}-C_{r}(\rho_{t}),\end{split}$$

$$\begin{split}C\bigl{(}(T\otimes{\operatorname{id}})\varphi)&-C(\varphi)\\ &=S\bigl{(}(\Delta\circ T\otimes\Delta)\varphi)-S\bigl{(}(\Delta\otimes\Delta)\varphi\bigr{)}\\ &=S\bigl{(}(\Delta\circ T\otimes{\operatorname{id}})\rho)-S\bigl{(}(\Delta\otimes{\operatorname{id}})\rho\bigr{)},\\ &\phantom{========}\bigl{[}\text{with }\rho=({\operatorname{id}}\otimes\Delta)\varphi\\ &\phantom{========[\text{with }\rho}=\sum_{i}p_{i}\varphi_{i}\otimes|i\rangle\!\langle i|\bigr{]},\\ &=\sum_{i}p_{i}\Bigl{(}S\bigl{(}\Delta(T(\varphi_{i}))\bigr{)}-S\bigl{(}\Delta(\varphi_{i})\bigr{)}\Bigr{)}\\ &\leq\max_{|\varphi\rangle\in A}S\bigl{(}\Delta(T(\varphi_{i}))\bigr{)}-S\bigl{(}\Delta(\varphi_{i})\bigr{)},\end{split}$$

$$\begin{split}C_{\text{gen}}(T)&\leq\sup_{\rho\text{ on }A\otimes C}C_{f}\bigl{(}(T\otimes{\operatorname{id}})\rho\bigr{)}-C_{f}(\rho)\\ &=\sup_{k}P_{f}(T\otimes{\operatorname{id}}_{k}),\end{split}$$

$${\mathbb{P}}_r(T)$$

$${\mathbb{P}}_f(T)$$

$${\tilde{\mathbb{P}}}_r(T)$$

$${\mathbb{P}}_r(T_1\otimes T_2)$$

$${\mathbb{P}}_f(T_1\otimes T_2)$$

$${\tilde{\mathbb{P}}}_r(T_i)={\tilde{P}}_r(T_i)=C_{\text{gen}}(T_i)$$

$${\tilde{P}}_r(T_1\otimes T_2)={\tilde{P}}_r(T_1)+{\tilde{P}}_r(T_2).$$

$$\begin{split}C_{X}\bigl{(}(&T_{1}\otimes T_{2}\otimes{\operatorname{id}}_{C})\rho\bigr{)}-C_{X}(\rho)\\ &=C_{X}\bigl{(}(T_{1}\otimes T_{2}\otimes{\operatorname{id}}_{C})\rho\bigr{)}-C_{X}\bigl{(}(T_{1}\otimes{\operatorname{id}}_{B_{2}}\otimes{\operatorname{id}}_{C})\rho\bigr{)}\\ &\phantom{==}+C_{X}\bigl{(}(T_{1}\otimes{\operatorname{id}}_{B_{2}}\otimes{\operatorname{id}}_{C})\rho\bigr{)}-C_{X}(\rho)\\ &=C_{X}\bigl{(}(T_{2}\otimes{\operatorname{id}}_{A_{1}C})\sigma\bigr{)}-C_{X}(\sigma)\\ &\phantom{==}+C_{X}\bigl{(}(T_{1}\otimes{\operatorname{id}}_{B_{2}C})\rho\bigr{)}-C_{X}(\rho)\\ &\leq\mathbb{P}_{X}(T_{2})+\mathbb{P}_{X}(T_{1}),\end{split}$$

$$\begin{split}C\bigl{(}(T_{1}\otimes T_{2})\varphi&\bigr{)}-C(\varphi)\\ &=C\bigl{(}(T_{1}\otimes T_{2})\varphi\bigr{)}-C\bigl{(}(T_{1}\otimes{\operatorname{id}}_{B_{2}})\varphi\bigr{)}\\ &\phantom{==}+C\bigl{(}(T_{1}\otimes{\operatorname{id}}_{B_{2}})\varphi\bigr{)}-C(\varphi)\\ &=C\bigl{(}({\operatorname{id}}_{A_{1}}\otimes T_{2})\psi\bigr{)}-C(\psi)\\ &\phantom{==}+C\bigl{(}(T_{1}\otimes{\operatorname{id}}_{B_{2}})\varphi\bigr{)}-C(\varphi)\\ &\leq\widetilde{\mathbb{P}}_{r}(T_{2})+\widetilde{\mathbb{P}}_{r}(T_{1})\\ &=\widetilde{P}_{r}(T_{2})+\widetilde{P}_{r}(T_{1}),\end{split}$$

$$K_{\alpha }=\sum _{i,j=0}^{d-1}{U}_{ij}|i⟩⟨j|\otimes |\alpha ⟩⟨i+\alpha \mathrm{mod}d|.$$

Full text

In particular, we discuss an information theoretic notion of coherence capacity of a quantum channel, and prove a single-letter formula for it in the case of unitaries. Then we move to the coherence cost of simulating a channel, and prove achievability results for unitaries and general channels acting on a $d$-dimensional system; we show that a maximally coherent state of rank $d$ is always sufficient as a resource if incoherent operations are allowed, and rank $d^{2}$ for “strictly incoherent” operations. We also show lower bounds on the simulation cost of channels that allow us to conclude that there exists bound coherence in operations, i.e. maps with non-zero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence.

I Introduction

Since its discovery to our days, quantum mechanics has provided a mathematical framework for the construction of physical theories. Indeterminism, interference, uncertainty, superposition and entanglement are concepts of quantum mechanics that distinguish it from classical physics, and which have become resources in quantum information processing.

Quantum resource theories aim at capturing the essence of these traits and quantifying them. Recently, quantum resource theories have been formulated in different areas of physics such as the resource theory of athermality in thermodynamics Brandao1 ; Brandao2 ; Horodecki ; Faist ; Gour ; Narasimhachar and the resource theory of asymmetry Gour1 ; Marvian . Furthermore, general structural frameworks of quantum resource theories have been proposed Brandao3 .

Resource theories using concepts of quantum mechanics have been developed since the appearance of the theory of entanglement Plenio1 ; Vedral ; HHHH . Very recently, Baumgratz et al. Baumgratz , following earlier work by Åberg Aaberg , have made quantum coherence itself, i.e. the concept of superposition

[TABLE]

the subject of a resource theory; see also Plenio-2 .

The present paper is concerned with this resource theory of coherence, and here we briefly recall its fundamental definitions, as well as some important coherence measures; for a comprehensive review, see Streltsov-rev . Let $\{|i\rangle\mathrel{\mathop{\mathchar 58\relax}}i=0,\ldots,d-1\}$ be a particular fixed basis of the $d$-dimensional Hilbert space ${\cal H}$; then all density matrices in this basis are “incoherent”, i.e. those of the form $\tilde{\delta}=\sum_{i=0}^{d-1}\delta_{i}|i\rangle\!\langle i|$. We denote by $\Delta\subset{\cal S}({\cal H})$ the set of such incoherent quantum states.

The definition of coherence monotones requires the identification of operations that are incoherent. These map the set of incoherent states to itself. More precisely, such a completely positive and trace preserving (cptp) map is specified by a set of Kraus operators $\{K_{\alpha}\}$ satisfying $\sum_{\alpha}K_{\alpha}^{\dagger}K_{\alpha}={\openone}$ and $K_{\alpha}{\Delta}K_{\alpha}^{\dagger}\subset{\Delta}$ for all $\alpha$. A Kraus operator with this property is called incoherent; we call it strictly incoherent if both $K$ and $K^{\dagger}$ are incoherent Winter ; Yadin . We distinguish two classes of incoherent operations (IO):

(i) Incoherent completely positive and trace preserving quantum operations (non-selective maps) $T$, which act as $T(\rho)=\sum_{\alpha}K_{\alpha}\rho K_{\alpha}^{\dagger}$ (note that this formulation implies the loss of information about the measurement outcome);

(ii) quantum operations for which measurement outcomes are retained, given by $\rho_{\alpha}=\frac{1}{p_{\alpha}}K_{\alpha}\rho K_{\alpha}^{\dagger}$ occurring with probability $p_{\alpha}=\operatorname{Tr}K_{\alpha}\rho K_{\alpha}^{\dagger}$. The latter can be modelled as a nonselective operation by explicitly introducing a new register to hold the (incoherent) measurement result:

[TABLE]

Here, we have made use of the convention that when composing systems, the incoherent states in the tensor product space are precisely the tensor products of incoherent states and their probabilistic mixtures (convex combinations) Aaberg . An operation is called strictly incoherent (SIO), if it can be written with strictly incoherent Kraus operators.

We define also the maximally coherent state on a $d$-dimensional system by $|\Psi_{d}\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i\rangle$, from which every state in dimension $d$ can be prepared Baumgratz ; Winter ; Du1 . Note that the definition of maximally coherent state is independent of a specific measure for the coherence Baumgratz ; Du2 .

A list of desirable conditions for any coherence measure, i.e. a functional from states to non-negative real numbers, was also presented Baumgratz :

$C(\rho)=0$ for all $\rho\in\Delta$; 2. 2.

Monotonicity under non-selective incoherent maps: $C(\rho)\geq C(T(\rho))$; 3. 3.

(Strong) monotonicity under selective incoherent maps: $C(\rho)\geq\sum_{\alpha}p_{\alpha}C(\rho_{\alpha})$; 4. 4.

Convexity: $\sum_{i}p_{i}C(\rho_{i})\geq C(\sum_{i}p_{i}\rho_{i})$.

The first two are definitely required to speak of a coherence measure, the third is sometimes demanded axiomatically, but often is really more of a convenience, and convexity should be thought of as nice if present but not absolutely necessary.

Among the most important examples are the following three measures: For pure states $\varphi=|\varphi\rangle\!\langle\varphi|$, the entropy of coherence is defined as

[TABLE]

where $\Delta$ is the dephasing (i.e. coherence-destroying) map $\Delta(\rho)=\sum_{i}|i\rangle\!\langle i|\rho|i\rangle\!\langle i|$; for mixed states, it is extended by the convex hull construction to the coherence of formation Aaberg ; Baumgratz

[TABLE]

Finally, the relative entropy of coherence Aaberg ; Baumgratz

[TABLE]

with the relative entropy $D(\rho\|\sigma)=\operatorname{Tr}\rho(\log\rho-\log\sigma)$. Both $C_{r}$ and $C_{f}$ satisfy all properties 1 through 4 above.

In the present paper, we expand our view from states as coherent resources to operations, showing how to extract pure state coherence from a given operation (section II), and how to implement operations using coherent states as a resource (section III). We briefly discuss the case of qubit unitaries as an example (section IV), and conclude in section V, where we observe that in operations, there is bound coherence, something that doesn’t exist for states. Most generally, we propose a definition for the rate of conversion between two channels using only incoherent operations, and present many open questions about this concept.

II Coherence generating capacity and coherence power of transformations

The free operations in our resource theory are the incoherent ones (IO), which means that in some sense, any other CPTP map represents a resource. How to measure it? Or better: How to assess its resource character? In the present section, we are focusing on how much pure state coherence can be created asymptotically, using a given operation $T\mathrel{\mathop{\mathchar 58\relax}}A\longrightarrow B$ a large number of times, when incoherent operations are for free.

The most general protocol to generate coherence must use the resource $T$ and incoherent operations according to some predetermined algorithm, in some order. We may assume that the channels $T$ are invoked one at a time; and we can integrate all incoherent operations in between one use of $T$ and the next into one incoherent operation, since IO is closed under composition. Thus, a mathematical description of the most general protocol is the following: One starts by preparing an incoherent state $\rho_{0}$ on $A\otimes A_{0}$, then lets act $T$, followed by an incoherent transformation $\mathcal{I}_{1}\mathrel{\mathop{\mathchar 58\relax}}B\otimes A_{0}\longrightarrow A\otimes A_{1}$, resulting in the state

[TABLE]

Iterating, given the state $\rho_{t}$ on $A\otimes A_{t}$ obtained after the action of $t$ realizations of $T$ and suitable incoherent operations, we let $T$ act and the incoherent transformation $\mathcal{I}_{t+1}\mathrel{\mathop{\mathchar 58\relax}}B\otimes A_{t}\longrightarrow A\otimes A_{t+1}$, resulting in the state

[TABLE]

At the end of $n$ iterations, we have a state $\rho_{n}$ on $A\otimes A_{n}$, and we call the above procedure a coherence generation protocol of rate $R$ and error $\epsilon$, if $|A_{n}|=2^{nR}$ and the reduced state $\rho_{n}^{A_{n}}=\operatorname{Tr}_{A}\rho_{n}$ has high fidelity with the maximally coherent state,

[TABLE]

The maximum number $R$ such that there exist coherence generating protocols for all $n$, with error going to zero and rates converging to $R$, is called the coherence generating capacity of $T$, and denoted $C_{\text{gen}}(T)$.

Theorem 1

For a general CPTP map $T\mathrel{\mathop{\mathchar 58\relax}}A\longrightarrow B$,

[TABLE]

where the supremum over all auxiliary systems $C$ and pure states $|\varphi\rangle\in A\otimes C$. Furthermore,

[TABLE]

where now the supremum is over mixed states $\rho$ on $A\otimes C$.

If $T$ is an isometry, i.e. $T(\rho)=V\rho V^{\dagger}$ for an isometry $V\mathrel{\mathop{\mathchar 58\relax}}A\hookrightarrow B$, the lower bound is an equality, and can be simplified:

[TABLE]

This result, the main one of the present section, should be compared to the formula, similar in spirit, for the entangling power of a bipartite unitary Bennett ; Collins . Furthermore, the above formulas for the coherence generating capacity are related to the coherence power (w.r.t.  the relative entropy measure)

[TABLE]

investigated by García Díaz et al. Garcia-Diaz and Bu et al. coherence-power . Let us also introduce the same maximization restricted to pure input states,

[TABLE]

Note that the only difference to our formulas is that we allow an ancilla system $C$ of arbitrary dimension. If we consider, for a general CPTP map $T$, the extension $T\otimes{\operatorname{id}}_{k}$ and

[TABLE]

then we have

[TABLE]

and in general

[TABLE]

Proof.

We start with the lower bound, Eq. (4): For a given ancilla $C$ and $|\varphi\rangle\in A\otimes C$, let R=C_{r}\bigl{(}(T\otimes{\operatorname{id}})\varphi\bigr{)}-C(\varphi). For any $\epsilon,\delta>0$, we can choose, by the results of Winter , a sufficiently large $n$ such that

[TABLE]

with $\rho=(T\otimes{\operatorname{id}})\varphi$, and where $\approx$ refers to approximation of the target state up to $\epsilon$ in trace norm. We only have to prove something when $R>0$, which can only arise if $T$ is not incoherent, meaning that there exists an initial state $|0\rangle$ mapped to a coherent resource $\sigma=T(|0\rangle\!\langle 0|)$, i.e. $C_{r}(\sigma)>0$. In the following, assume $R>2\delta$. Now, we may assume that $n$ is large enough so that with $R_{0}=\frac{C(\varphi)+\delta}{C_{r}(\sigma)}+\delta$,

[TABLE]

The protocol consists of the following steps:

0: Use $\lfloor nR_{0}\rfloor$ instances of $T$ to create as many copies of $\sigma$, and convert them into $\Psi_{2}^{\otimes\lfloor nC(\varphi)+n\delta\rfloor}$ (up to trace norm $\epsilon$).

1-k (repeat): Convert first $\lfloor nC(\varphi)+n\delta\rfloor$ of the already created copies of $\Psi_{2}$ into $n$ copies of $\varphi$; then apply $T$ to each of them to obtain $\rho=(T\otimes{\operatorname{id}})\varphi$; and convert the $n$ copies of $\rho$ to $\Psi_{2}^{\otimes\lceil nC_{r}(\rho)-n\delta\rceil}$, incurring an error of $2\epsilon$ in trace norm in each repetition.

At the end, we have $(k-1)n(R-2\delta)+nC(\varphi)$ copies of $\Psi_{2}$, up to trace distance $O(k^{2}\epsilon)$, using the channel a total of $kn+nR_{0}$ times, i.e. the rate is $\geq(R-2\delta)\frac{k-1}{k+R_{0}}$, which can be made arbitrarily close to $R$ by choosing $\delta$ small enough and $k$ large enough (which in turn can be effected by sufficiently small $\epsilon$).

For the upper bound, Eq. (5), consider a generic protocol using the channel $n$ times, starting from $\rho_{0}$ (incoherent) and generating $\rho_{1},\ldots,\rho_{n}$ step by step along the way, such that $\rho_{n}$ has fidelity $\geq 1-\epsilon$ with $\Psi_{2}^{\otimes nR}$. By the asymptotic continuity of $C_{r}$ (Winter, , Lemma 12), $C_{r}(\rho_{n})\geq nR-2\delta n-2$, with $\delta=\sqrt{\epsilon(2-\epsilon)}$, so we can bound

[TABLE]

where we have used the fact that $\rho_{0}$ is incoherent and that \rho_{t+1}=\mathcal{I}_{t+1}\bigl{(}(T\otimes{\operatorname{id}})\rho_{t}\bigr{)}, with an incoherent operation $\mathcal{I}_{t+1}$, which can only decrease relative entropy of coherence. However, each term on the right hand sum is of the form C_{r}\bigl{(}(T\otimes{\operatorname{id}})\rho\bigr{)}-C_{r}(\rho) for a suitable ancilla $C$ and a state $\rho$ on $A\otimes C$. Thus, dividing by $n$ and letting $n\rightarrow\infty$, $\epsilon\rightarrow 0$ shows that R\leq\sup_{\rho\text{ on }A\otimes C}C_{r}\bigl{(}(T\otimes{\operatorname{id}})\rho\bigr{)}-C_{r}(\rho).

For an isometric channel $T(\rho)=V\rho V^{\dagger}$, note that the initial state $\rho_{0}$ in a general protocol is without loss of generality pure, and that $T$ maps pure states to pure states. The incoherent operations $\mathcal{I}_{t}$ map pure states to ensembles of pure states, so that following the same converse reasoning as above, we end up upper bounding $R$ by an average of expressions C_{r}\bigl{(}(T\otimes{\operatorname{id}})\rho\bigr{)}-C_{r}(\rho), with pure states $\rho$, i.e. Eq. (1), since we also have C_{\text{gen}}(T)\geq C\bigl{(}(T\otimes{\operatorname{id}})\varphi\bigr{)}-C(\varphi) from the other direction. The fact that no ancilla system is needed, is an elementary calculation. Indeed, for a pure state $|\varphi\rangle\in A\otimes C$,

[TABLE]

and we are done.     \sqcap$$\sqcup

**Remark ** We do not know, at this point, whether the suprema over the ancillary systems in the upper and lower bounds in Eq. (11) are necessary in general, i.e. it might be that $P_{r}(T)=P_{r}^{(k)}$ and/or $\widetilde{P}_{r}(T)=\widetilde{P}_{r}^{(k)}$; note that the latter is the case for unitaries, even though it seems unlikely in general, cf. Bennett ; Collins . We do know, however, that $P_{r}$ is convex and non-increasing under composition of the channel with incoherent operations (Garcia-Diaz, , Cor. 1 and  2).

It should be appreciated that even the calculation of $P_{r}(T)$ appears to be a hard problem. The investigation of further questions, such as the additivity of $P_{r}$, $\widetilde{P}_{r}$ or $C_{\text{gen}}$, may depend on making progress on that problem.

**Remark ** The same reasoning as in the proof of Theorem 1, replacing $C_{r}$ with $C_{f}$, shows that

[TABLE]

with the coherence of formation power, given by P_{f}(T)\mathrel{\mathop{\mathchar 58\relax}}=\max_{\rho}C_{f}\bigl{(}T(\rho)\bigr{)}-C_{f}(\rho).

Despite the fact that $C_{f}(\rho)\geq C_{r}(\rho)$, since the upper bound is given by a difference of two coherence measures, it might be that for certain channels, the bound (12) is better than (5), and vice versa for others.

Since the supremum over $k$ of the coherence powers of $T\otimes{\operatorname{id}}_{k}$ play such an important role in our bounds, we introduce notation for them,

[TABLE]

and call them the complete coherence powers with respect to relative entropy of coherence and coherence of formation, respectively. For these parameters, and the coherence generation capacity of isometries, we note the following additivity formulas.

Proposition 2

For any CPTP maps $T_{1}\mathrel{\mathop{\mathchar 58\relax}}A_{1}\longrightarrow B_{1}$ and $T_{2}\mathrel{\mathop{\mathchar 58\relax}}A_{2}\longrightarrow B_{2}$,

[TABLE]

Furthermore, for isometries $T_{i}(\rho)=V_{i}\rho V_{i}^{\dagger}$,

[TABLE]

and

[TABLE]

In other words, the coherence generating capacity of isometries is additive.

Proof.

For $X\in\{r,f\}$, any ancilla system $C$ and any state $\rho$,

[TABLE]

where we have introduced the state $\sigma=(T_{1}\otimes{\operatorname{id}}_{B_{2}}\otimes{\operatorname{id}}_{C})\rho$. By taking the supremum of the left hand side over all ancillas $C$ and all states $\rho$, we obtain $\mathbb{P}_{X}(T_{1}\otimes T_{2})\leq\mathbb{P}_{X}(T_{1})+\mathbb{P}_{X}(T_{2})$; since the opposite inequality is trivial, using tensor product ancillas and tensor product input states, and employing the additivity of $C_{r}$ and $C_{f}$ Winter , we have proved the equality.

In the case of isometries, Eq. (1) in Theorem 1 already shows $\widetilde{\mathbb{P}}_{r}(T_{i})=\widetilde{P}_{r}(T_{i})=C_{\text{gen}}(T_{i})$. For the tensor product, we again have trivially $\widetilde{P}_{r}(T_{1}\otimes T_{2})\geq\widetilde{P}_{r}(T_{1})+\widetilde{P}_{r}(T_{2})$, by using tensor product input states. To get the opposite inequality, we proceed as above: for any pure state $|\varphi\rangle\in A_{1}\otimes A_{2}$,

[TABLE]

with the (pure) state $\psi=(T_{1}\otimes{\operatorname{id}}_{B_{2}})\varphi$, and we are done.     \sqcap$$\sqcup

III Implementation of channels:

Coherence cost of simulation

We have seen that a CPTP map can be a resource for coherence because one can use it to generate coherence from scratch, in the form of maximally coherent qubit states. True to the resource paradigm, we have to ask immediately the opposite question: Is it possible to create the resource using pure coherent states and only incoherent operations? Here we show that the answer is generally yes, and we define the asymptotic coherence cost $C_{\text{sim}}(T)$ as the minimum rate of pure state coherence necessary to implement many independent instances of $T$ using only incoherent operations otherwise.

We start by recalling the implementation of an arbitrary unitary operation $U=\sum_{ij=0}^{d-1}U_{ij}|i\rangle\!\langle j|$ by means of an incoherent operation with Kraus operators $\{K_{\alpha}\}$, and using the maximally coherent state $|\Psi_{d}\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i\rangle$ as resource (Chitambar, , Lemma 2). Since it is important for us that the implementation is in fact by means of a strictly incoherent operation, and for self-containedness, we include the proof.

Proposition 3** **(Chitambar/Hsieh Chitambar )

Consuming one copy of $\Psi_{d}$ as a resource, any unitary $U=\sum_{i,j=0}^{d-1}U_{ij}|i\rangle\!\langle j|$ acting on ${{\mathbb{C}}}^{d}$ can be simulated using strictly incoherent operations.

Proof.

We essentially follow Chitambar ; see also Baumgratz for the qubit case. Let the Kraus operators have the following form:

[TABLE]

It is easy to see that these Kraus operators satisfy $K_{\alpha}\Delta K_{\alpha}^{\dagger}\subset\Delta$, and that they are indeed strictly incoherent, for all $\alpha=0,1,\dots,d-1$. Furthermore, with the notation $q=i+\alpha\!\mod\!d$,

[TABLE]

Now, let $|\phi\rangle=\sum_{k=0}^{d-1}\phi_{k}|k\rangle$, then

[TABLE]

Thus, under this incoherent operation, and tracing out the ancilla afterwards, the system will be in the desired state $U|\phi\rangle\!\langle\phi|U^{\dagger}$ with certainty.     \sqcap$$\sqcup

Now we pass to the general case of CPTP maps, which extends the above result for unitaries, with two different protocols.

Theorem 4

Any CPTP map $T\mathrel{\mathop{\mathchar 58\relax}}A\longrightarrow B$ can be implemented by incoherent operations, using a maximally coherent resource state $\Psi_{d}$, where $d=|B|$.

Proof.

Let $T(\rho)=\sum_{\alpha}K_{\alpha}\rho K_{\alpha}^{\dagger}$ be a Kraus decomposition of $T$, with Kraus operators $K_{\alpha}\mathrel{\mathop{\mathchar 58\relax}}A\longrightarrow B$. The idea of the simulation is to use teleportation of the output of $T$, which involves a maximally entangled state $\Phi_{D}$ on $B^{\prime}\otimes B^{\prime\prime}$, a Bell-measurement on system $B\otimes B^{\prime}$ with outcomes $jk\in\{0,1,\dots,d-1\}^{2}$, and unitaries $U_{jk}$ on $B^{\prime\prime}$. The unitaries $U_{jk}$ can be written as $U_{jk}=Z^{j}X^{k}$, with the phase and cyclic shift unitaries

[TABLE]

where $\omega=e^{\frac{2\pi i}{d}}$ is the $d$-th root of unity. This scheme can be reduced to a destructive (hence incoherent) POVM on $A\otimes B^{\prime}$ with outcomes $jk\alpha$, followed by the application of the incoherent(!) $U_{jk}$. In detail, the probability of getting outcome $jk$ is

[TABLE]

where $\sigma$ is a state on $A\otimes B^{\prime}$ and the

[TABLE]

are the Bell states. We can define the POVM elements $M_{jk\alpha}=(K_{\alpha}\otimes{\openone})^{\dagger}\Phi^{(jk)}(K_{\alpha}\otimes{\openone})$, so that

[TABLE]

with $M_{jk}=\sum_{\alpha}M_{jk\alpha}$. This leads to a new equivalent scheme in which, given a state $\rho$ on $A$ and a maximally entangled state on $B^{\prime}\otimes B^{\prime\prime}$, we can apply the measurement $M_{jk}$ on $A\otimes B^{\prime}$ with outcomes $jk$, and unitaries $U_{jk}$ acting on $B^{\prime\prime}$. Formally, let us define the Kraus operators of the protocol by letting

[TABLE]

It can be checked readily that they satisfy the normalization condition

[TABLE]

Applying the Kraus operators $L_{jk\alpha}$ on all the system we get

[TABLE]

Hence, $\sum_{jk\alpha}L_{jk\alpha}(\rho\otimes\Phi_{d})L_{jk\alpha}^{\dagger}=\sum_{\alpha}K_{\alpha}\rho K_{\alpha}^{\dagger}=T(\rho)$, and the proof is complete.     \sqcap$$\sqcup

Theorem 5

Any CPTP map $T\mathrel{\mathop{\mathchar 58\relax}}A\longrightarrow B$ can be implemented by strictly incoherent operations and a maximally coherent state $\Psi_{d}$, where $d\leq|A||B|$.

Proof.

The channel $T$ is, first of all, a convex combination of extremal CPTP maps $T_{\lambda}$, each of which has at most $|A|$ Kraus operators Choi : $T=\sum_{\lambda}p_{\lambda}T_{\lambda}$. Clearly, we only have to prove the claim for the $T_{\lambda}$. Because of the bound on the Kraus operators, each $T_{\lambda}$ has a unitary dilation $U_{\lambda}\mathrel{\mathop{\mathchar 58\relax}}A\otimes B\rightarrow B\otimes A$, such that $T_{\lambda}(\rho)=\operatorname{Tr}_{A}U_{\lambda}(\rho\otimes|0\rangle\!\langle 0|_{B})U_{\lambda}^{\dagger}$, with a fixed incoherent state $|0\rangle\in B$.

As the $U_{\lambda}$ act on a space of dimension $d=|A||B|$, we can invoke the simulation according to Proposition 3.     \sqcap$$\sqcup

**Remark ** Comparing Theorems 4 and 5, we note that the latter always consumes more resources, but it is guaranteed to be implemented by strictly incoherent operations, a much narrower class than the incoherent operations.

We leave it as an open question whether the resource consumption of Theorem 4 can be achieved by strictly incoherent operations, or whether there is a performance gap between incoherent and strictly incoherent operations.

Note that these two classes, incoherent operations and strictly incoherent operations, are distinct as sets of CPTP maps, although it is known that they induce the same possible state transformations of a given state into a target state for qubits Chitambar-2 and for pure states in arbitrary dimension Zhu (correcting the earlier erroneous proof of the claim by Du1 ; the SIO part of the pure state transformations is due to Winter ). However, for the distillation of pure coherence at rate $C_{r}(\rho)$ Winter , IO are needed and it remains unknown whether SIO can attain the same rate. Crucially, any destructive measurement, i.e. any POVM followed by an incoherent state preparation, is IO, but the only measurements allowed under SIO are of diagonal, i.e. incoherent, observables.

The results so far are about the resources required for the exact implementation of a single instance of a channel, in the worst case. It is intuitively clear that some channels are easier to implement in the sense that fewer resources are needed; e.g. for the identity or any incoherent channel, no coherent resource is required.

In the spirit of the previous section, we are interested in the minimum resources required to implement many independent instances of $T$.

Definition 6

An $n$-block incoherent simulation of a channel $T\mathrel{\mathop{\mathchar 58\relax}}A\longrightarrow B$ with error $\epsilon$ and coherent resource $\Psi_{d}$ (on space $D$) is an incoherent operation ${\cal I}\mathrel{\mathop{\mathchar 58\relax}}A^{n}\otimes D\longrightarrow B^{n}$, such that $T^{\prime}(\rho)\mathrel{\mathop{\mathchar 58\relax}}={\cal I}(\rho\otimes\Psi_{d})$ satisfies

[TABLE]

Here, $C$ is an arbitrary ancilla system; the error criterion of the simulation is known as diamond norm diamond or completely bounded trace norm Paulsen , see also Watrous .

The rate of the simulation is $\frac{1}{n}\log d$, and the simulation cost of $T$, denoted $C_{\text{sim}}(T)$, is the smallest $R$ such that there exist $n$-block incoherent simulations with error going to [math] and rate going to $R$ as $n\rightarrow\infty$.

The best general bounds we have on the simulation cost are contained in the following theorem.

Theorem 7

For any CPTP map $T\mathrel{\mathop{\mathchar 58\relax}}A\longrightarrow B$,

[TABLE]

Furthermore,

[TABLE]

where we recall the definitions of the relative entropy coherence power, P_{r}(T)=\max_{\rho}C_{r}\bigl{(}T(\rho)\bigr{)}-C_{r}(\rho), and of the coherence of formation power, P_{f}(T)=\max_{\rho}C_{f}\bigl{(}T(\rho)\bigr{)}-C_{f}(\rho).

Proof.

We start with Eq. (17): The upper bound is a direct consequence of Theorem 4. The lower bound follows from the fact that $T$ is implemented using maximally coherent states at rate $R=C_{\text{sim}}(T)$ and incoherent operations. Generation of entanglement on the other hand uses $T$ and some more incoherent operations. Since incoherent operations cannot increase the amount of entanglement, the overall process of simulation and generation cannot result in a rate of coherence of more than $R$.

Regarding Eq. (18), the idea is that for $\epsilon>0$ and $n$ large enough, since the simulation implements a CPTP map $T^{\prime}$ that is within diamond norm $\epsilon$ from $T^{\otimes n}$, using incoherent operations and $\Psi_{2}^{\otimes n(R+\epsilon)}$ as a resource. Applying the simulation to the state $\rho^{\otimes n}$, results in (T^{\prime}\otimes{\operatorname{id}}_{k^{n}})\rho^{\otimes n}\approx\bigl{(}(T\otimes{\operatorname{id}}_{k})\rho\bigr{)}^{\otimes n}, hence we have an overall incoherent operation

[TABLE]

By monotonicity of $C_{X}$ ($X\in\{r,f\}$) under IO, and $C_{X}(\Psi_{2})=1$, this means

[TABLE]

where we have used additivity of $C_{r}$ and $C_{f}$ Winter . Since this holds for all $\rho$, we obtain

[TABLE]

invoking in the second line Lemma 8 below, with $\kappa_{r}=4\log|B|$ and $\kappa_{f}=\log|B|+\log k$, and in the third a tensor power test state. Since $\epsilon$ can be made arbitrarily small, and $n$ as well as $k$ arbitrarily large, the claim follows.     \sqcap$$\sqcup

Here we state the technical lemma required in the proof of Theorem 7.

Lemma 8

The relative entropy coherence power and the coherence of formation power are asymptotically continuous with respect to the diamond norm metric on channels. To be precise, for $T_{1},T_{2}\mathrel{\mathop{\mathchar 58\relax}}A\longrightarrow B$ with $\frac{1}{2}\|T_{1}-T_{2}\|_{\diamond}\leq\epsilon$,

[TABLE]

where $g(x)=(1+x)h_{2}\!\left(\!\frac{x}{1+x}\!\right)=(1+x)\log(1+x)-x\log x$.

Proof.

For the first bound, observe

[TABLE]

where in the first line we insert the same variable $\rho^{AC}$ to maximise $P_{r}(T_{j}\otimes{\operatorname{id}}_{k})$ and notice that in that case, the term $C_{r}(\rho)$ cancels; then in the second line, we use the definition of the relative entropy of coherence and in the third we use chain rule $S(BC)=S(B|C)+S(C)$ for the entropy, allowing us to cancel matching $S(C)$ terms; in the fourth line we invoke the triangle inequality, and finally the Alicki-Fannes bound for the conditional entropy AlickiFannes in the form given in (Winter-S-cont, , Lemma 2).

For the second bound, we start very similarly:

[TABLE]

where the last line comes directly from the asymptotic continuity of the coherence of formation (Winter, , Lemma 15). We close the proof expressing our belief that it is possible to prove a version of Eq. (20) where $k$ does not appear on the right hand side.     \sqcap$$\sqcup

**Remark ** As a consequence, while for a channel $T$ that is close to an incoherent operation (in diamond norm), or in fact close to a MIO operation, the coherence generating capacity $C_{\text{gen}}(T)$ is also close to [math], we do not know at the moment whether the same holds for the simulation cost $C_{\text{sim}}(T)$.

IV Qubit unitaries

In this section, we want to have a closer look at qubit unitaries, for which we would like to find the coherence generating capacity and simulation cost.

To start our analysis, we note that a general $2\times 2$-unitary has four real parameters, but we can transform unitaries into each other at no cost by preceding or following them by incoherent unitaries, i.e. combinations of the bit flip $\sigma_{x}$ and diagonal (phase) unitaries $\begin{pmatrix}e^{i\alpha}&0\\ 0&e^{i\beta}\end{pmatrix}$. This implies an equivalence relation among qubit unitaries up to incoherent unitaries. A unique representative of each equivalence class is given by

[TABLE]

where $c=\cos\theta$ and $s=\sin\theta$ and with $0\leq\theta\leq\frac{\pi}{4}$, so that $c\geq s\geq 0$.

One can calculate $C_{\text{gen}}(U(\theta))$, using the formula from Theorem 1.

Clearly, by choosing the test state $\varphi$ to be pure incoherent,

[TABLE]

with $h_{2}(x)=-x\log x-(1-x)\log(1-x)$ the binary entropy. Perhaps surprisingly, however, this is in general not the optimal state (Maria:MSc, , Cor. 5) (see also Garcia-Diaz ), meaning that $\widetilde{P}_{r}(U(\theta))$ is attained at a coherent test state $\varphi$, although no closed form expression seems to be known. In fact, simple manipulations show that we only need to optimise $C\!\left(U(\theta)\varphi U(\theta)^{\dagger}\right)-C(\varphi)$ over states $|\varphi\rangle=U(\alpha)|0\rangle=\cos\alpha|0\rangle+\sin\alpha|1\rangle$, $0\leq\alpha\leq\pi$ (i.e. no phases are necessary). The function to optimise becomes h_{2}\bigl{(}\cos^{2}(\alpha+\theta)\bigr{)}-h_{2}\bigl{(}\cos^{2}\alpha\bigr{)}. Its critical points satisfy the transcendental equation

[TABLE]

which can be solved numerically. Fig. 1 shows that $C_{\text{gen}}(U(\theta))=\widetilde{P}_{r}(U(\theta))>h_{2}(\cos^{2}\theta)$ for across the whole interval, except at the endpoints $\theta=0,\,\frac{\pi}{4}$; in Fig. 2 we plot the optimal $\alpha$ for $U(\theta)$.

On the other hand, regarding the implementation of these unitary channels, all we can say for the moment is that $C_{\text{sim}}(U(\theta))\leq 1$, because we can implement each instance of the qubit unitary using a qubit maximally coherent state $\Psi_{2}$. It is perhaps natural to expect that one could get away with a smaller amount of coherence, but it turns out that with a two-dimensional resource state this is impossible.

Proposition 9

The only qubit coherent resource state $|r\rangle\in{{\mathbb{C}}}^{2}$ that permits the implementation of $U(\theta)$, $0<\theta\leq\frac{\pi}{4}$, is the maximally coherent state.

Furthermore, any two-qubit incoherent operation $\mathcal{I}$ such that $\mathcal{I}(\rho\otimes|r\rangle\!\langle r|)=U(\theta)\rho U(\theta)^{\dagger}\otimes\sigma$ for general $\rho$, is such that the state $\sigma$ left behind in the ancilla is necessarily incoherent.

Proof.

We want to know for which state $|r\rangle=c^{\prime}|0\rangle+s^{\prime}|1\rangle$ the transformation $|\psi\rangle|r\rangle\xrightarrow{IO}(U(\theta)|\psi\rangle)|0\rangle$ is possible, for a general state $|\psi\rangle$. Without loss of generality, the incoherent Kraus operators achieving the transformation have the following general form:

[TABLE]

where $|r^{\perp}\rangle=s^{\prime}|0\rangle-c^{\prime}|1\rangle$ is the vector orthogonal to $|r\rangle$. We now need to find the form of $R$ such that $K$ is incoherent. For that, we impose incoherence of $K$ when tracing out the ancillary part: $\langle 0|^{A}K|0\rangle^{A}=\mathrel{\mathop{\mathchar 58\relax}}T_{0}$ and $\langle 1|^{A}K|1\rangle^{A}=\mathrel{\mathop{\mathchar 58\relax}}T_{1}$, where $T_{0}$ and $T_{1}$ must be $2$-dimensional incoherent operators. We then obtain that $R=s^{\prime}T_{0}-c^{\prime}T_{1}$ and $\lambda U=c^{\prime}T_{0}+s^{\prime}T_{1}$. The latter condition enforces that either $T_{0}\propto\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ and $T_{1}\propto\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ or viceversa; or $T_{0}\propto\begin{pmatrix}1&-1\\ 0&0\end{pmatrix}$ and $T_{1}\propto\begin{pmatrix}0&0\\ 1&1\end{pmatrix}$ or viceversa. From these possibilities, we get $4$ possible $R$ matrices, which define $4$ different Kraus operators $K_{i}$ defined, according to Eq. (24), by $R_{i}$ matrices as follows:

[TABLE]

and the general incoherent Kraus operator is $K=\lambda_{i}K_{i}$ ($i=1,2,3,4$). Finally, after imposing $\sum_{i}|\lambda_{i}|^{2}K^{\dagger}_{i}K_{i}={\openone}$, we obtain the following conditions on $R_{i}$ and $\lambda_{i}$:

[TABLE]

It can be verified that these conditions are only fulfilled when $|c^{\prime}|=|s^{\prime}|=\frac{1}{\sqrt{2}}$, i.e. $|r\rangle$ is maximally coherent.

If the incoherent implementation of the unitary, instead of mapping two qubits (input and resource state) to one (output), but to two (output plus residual resource), i.e. $\mathcal{I}(\rho\otimes|r\rangle\!\langle r|)=U(\theta)\rho U(\theta)^{\dagger}\otimes\sigma$, then first of all $\sigma$ has to be the same irrespective of the state $\rho$. Otherwise we would be able, by measuring $\sigma$, to learn some information about $\rho$ without disturbing it. Now consider a pure incoherent input state $\rho=|0\rangle\!\langle 0|$, and note that the desired output state $U(\theta)|\psi\rangle$ has nontrivial coherence. But now observe that $\mathcal{I}$ takes in a state of coherence rank $2$ Winter , and produces a product of a pure state of coherence rank $2$ with another state. Since the coherence rank cannot increase, even under individual Kraus operators Winter , it must be the case that $\sigma$ is incoherent.     \sqcap$$\sqcup

This result might suggest an irreversibility between simulation and coherence generation for these unitaries, but we point out that it does not preclude the possibility of simulations using a higher rank, yet less coherent, resource state (cf. stahlke , where the analogue is demonstrated for LOCC implementation of bipartite unitaries using entangled resources); or of a simulation of many instances of $U(\theta)$ at a cost lower than $1$ per unitary.

V Conclusion and Outlook

We have shown that using a maximally coherent state and strictly incoherent operations, we can implement any unitary on a system (using $\Psi$ for a $d$-dimensional unitary), and via the Stinespring dilation any CPTP map (using $\Psi_{d^{2}}$ for a channel acting on a $d$-dimensional system). By teleportation, we prove that for incoherent operations and a $d$-dimensional maximally coherent state any noisy channel can be implemented.

Vice versa, every incoherent operation gives rise to some capacity of generating pure coherence by using it asymptotically many times, and we have given capacity bounds in general, and a single-letter formula for the case of unitaries. We also found the additive upper bounds $\mathbb{P}_{r}(T)$ and $\mathbb{P}_{f}(T)$ on the coherence generation capacity $C_{\text{gen}}(T)$, even though we do not know whether these numbers are efficiently computable due to the presence of the extension $\otimes{\operatorname{id}}_{k}$, nor whether these extensions are even necessary. It is open at the moment whether the coherence generation capacity $C_{\text{gen}}(T)$ itself is additive for general tensor product channels, and likewise the lower bound given in Theorem 1, $\widetilde{\mathbb{P}}_{r}(T)=\sup_{k}\widetilde{P}_{r}(T\otimes{\operatorname{id}}_{k})$; at least for isometric channels they are.

The coherence generating capacity is never larger than the simulation cost, but in general these two numbers will be different. As an extreme case, consider any CPTP map $T$ that is not incoherent, but is a so-called “maximally incoherent operation” (MIO, cf. Streltsov-rev ), meaning that $T(\rho)\in\Delta$ for all $\rho\in\Delta$. This class was considered in Brandao3 , and it makes coherence theory asymptotically reversible Winter , all states $\rho$ being equivalent to $C_{r}(\rho)$ maximally coherent qubit states. Such maps exist even in qubits, cf. Chitambar-erratum correcting (Chitambar-2, , Thm. 21). We expect the simulation cost of any such $T$ to be positive, $C_{\text{sim}}(T)>0$. At the same time, $C_{\text{gen}}(T)=0$ by Theorem 1, because the relative entropy of coherence is a MIO monotone, and the tensor product of MIO transformations is MIO. To obtain an example, we can take any MIO channel for which there exists a state $\rho$ such that C_{f}\bigl{(}T(\rho)\bigr{)}>C_{f}(\rho), since by Theorem 7, $C_{\text{sim}}(T)$ is lower bounded by the difference of the two. (As an aside, we note that this cannot be realised in qubits, because for qubits, any state transformation possible under MIO is already possible under IO, for which $C_{f}$ is a monotone Chitambar-2 ; Hu .) Concretely, consider the following states on a $2d$-dimensional system $A$, which could be called coherent flower states, since their corresponding maximally correlated states (cf. Adesso-corr ; Winter ) are the well-known flower states HHHO . We write them as $2\times 2$-block matrices,

[TABLE]

where $U$ is the $d$-dimensional discrete Fourier transform matrix. Via the correspondence between $C_{r}$ and the relative entropy of entanglement, and between $C_{f}$ and the entanglement of formation, respectively, of the associated state, we know that $C_{r}(\rho_{d})=1$ and $C_{f}(\rho_{d})=1+\frac{1}{2}\log d$ HHHO . By the results of Brandao3 , however, for every $\epsilon>0$ and sufficiently large $n$, there exists a MIO transformation $T^{(n)}\mathrel{\mathop{\mathchar 58\relax}}({{\mathbb{C}}}^{2})^{\otimes n(1+\epsilon)}\longrightarrow A^{n}$ with

[TABLE]

By the asymptotic continuity of $C_{f}$ Winter , we have $C_{f}(\rho^{(n)})\geq n\left(1+\frac{1}{2}\log d\right)-n\epsilon\log(2d)-g(\epsilon)$, while of course the preimage $\rho_{0}=\Psi_{2}^{\otimes n(1+\epsilon)}$ has $C_{f}(\rho_{0})\leq n+n\epsilon$, so for $\epsilon$ small enough and $n$ large enough, we have a gap:

[TABLE]

invoking Theorem 7. Thus, while states in the resource theory of coherence cannot exhibit bound resource — indeed, it was observed in Winter that vanishing distillable coherence, $C_{r}(\rho)=0$, implies vanishing coherence cost $C_{f}(\rho)=0$ —, operations can have bound coherence, and the gap can be large on the scale of the logarithm of the channel dimension. We observe that this effect can only occur for maximally incoherent operations, which are precisely the ones with $C_{\text{gen}}(T)=0$. We argued already that MIO channels have zero coherence generating capacity; in the other direction, if $T$ is not MIO, it means that there exists an incoherent state $\rho$ such that $T(\rho)$ has coherence, and this can be distilled at rate $C_{r}(T(\rho))$ Winter . As MIO are closed under forming tensor products, it also follows that $C_{\text{gen}}$ does not exhibit superactivation.

This example and the subsequent considerations raise the question of how our theory would change if we considered all MIO transformations as free operations. By definition, the above example – by virtue of being MIO – has zero MIO-simulation cost, so there is no bound coherence any more. It may still be the case that there is in general a difference between MIO-simulation cost and MIO-coherence generating capacity, but deciding this possibility is beyond the scope of the present investigation. We only note that Theorem 1 gives us a single-letter formula for the MIO-coherence generating capacity, namely

[TABLE]

the complete relative entropy coherence power of $T$. The supremum is over all auxiliary systems $C$ and mixed states $\rho$ on $A\otimes C$. Indeed, the upper bound of Eq. (5) still applies, because $C_{r}$ is MIO monotone, which is all we needed in the proof of Theorem 1. For the lower bound, that it is attainable follows the same idea as the proof of Eq. (4), only that we can now use an arbitrary mixed state $\rho$ in the argument, since its coherence cost under MIO equals $C_{r}(\rho)$ Brandao3 ; Winter .

One of the most exciting possibilities presented by the point of view of channels as coherence resources is the transformation of channels into channels by means of preceding and post-processing a given one by incoherent operations to obtain a different one. The fundamental question one can ask here is how efficiently, i.e. at which rate $R(T_{1}\!\rightarrow\!T_{2})$ one can transform asymptotically many instances of $T_{1}$ into instances of $T_{2}$, with asymptotically vanishing diamond norm error. To make nontrivial statements about these rates, one would need to extend some of the various coherence monotones that have been studied for states to CPTP maps.

Acknowledgements.

The present project was started during two visits of KBD to the Quantum Information Group at the Universitat Autònoma de Barcelona. The authors thank Ludovico Lami, Swapan Rana, Alexander Streltsov, Michalis Skotiniotis, and Mark Wilde for interesting discussions and comments on the coherence of channels. MGD is supported by a doctoral studies fellowship of the Fundación “la Caixa”. AW is or was supported by the European Commission (STREP “RAQUEL”) and the ERC (Advanced Grant “IRQUAT”). The authors acknowledge furthermore funding by the Spanish MINECO (grants FIS2013-40627-P and FIS2016-86681-P), with the support of FEDER funds, and by the Generalitat de Catalunya CIRIT, project 2014-SGR-966.