All States are Universal Catalysts in Quantum Thermodynamics

Quantum catalysis is a fascinating concept that demonstrates how certain transformations can only become possible when given access to a specific resource that has to be returned unaffected. It was first discovered in the context of entanglement theory, and since then, it has been applied in a number of resource-theoretic frameworks, including quantum thermodynamics. Although, in that case, the necessary (and sometimes also sufficient) conditions on the existence of a catalyst are known, almost nothing is known about the precise form of the catalyst state required by the transformation. In particular, it is not clear whether it has to have some special properties or be finely tuned to the desired transformation. In this work, we describe a surprising property of multicopy states: We show that in resource theories governed by majorization, all resourceful states are catalysts for all allowed transformations. In quantum thermodynamics, this means that the so-called “second laws of thermodynamics” do not require a fine-tuned catalyst; rather, any state, given sufficiently many copies, can serve as a useful catalyst. These analytic results are accompanied by several numerical investigations that indicate that neither a multicopy form nor a very-large-dimension catalyst is required to activate most allowed transformations catalytically.

Popular Summary

Quantum catalysts make previously impossible quantum transformations possible. These special quantum resources—such as ancillary entangled states—open up new possibilities for manipulating objects without consuming or degrading the new resource. Moreover, quantum catalysts can be reused indefinitely, making them highly desirable resources. While intuition suggests catalysts should be rare or finely tuned, we show that is not the case. Rather, any state can act as a quantum catalyst for any transformation, provided that enough copies are supplied.

Our interactions with the macroscopic world suggest that quantum catalysts must, at the very least, be finely tuned for a particular purpose. This intuition comes from our everyday experience: Since quantum catalysts model the behavior of thermal machines or ancillary experimental instruments, they need to be carefully tuned before they can aid in performing the desired transformation. However, the quantum world often behaves contrary to our intuition—and quantum catalysts are no different.

Our mathematical and numerical analysis draws from quantum thermodynamics to prove that quantum catalysis can come from any state. In this way, quantum catalysis is seen to be completely different from macroscopic catalysis, with its own rich structure, much of which is still to be understood.

Figure 1

Main protocol. The catalyst C is processed using the unitary $U_{\mathrm{CB}}$, and then, along with the system S, it is used as an input for the operation ${\mathcal{E}}_{\mathrm{SC}}(·)≔{\mathrm{Tr}}_{\mathrm{B}}[V_{\mathrm{SCB}}\mathbf{(}(·{)}_{\mathrm{SC}}\otimes {\tau }_{\mathrm{B}}\mathbf{)}{V}_{\mathrm{SCB}}^{†}]$. The resulting state of the catalyst is then transformed back using the unitary $U_{\mathrm{CB}}^{†}$. As long as the backaction of the map ${\mathcal{E}}_{\mathrm{SC}}$ on the catalyst is small, the protocol as a whole leaves the state of the catalyst approximately undisturbed.

Figure 2

An estimate of the volume of all block-diagonal pairs of states $({\rho }_{\mathrm{S}},{\sigma }_{\mathrm{S}})$ with $H_{\mathrm{S}}\propto {\mathbb{1}}_{\mathrm{S}}$ for which there is a transformation taking ${\rho }_{\mathrm{S}}^{\otimes i}$ into ${\sigma }_{\mathrm{S}}^{\otimes i}$ for some $i\le k$, divided by the volume of all block-diagonal pairs that can be catalytically activated. Catalytically activated pairs are such that (i) they satisfy the respective second laws and (ii) ${\rho }_{\mathrm{S}}$ cannot be directly converted into ${\sigma }_{\mathrm{S}}$.

Figure 3

Probability $p_{\text{succ}}({\mathit{p}}^{\star },{\mathit{q}}^{\star },{\epsilon }_{\mathrm{C}})$ that a random state of dimension $d_{\mathrm{C}}$ drawn from probability distribution ${\mathcal{P}}_{\text{dist}}$ can catalyze a fixed-state transformation ${\mathit{p}}^{\star }\to {\mathit{q}}^{\star }$. Each panel corresponds to a different distribution: (a) Rayleigh ${\mathcal{P}}_{\mathrm{ray}}$, (b) uniform ${\mathcal{P}}_{\text{unif}}$, and (c) exponential ${\mathcal{P}}_{\exp }$. The insets illustrate an exemplary distribution of eigenvalues $c_k^{(i)}$ of a random catalyst ${\mathit{c}}^{(i)}$ drawn according to a respective distribution and then normalized.

Figure 4

Probability $p_{\text{succ}}({\mathit{p}}^{\star },\mathit{q},{\epsilon }_{\mathrm{C}})$ that a random state can be used to enable the transformation from ${\mathit{p}}^{\star }$ to $\mathit{q}$ approximately catalytically, i.e., with an error ${\epsilon }_{\mathrm{C}}=\mu {\epsilon }_{\mathrm{C}}^{\mathrm{bnd}}$ and $\mu =0.1$. Plots (a)–(c) correspond to different dimensions of the random catalyst ($d_{\mathrm{C}}=2^4$, $2^6$, and $2^8$, respectively). States inside the region bounded by dashed lines define the thermal set $\mathsf{S}({\mathit{p}}^{\star })$, consisting of all states that can be reached from ${\mathit{p}}^{\star }$ using thermal operations. The solid line corresponds to the catalytic-thermal set $\mathsf{T}({\mathit{p}}^{\star })$, consisting of all states that can be reached from ${\mathit{p}}^{\star }$ with the help of some (potentially finely tuned) catalyst. Note that the probability of success $p_{\text{succ}}({\mathit{p}}^{\star },\mathit{q},{\epsilon }_{\mathrm{C}})$ is generally very close to 1 for most final states $\mathit{q}$ inside $\mathsf{D}({\mathit{p}}^{\star })≔\mathsf{T}({\mathit{p}}^{\star })\backslash \mathsf{S}({\mathit{p}}^{\star })$, even when the dimension of the catalyst is relatively small. Plot (d) illustrates the cumulative distribution of $p_{\text{succ}}({\mathit{p}}^{\star },\mathit{q},{\epsilon }_{\mathrm{C}})$. To simplify the interpretation, an exemplary point is drawn in red. It corresponds to the case when 30% of all states $\mathit{q}\in \mathsf{D}({\mathit{p}}^{\star })$ can be reached using random catalysts of dimension $d_{\mathrm{C}}=2^4$, with probability less than 0.4. In other words, 70% of all possible catalytic transformations can be realized using random catalysts with probability 0.4 or higher.

Figure 5

Fraction $f(\mathit{p})$ of all states inside the catalytic activation set $\mathsf{D}(\mathit{p})$ (CAS) for which the probability $p_{\text{succ}}(\mathit{p},\mathit{q},{\epsilon }_{\mathrm{C}})$ that a random state can be used as a catalyst is larger than the threshold value ${\gamma }_{\mathrm{thd}}=0.9$. Plots (a)–(c) correspond to random catalysts of dimensions $2^4$, $2^6$, and $2^8$, respectively. Plot (d) illustrates the cumulative distribution of $f(\mathit{p})$. As an example, the red point corresponds to the situation where random states of dimension $d_{\mathrm{C}}=2^4$ are used to catalyze possible transformations. These random catalysts are not useful for roughly 25% of all possible initial states $\mathit{p}$; i.e., for each such state, they allow, at most, approximately 30% of all output states $\mathit{q}$ in CAS to be reached with probability equal to or greater than ${\gamma }_{\text{thld}}$. In other words, such random catalysts can be used to reach more than 30% of all possible output states $\mathit{q}\in \mathsf{D}(\mathit{p})$, with probability at least ${\gamma }_{\text{thld}}$, for at least 75% of all initial states $\mathit{p}$.

Figure 6

Fraction $f(\mathit{p})$ of all states inside the catalytic activation set of $\mathit{p}$ that can be catalyzed using a multicopy catalyst composed of $n\in \{4,8,10\}$ qubits ${\omega }_{\mathrm{C}}=\text{diag}(1-r,r)$, for different values of parameter $r$. Column 4 shows a cumulative distribution of $f(\mathit{p})$; i.e., it illustrates the fraction of catalytic transformations that can be activated using a multicopy catalyst. For example, the red exemplary point indicates that for around 0.25 of all possible input states $\mathit{p}$, a quantum system composed of $n=10$ qubits in a state with $r=0.3$ acts as a catalyst for less than 0.5 of all possible output states $\mathit{q}$ in CAS. Equivalently, for this choice of parameters, the multicopy catalyst can activate more than 0.5 of all possible transformations for approximately 0.75 of all possible input states.

Figure 7

Cumulative distribution of $f(\mathit{p})$ for different choices of parameters: (a) $[d_{\mathrm{S}}=4,\mu =0.05,{\gamma }_{\text{thld}}=0.9]$, (b) $[d_{\mathrm{S}}=5,\mu =0.05,{\gamma }_{\text{thld}}=0.9]$, (c) $[d_{\mathrm{S}}=4,\mu =0.1,{\gamma }_{\text{thld}}=0.8]$, and (d) $[d_{\mathrm{S}}=5,\mu =0.1,{\gamma }_{\text{thld}}=0.8]$.