Dynamical Resources

Physical systems and their corresponding Hilbert spaces will be denoted by $A$, $B$, $C$, etc, where we will use the notation $AB$ to mean $A\otimes B$. Dimensions will be denoted with vertical bars; e.g. the dimension of system $A$ will be denoted by $\left|A\right|$. The tilde symbol will be reserved to indicate a replica of a system. For example, $\widetilde{A}$ denotes a replica of $A$, i.e. $\left|A\right|=\left|\widetilde{A}\right|$. Density matrices acting on Hilbert spaces will be denoted by lowercase Greek letter $\rho$, $\sigma$, etc, with one exception for the maximally mixed state (i.e. the uniform state), which will be denoted by $u_{A}:=\frac{1}{\left|A\right|}I_{A}$.

The set of all bounded operators acting on system $A$ is denoted by $\mathfrak{B}\left(A\right)$, the set of all Hermitian matrices acting on $A$ by $\mathrm{Herm}\left(A\right)$, and the set of all density matrices acting on system $A$ by $\mathfrak{D}\left(A\right)$. Note that $\mathfrak{D}\left(A\right)\subset\mathrm{Herm}\left(A\right)\subset\mathfrak{B}\left(A\right)$. We use the calligraphic letters $\mathcal{D}$, $\mathcal{E}$, $\mathcal{F}$, etc. to denote quantum channels, reserving $\mathcal{V}$ to represent an isometry map. The identity map on a system $A$ will be denoted by $\mathsf{id}_{A}$. The set of all linear maps from $\mathfrak{B}\left(A\right)$ to $\mathfrak{B}\left(B\right)$ is denoted by $\mathfrak{L}\left(A\to B\right)$, the set of all completely positive (CP) maps by $\mathrm{CP}\left(A\to B\right)$, and the set of quantum channels by $\mathrm{CPTP}\left(A\to B\right)$. Note that $\mathrm{CPTP}\left(A\to B\right)\subset\mathrm{CP}\left(A\to B\right)\subset\mathfrak{L}\left(A\to B\right)$. $\mathrm{Herm}\left(A\to B\right)$ will denote the real vector space of all Hermitian-preserving maps in $\mathfrak{L}\left(A\to B\right)$. We will write $\mathcal{N}\geq 0$ to mean that the map $\mathcal{N}\in\mathrm{Herm}\left(A\to B\right)$ is completely positive.

Since in this paper we focus on dynamical resources in the form of quantum channels, unless otherwise specified, it will be convenient to associate two subsystems $A_{0}$ and $A_{1}$ with every physical system $A$, referring, respectively, to the input and output of the resource. Hence, any physical system will be comprised of two subsystems $A=\left(A_{0},A_{1}\right)$, even those representing a static resource, in which case we simply have $\left|A_{0}\right|=1$. For simplicity, we will denote a channel with a subscript $A$, e.g. $\mathcal{N}_{A}$, to mean that it is an element of $\mathrm{CPTP}\left(A_{0}\to A_{1}\right)$. Similarly, a bipartite channel in $\mathrm{CPTP}\left(A_{0}B_{0}\to A_{1}B_{1}\right)$ will be denoted by $\mathcal{N}_{AB}$. This notation makes the analogy with bipartite states more transparent.

In this setting, when we consider $A=\left(A_{0},A_{1}\right)$, $B=\left(B_{0},B_{1}\right)$, etc. comprised of input and output subsystems, the symbol $\mathfrak{L}\left(A\to B\right)$ refers to all linear maps from the vector space $\mathfrak{L}\left(A_{0}\to A_{1}\right)$ to the vector space $\mathfrak{L}\left(B_{0}\to B_{1}\right)$. Similarly, $\mathrm{Herm}\left(A\to B\right)\subset\mathfrak{L}\left(A\to B\right)$ is a real vector space consisting of all the linear maps that take elements in $\mathrm{Herm}\left(A_{0}\to A_{1}\right)$ to elements in $\mathrm{Herm}\left(B_{0}\to B_{1}\right)$. In other terms, maps in $\mathrm{Herm}\left(A\to B\right)$ take Hermitian-preserving maps to Hermitian-preserving maps. Linear maps in $\mathfrak{L}\left(A\to B\right)$ and $\mathrm{Herm}\left(A\to B\right)$ will be called supermaps, and will be denoted by capital Greek letters $\Theta$, $\Upsilon$, $\Omega$, etc. The identity supermap in $\mathfrak{L}\left(A\to A\right)$ will be denoted by $\mathbbm{1}_{A}$.

We will use square brackets to denote the action of a supermap $\Theta_{A\to B}\in\mathfrak{L}\left(A\to B\right)$ on a linear map $\mathcal{N}_{A}\in\mathfrak{L}\left(A_{0}\to A_{1}\right)$. For example, $\Theta_{A\to B}\left[\mathcal{N}_{A}\right]$ is a linear map in $\mathfrak{L}\left(B_{0}\to B_{1}\right)$ obtained from the action of the supermap $\Theta$ on the map $\mathcal{N}$. Moreover, for a simpler notation, the identity supermap will not appear explicitly in equations; e.g. $\Theta_{A\to B}\left[\mathcal{N}_{RA}\right]$ will mean $\left(\mathbbm{1}_{R}\otimes\Theta_{A\to B}\right)\left[\mathcal{N}_{RA}\right]$. Instead, the action of linear map (e.g. quantum channel) $\mathcal{N}_{A}\in\mathfrak{L}\left(A_{0}\to A_{1}\right)$ on a matrix $\rho\in\mathfrak{B}\left(A_{0}\right)$ is written with round brackets, i.e. $\mathcal{N}_{A}\left(\rho_{A_{0}}\right)\in\mathfrak{B}\left(A_{1}\right)$.

Finally, we adopt the following convention concerning partial traces: when a system is missing, we take the partial trace over the missing system. This applies to matrices as well as to maps. For example, if $M_{AB}$ is a matrix on $A_{0}A_{1}B_{0}B_{1}$, $M_{AB_{0}}$ denotes the partial trace on the missing system $B_{1}$: $M_{AB_{0}}:=\mathrm{Tr}_{B_{1}}\left[M_{AB}\right]$.

The space $\mathfrak{L}\left(A_{0}\to A_{1}\right)$ is equipped with an inner product given by

$$\left\langle\mathcal{N}_{A},\mathcal{M}_{A}\right\rangle:=\sum_{i,j}\left\langle\mathcal{N}_{A}\left(\left|i\right\rangle\left\langle j\right|_{A_{0}}\right),\mathcal{M}_{A}\left(\left|i\right\rangle\left\langle j\right|_{A_{0}}\right)\right\rangle_{\mathrm{HS}},$$

(1)

where $\left\langle X,Y\right\rangle_{\mathrm{HS}}:=\mathrm{tr}\left[X^{\dagger}Y\right]$ is the Hilbert-Schmidt inner product between matrices $X,Y\in\mathfrak{B}\left(A_{1}\right)$. The inner product above can be expressed in terms of the Choi matrices of $\mathcal{N}$ and $\mathcal{M}$. Denote by $J_{A}^{\mathcal{N}}:=\mathcal{N}_{\widetilde{A}_{0}\to A_{1}}\left(\phi_{A_{0}\widetilde{A}_{0}}^{+}\right)$ the Choi matrix of $\mathcal{N}_{A}$, where $\phi_{A_{0}\widetilde{A}_{0}}^{+}:=\left|\phi^{+}\right\rangle\left\langle\phi^{+}\right|_{A_{0}\widetilde{A}_{0}}$ and $\left|\phi^{+}\right\rangle_{A_{0}\widetilde{A}_{0}}=\sum_{i}\left|ii\right\rangle_{A_{0}\widetilde{A}_{0}}$ is the unnormalized maximally entangled state. With this notation, the inner product between $\mathcal{N}_{A}$ and $\mathcal{M}_{A}$ can be expressed as [43]

$$\left\langle\mathcal{N}_{A},\mathcal{M}_{A}\right\rangle=\left\langle J_{A}^{\mathcal{N}},J_{A}^{\mathcal{M}}\right\rangle_{\mathrm{HS}}=\mathrm{tr}\left[\left(J_{A}^{\mathcal{N}}\right)^{\dagger}J_{A}^{\mathcal{M}}\right].$$

The canonical orthonormal basis (relative to the above inner product) is given by $\left\{\mathcal{E}_{A}^{ijk\ell}\right\}$, where

$$\mathcal{E}_{A}^{ijk\ell}\left(\rho_{A_{0}}\right):=\left\langle i\middle|\rho_{A_{0}}\middle|j\right\rangle\left|k\right\rangle\left\langle\ell\right|_{A_{1}}\quad\forall\rho\in\mathfrak{B}\left(A_{0}\right).$$

The space $\mathfrak{L}\left(A\to B\right)$ with $A=\left(A_{0},A_{1}\right)$ and $B=\left(B_{0},B_{1}\right)$ is also equipped with the following inner product: given $\Theta,\Omega\in\mathfrak{L}(A\to B)$

$$\left\langle\Theta_{A\to B},\Omega_{A\to B}\right\rangle:=\sum_{i,j,k,\ell}\left\langle\Theta_{A\to B}\left[\mathcal{E}_{A}^{ijk\ell}\right],\Omega_{A\to B}\left[\mathcal{E}_{A}^{ijk\ell}\right]\right\rangle,$$

where the inner product on the right-hand side is the inner product between maps as defined in Eq. (1). Similarly to the inner product between maps, the inner product between supermaps can also be expressed in terms of Choi matrices. We define the Choi matrix of a supermap $\Theta\in\mathfrak{L}\left(A\to B\right)$ to be [43]

$$\mathbf{J}_{AB}^{\Theta}:=\sum_{i,j,k,\ell}J_{A}^{\mathcal{E}^{ijk\ell}}\otimes J_{B}^{\Theta\left[\mathcal{E}_{A}^{ijk\ell}\right]}.$$

Then, with this notation, the inner product between two supermaps $\Theta$ and $\Omega$ can be expressed as

$$\left\langle\Theta_{A\to B},\Omega_{A\to B}\right\rangle=\left\langle\mathbf{J}_{AB}^{\Theta},\mathbf{J}_{AB}^{\Omega}\right\rangle_{\mathrm{HS}}=\mathrm{tr}\left[\left(\mathbf{J}_{AB}^{\Theta}\right)^{\dagger}\mathbf{J}_{AB}^{\Omega}\right].$$

The Choi matrix of a supermap $\Theta\in\mathfrak{L}\left(A\to B\right)$ can also be expressed in other three alternative ways [43]. First, from its definition, $\mathbf{J}_{AB}^{\Theta}$ can be expressed as the Choi matrix of the map

$$\mathcal{P}_{AB}^{\Theta}:=\Theta_{\widetilde{A}\to B}\left[\Phi_{A\widetilde{A}}^{+}\right],$$

where the map $\Phi_{A\widetilde{A}}^{+}$ is defined as

$$\Phi_{A\widetilde{A}}^{+}:=\sum_{i,j,k,\ell}\mathcal{E}_{A}^{ijk\ell}\otimes\mathcal{E}_{\widetilde{A}}^{ijk\ell}.$$

A simple calculation shows that $\Phi_{A\widetilde{A}}^{+}$ is completely positive, and acts on $\rho\in\mathfrak{B}\left(A_{0}\widetilde{A}_{0}\right)$ as

$$\Phi_{A\widetilde{A}}^{+}\left(\rho_{A_{0}\widetilde{A}_{0}}\right)=\mathrm{tr}\left[\rho_{A_{0}\widetilde{A}_{0}}\phi_{A_{0}\widetilde{A}_{0}}^{+}\right]\phi_{A_{1}\widetilde{A}_{1}}^{+}.$$

In other terms, the CP map $\Phi_{A\widetilde{A}}^{+}$ can be viewed as a generalization of the (unnormalized) maximally entangled state $\phi_{A_{0}\widetilde{A}_{0}}^{+}$.

A supermap $\Theta\in\mathfrak{L}\left(A\to B\right)$ can also be characterized by its action on Choi matrices. One can define a linear map $\mathcal{R}^{\Theta}:\mathfrak{B}\left(A\right)\to\mathfrak{B}\left(B\right)$ as

$$\mathcal{R}_{A\to B}^{\Theta}\left(\rho_{A}\right):=\mathrm{tr}_{A}\left[\mathbf{J}_{AB}^{\Theta}\left(\rho_{A}^{T}\otimes I_{B}\right)\right]\quad\forall\rho\in\mathfrak{B}\left(A\right).$$

With this definition, $\mathbf{J}_{AB}^{\Theta}$ can be viewed as the Choi matrix of $\mathcal{R}_{A\to B}^{\Theta}$. Note that although $\mathcal{P}_{AB}^{\Theta}$ and $\mathcal{R}_{A\to B}^{\Theta}$ have the same Choi matrix $\mathbf{J}_{AB}^{\Theta}$, $\mathcal{P}_{AB}^{\Theta}$ takes systems $A_{0}B_{0}$ to $A_{1}B_{1}$, whereas the map $\mathcal{R}^{\Theta}$ takes system $A=\left(A_{0},A_{1}\right)$ to system $B=\left(B_{0},B_{1}\right)$. This brings us to the last representation of a supermap in terms of a linear map $\mathcal{Q}^{\Theta}:\mathfrak{B}\left(A_{1}B_{0}\right)\to\mathfrak{B}\left(A_{0}B_{1}\right)$, which is defined as the map satisfying

$$\mathbf{J}_{AB}^{\Theta}:=\mathcal{Q}_{\widetilde{A}_{1}\widetilde{B}_{0}\to A_{0}B_{1}}^{\Theta}\left(\phi_{A_{1}\widetilde{A}_{1}}^{+}\otimes\phi_{B_{0}\widetilde{B}_{0}}^{+}\right),$$

(2)

or as $\mathcal{Q}^{\Theta}:=\mathbbm{1}_{A}\otimes\Theta_{A\rightarrow B}\left[\mathcal{S}_{A}\right]$, where $\mathcal{S}_{A}$ is the swap from $A_{1}$ to $A_{0}$. All these three representations of a supermap, $\mathcal{P}^{\Theta}$, $\mathcal{Q}^{\Theta}$, and $\mathcal{R}^{\Theta}$, play a useful role in the study of quantum resource theories, as shown in Ref. [74] in the case of the entanglement of bipartite channels.

A superchannel is a supermap $\Theta_{A\to B}\in\mathfrak{L}\left(A\to B\right)$ that takes quantum channels to quantum channels even when tensored with the identity supermap [35, 67, 66, 68, 69, 43, 70]. More precisely, $\Theta_{A\to B}\in\mathfrak{L}\left(A\to B\right)$ is called a superchannel if it satisfies the following two conditions:

1. 1.

   For any trace-preserving map $\mathcal{N}_{A}\in\mathfrak{L}\left(A_{0}\to A_{1}\right)$, the map $\Theta_{A\to B}\left[\mathcal{N}_{A}\right]$ is a trace-preserving map in $\mathfrak{L}\left(B_{0}\to B_{1}\right)$.

2. 2.

   For any system $R=\left(R_{0},R_{1}\right)$ and any bipartite CP map $\mathcal{N}_{RA}\in\mathrm{CP}\left(R_{0}A_{0}\to R_{1}A_{1}\right)$, the map $\Theta_{A\to B}\left[\mathcal{N}_{RA}\right]$ is also CP.

We will also say that a supermap $\Theta_{A\to B}\in\mathfrak{L}\left(A\to B\right)$, is positive if it takes CP maps to CP maps, and completely positive (CP), if it satisfies the second condition above [35, 43]. Therefore, a superchannel is a CP supermap that takes trace-preserving maps to trace-preserving maps [43, 70]. We will denote the set of superchannels from $A$ to $B$ by $\mathfrak{S}\left(A\rightarrow B\right)$. Note that $\mathfrak{S}\left(A\rightarrow B\right)\subset\mathfrak{L}\left(A\rightarrow B\right)$.

The above definition is axiomatic and minimalist, in the sense that any physical evolution (or simulation) of a quantum channel must satisfy these two basic conditions. The third part of the following theorem shows that these two conditions are sufficient to ensure that superchannels are indeed physical processes.

In general, the realization of a superchannel as given in Fig. 1 is not unique. This is due to the presence of a memory system, described in Fig. 1 with the letter $E$. To see why, consider an isometry channel $\mathcal{V}_{E\to E^{\prime}}$ defined for all $\rho\in\mathfrak{B}\left(E\right)$ by $\mathcal{V}_{E\to E^{\prime}}\left(\rho\right)=V\rho_{E}V^{\dagger}$, where $V:E\to E^{\prime}$ is an isometry matrix satisfying $V^{\dagger}V=I_{E}$. Then, this isometry channel has many left inverses given by

$$\mathcal{V}_{E^{\prime}\to E}^{-1}\left(\sigma_{E^{\prime}}\right)=V^{\dagger}\sigma_{E^{\prime}}V+\mathrm{tr}\left[\left(I_{E^{\prime}}-VV^{\dagger}\right)\sigma_{E^{\prime}}\right]\tau_{E},$$

where $\tau\in\mathfrak{D}\left(E\right)$ is an arbitrary fixed density matrix. We can easily check that $\mathcal{V}^{-1}\circ\mathcal{V}=\mathsf{id}$. In Fig. 2 we use this map to show that the realization of a superchannel in terms of pre- and post-processing is not unique.

Moreover, there is another way in which the realization of a superchannel can be non-unique, namely by appending a state in the pre-processing, and then discarding it in the post-processing. To see how this works, let $\mathcal{F}_{B_{0}\to EA_{0}}$ and $\mathcal{E}_{EA_{1}\to B_{1}}$ be the pre-processing and the post-processing in a realization of a superchannel $\Theta\in\mathfrak{S}\left(A\to B\right)$, respectively. Now consider the new pre-processing $\mathcal{F}^{\prime}_{B_{0}\to E^{\prime}EA_{0}}:=\rho_{E^{\prime}}\otimes\mathcal{F}_{B_{0}\to EA_{0}}$, where $\rho\in\mathfrak{D}\left(E^{\prime}\right)$, and the new post-processing $\mathcal{E}_{A_{1}EE^{\prime}\to B_{1}}:=\mathrm{tr}_{E^{\prime}}\otimes\mathcal{E}_{EA_{1}\to B_{1}}$. It is straightforward to check that $\mathcal{F}^{\prime}$ and $\mathcal{E}^{\prime}$ realize exactly the same superchannel $\Theta$, as $\mathcal{F}$ and $\mathcal{E}$.

Although the realization of a superchannel is not unique, if we restrict the dimension of system $E$ to be the smallest possible, and the map $\mathcal{F}$ to be an isometry, we can obtain a new uniqueness result, expressed by the following theorem, which subsumes some of the results in Ref. [75].

A quantum instrument is a collection of CP maps $\left\{\mathcal{E}_{x}\right\}$ such that their sum $\sum_{x}\mathcal{E}_{x}$ is a CPTP map. Note that each $\mathcal{E}^{x}$ is trace non-increasing, and that every CP map that is trace non-increasing can be completed to a full quantum instrument. Quantum instruments are used to characterize the most general measurements that can be performed on a physical system, including, as special cases, projective von Neumann measurements, POVMs, and generalized measurements. Therefore, we discuss the generalization of a quantum instrument to a collection of objects that act on quantum channels. We call this generalization a superinstrument [70].

A superinstrument is a collection of supermaps $\left\{\Theta_{x}\right\}$, where each $\Theta_{x}\in\mathfrak{L}\left(A\to B\right)$ is CP (i.e. $\mathbf{J}_{AB}^{\Theta_{x}}\geq 0$), and the sum $\sum_{x}\Theta_{x}$ is a superchannel. Similar to the state domain, every $\Theta_{x}$ maps quantum channels to CP trace non-increasing maps. However, in the channel domain not every supermap $\Theta\in\mathfrak{L}\left(A\to B\right)$ with a positive semi-definite Choi matrix, and that takes channels to CP trace non-increasing maps, can be completed to a superchannel. In Ref. [70] a counterexample was given, and it was also shown that a CP supermap $\Theta\in\mathfrak{L}\left(A\to B\right)$ can be completed to a superchannel (i.e. there exists a CP supermap $\Omega\in\mathfrak{L}\left(A\to B\right)$ such that $\Theta+\Omega$ is a superchannel) if and only if for any system $R$, the supermap $\mathbbm{1}_{R}\otimes\Theta$ takes quantum channels to CP trace non-increasing maps. In Ref. [70] it was shown that this phenomenon is associated with the existence of signaling bipartite channels.

While the above discussion is subtle, it demonstrates (see details in Ref. [70]) that every element $\Theta_{x}$ of a superinstrument $\left\{\Theta_{x}\right\}$ satisfies $\mathrm{tr}\left[\mathbf{J}_{AB_{0}}^{\Theta_{x}}\alpha_{AB_{0}}\right]\leq 1$ for every $\alpha_{AB_{0}}\geq 0$ such that $\alpha_{A_{0}B_{0}}=I_{A_{0}}\otimes\rho_{B_{0}}$, where $\rho\in\mathfrak{D}\left(B_{0}\right)$. Moreover, every superinstrument can be realized as in Fig. 3, with an isometry pre-processing and a quantum instrument as the post-processing [35, 70].

Like quantum instruments, any superinstrument $\left\{\Theta_{x}\right\}$ in $\mathfrak{L}\left(A\to B\right)$ can be viewed as a superchannel $\Theta\in\mathfrak{L}\left(A\to BX\right)$, where system $X=\left(X_{0},X_{1}\right)$ has trivial input dimension $\left|X_{0}\right|=1$, and the output system $X_{1}$ is classical. Hence, a superinstrument can be expressed as

$$\Theta_{A\to BX}=\sum_{x}\Theta_{A\to B}^{x}\otimes\left|x\right\rangle\left\langle x\right|_{X},$$

where $X\equiv X_{1}$. This characterization of a superinstrument is particularly useful in the context of quantum resource theories, since the above relation demonstrates that the set of free superinstruments can be viewed as a subset of the set of free superchannels.

Quantum combs are multipartite channels with a well-defined causal structure (see Fig. 4(a)) [76, 65, 66, 77, 78, 79].

They generalize the notion of superchannels to objects that take several channels as input, and output a channel (see Refs. [65, 66] for more details, and a for a further generalization where the input and the output of combs are combs themselves). A comb acting on $n$ channels is depicted in Fig. 4(b). We will denote a comb with $n$ channel-slots as input by $\mathscr{C}_{n}$, and its action on $n$ channels by $\mathscr{C}_{n}\left[\mathcal{N}_{1},\dots,\mathcal{N}_{n}\right]$. The causal relation between the different slots ensures that each such comb can be realized with $n+1$ channels $\mathcal{E}_{1},\dots,\mathcal{E}_{n+1}$ as in Fig. 4(b). We therefore associate a quantum channel

$$\mathcal{Q}^{\mathscr{C}_{n}}:=\mathcal{E}_{n+1}\circ\mathcal{E}_{n}\circ\dots\circ\mathcal{E}_{1}$$

with every comb. Note that the quantum channel $\mathcal{Q}^{\mathscr{C}_{n}}$ has a causal structure in the sense that the input to $\mathcal{E}_{k}$ cannot affect the output of $\mathcal{E}_{k-1}$ for any $k=2,\dots,n+1$. The Choi matrix of the comb is defined as the Choi matrix of $\mathcal{Q}^{\mathscr{C}_{n}}$. Owing to the causal structure of $\mathcal{Q}^{\mathscr{C}_{n}}$, the marginals of the Choi matrix of $\mathscr{C}_{n}$ satisfy similar relations to Eq. (3) (see Refs. [65, 66] for more details).

Note that there are other ways to manipulate multiple quantum channels where we do not require any causal structure on the different channel-slots [67, 80, 81], but we will not use them in our analysis.

For every pair of physical systems $A$ and $B$, consider a subset of CPTP maps $\mathfrak{F}\left(A\to B\right)\subset\mathrm{CPTP}\left(A\to B\right)$. $\mathfrak{F}$ identifies a quantum resource theory (QRT) if the following two conditions hold [14]:

1. 1.

   For every physical system $A$, the set $\mathfrak{F}\left(A\to A\right)$ contains the identity map $\mathsf{id}_{A}$.

2. 2.

   For any three systems $A$, $B$, $C$, if $\mathcal{M}\in\mathfrak{F}\left(A\to B\right)$ and $\mathcal{N}\in\mathfrak{F}\left(B\to C\right)$, then $\mathcal{N}\circ\mathcal{M}\in\mathfrak{F}\left(A\to C\right)$.

The elements in each set $\mathfrak{F}\left(A\to B\right)$ are called free operations. The set $\mathfrak{F}\left(A\right):=\mathfrak{F}\left(1\to A\right)$, where the $1$ stands for the trivial (i.e. 1-dimensional) system, will be used to denote the set of free states.

In any QRT we can consider either static or dynamical inter-conversions. In a static inter-conversion we look for conditions under which a conversion from one resource state (i.e. not in $\mathfrak{F}\left(A\right)$) to another is possible by free operations. In a dynamical inter-conversion we are interested in the conditions under which a conversion from one resource channel (i.e. not in $\mathfrak{F}\left(A\to B\right)$) to another is possible with free superchannels. Clearly, static inter-conversions can be viewed as a special type of dynamical ones.

In this article we will consider QRTs that admit a tensor product structure. That is, the set of free operations $\mathfrak{F}$ satisfies the following additional conditions:

1. 3.

   Free operations are “completely free”: for any three physical systems $A$, $B$, and $C$, if $\mathcal{M}\in\mathfrak{F}\left(A\to B\right)$ then $\mathsf{id}_{C}\otimes\mathcal{M}\in\mathfrak{F}\left(CA\to CB\right)$.

2. 4.

   Discarding a system (i.e. the trace) is a free operation: for every system $A$, the set $\mathfrak{F}\left(A\to 1\right)$ is not empty.

The above additional conditions are very natural, and satisfied by almost all QRTs studied in literature [14]. These conditions imply that if $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ are free channels, then also $\mathcal{M}_{1}\otimes\mathcal{M}_{2}$ is free. In addition, they also imply that appending free states is a free operation; i.e. for any given free state $\sigma\in\mathfrak{F}\left(B\right)$, the CPTP map $\mathcal{F}_{\sigma}\left(\rho\right):=\rho\otimes\sigma$ is a free map, i.e. it belongs to $\mathfrak{F}\left(A\to AB\right)$. This in turn implies that the replacement map $\mathcal{R}_{\sigma}$ is free, where $\mathcal{R}_{\sigma}\left(\rho\right)=\mathrm{tr}\left[\rho\right]\sigma$, for every density matrix $\rho$, and some fixed free state $\sigma$. In the following we will also assume that $\mathfrak{F}\left(A\to B\right)$ is topologically closed for all systems $A$ and $B$, as it is natural to assume that arbitrarily good approximations of free operations are free as well.

The examples of dynamical monotones presented in the previous subsection are typically very hard to compute due to the optimizations involved. Here for the first time we introduce a family of dynamical resource montones for convex resource theories that in some cases (e.g. NPT entanglement, Ref. [59]) can be computed with SDPs. Furthermore, each member of the family is convex, and the family itself is complete, in the sense that the monotones provide both necessary and sufficient conditions for the conversion of a dynamical resource into another with free superchannels. In this sense, this family of monotones fully captures the resourcefulness of a dynamical resource.

An example of a complete family of static resource monotones is known for pure-state entanglement theory [93, 94, 95]. There, the family of entanglement monotones is given in terms of Ky-Fan norms, and due to Nielsen majorization theorem [93], this family provides both necessary and sufficient conditions for the convertibility of pure bipartite states. The fact that the family consists of a finite number of monotones makes it easy to determine the convertibility of bipartite pure states under LOCC. However, for mixed states it is known that, already in local dimension 4, a finite number of monotones is insufficient to fully determine the exact interconversions between bipartite mixed states [96]. Therefore, in general, one cannot expect to find a finite and complete family for a generic QRT.