1 Introduction↩︎

Correlations between separated systems represent a fundamental concept in many scientific disciplines and a resource in various tasks. To understand correlations from an information-theoretic perspective one studies how they can be established and transformed under various operational restrictions, termed transformation classes. Arguably, local operations (LO) represent the minimal such class of operations: applied separately on different systems, they preserve their independence and are thus expected not to increase measures of correlations based on information as, e.g., captured by various data processing inequalities.

In quantum information theory these ideas have been structured in the resource-theoretic approach [1], explicitly specifying which operations (e.g. LO) and which state (e.g. product states) are free. Correlation measures must then be monotones under those operations. In this context, a central question is to characterize genuinely quantum forms of correlations, and separate them from their classical counterpart (see e.g. [2]–[5]). The most prominent example is entanglement theory, where LO are considered together with additional classical communication between the parties, resulting in the LOCC transformation class [6]. Here, the addition of unbounded classical communication in particular guarantees that any amount of classical correlations comes for free.

Remarkably, it was discovered that the set of state transformations possible with LOCC is drastically enlarged by allowing an extra system to act as a catalyst [7]–[14]. Quantum catalysis, which has also been studied in different contexts [15]–[19], shows that having access to an additional entangled state (a catalyst), which must be returned in its initial state, can enable transformations that would otherwise be impossible. Conceptually, this reveals a novel way of exploiting entanglement: rather than acting as a consumable, it can also serve as a reusable resource. This also makes quantum catalysis a natural extension of the LOCC framework. In particular, this effect has operational consequences; for example, catalysis can improve teleportation [20], state merging [11], and entanglement distillation [11].

This raises a number of fundamental questions: what is the real power of the catalyst in enabling state transformations? In particular, what state transformations are possible in a setting where only LO and catalysis are considered, without involving classical communication?

In order to disentangle the effect of the catalyst from that of classical communication, we explore a class of state transformations termed catalytic local operations (CLO). Here, distant parties share a system and a catalyst to which they apply local operations (represented by completely positive trace-preserving (CPTP) maps) and must return the catalyst exactly unchanged. Yet, the system and catalyst can become correlated at the end of the process. Importantly, classical communication is not allowed in this setting. Intuitively, CLO allows the parties to exploit pre-established quantum correlations encoded in the catalyst, but since the catalyst is returned in the same state, it cannot be consumed as a resource. We investigate which state transformations are possible in CLO, and in particular highlight its fundamental difference with the LOCC class. We also investigate how CLO relates to LOCC transformations assisted by bounded quantum communication.

More formally, we define a class of operations CLO\(^{(d)}\), where local operations are augmented with the use of a catalyst featuring quantum correlations of bounded dimension, i.e., assuming its Schmidt number is upper bounded by a parameter \(d\). First, we note that there are transformations in \(\text{LOCC}\) that cannot be implemented using \(\text{CLO}^{(d)}\). Indeed, classical communication cannot be replaced by a catalyst in general. Then we compare \(\text{CLO}^{(d)}\) with the class \(\text{SLOCCQ}^{(d)}\) of protocols that allow stochastic local operations, classical communication, and bounded quantum communication, such that the product of the dimensions of the transmitted quantum systems is at most \(d\). Intuitively, sending a quantum system of dimension \(d\) is at least as powerful as preparing a catalyst with Schmidt number \(d\), so \(\text{CLO}^{(d)}\) is a subset of \(\text{SLOCCQ}^{(d)}\) by construction. Surprisingly, however, catalytic local operations can sometimes outperform SLOCCQ with a respectively smaller budget for quantum communication, showing strict separations between the classes, that is, \(\text{CLO}^{(d)} \not\subset \text{SLOCCQ}^{(d-1)}\). These results show that quantum catalysis, classical communication, and bounded quantum communication are incomparable resources and capture genuinely distinct aspects of bipartite quantum state manipulation.

2 Definitions and preliminary results↩︎

We investigate different transformation classes from bipartite quantum states \(\rho_{AB} \in \mathcal{L(\mathcal{H}_A\otimes \mathcal{H}_B)}\) to bipartite quantum states of potentially different systems \(\sigma_{A'B'}\in \mathcal{L}(\mathcal{H}_{A'}\otimes \mathcal{H}_{B'})\). A given transformation class \(\tt T\), associates with each input state \(\rho_{AB}\) and given output systems (Hilbert spaces \(\mathcal{H}_{A'}\) and \(\mathcal{H}_{B'})\) a set of density operators to which that state can be transformed, namely \[\label{eq:32state32transform} {\tt T}_{A'B'} : \rho_{AB} \mapsto {\tt T}_{A'B'}(\rho_{AB})\subset \mathcal{L}(\mathcal{H}_{A'}\otimes \mathcal{H}_{B'}).\tag{1}\] When the output systems are not specified, we write \({\tt T}(\rho_{AB})\) to denote the union of reachable sets of output states for all dimensions of \(A'\) and \(B'\).

Since the world is not limited to the two systems \(A(A')\) and \(B(B')\), we naturally include in the study of transformation classes input state \(\rho_{ABR}\in \mathcal{L}(\mathcal{H}_A\otimes \mathcal{H}_B \otimes \mathcal{H}_R)\) that may share quantum correlations with an auxiliary system \(R\), to which Alice and Bob do not have access. In this case the reachable set is \({\tt T}_{A'B'}(\rho_{ABR})\subset \mathcal{L}(\mathcal{H}_{A'}\otimes \mathcal{H}_{B'} \otimes \mathcal{H}_{R})\). To keep the notation simple, we omit \(R\) where it is trivial or irrelevant for the discussion.

In quantum information it is common that such transformations can be described irrespectively of \(\rho_{AB}\), i.e. \({\tt T}_{A'B'}(\rho_{AB})\) contains the images of a fixed set of CP(TP) maps on \(\rho_{AB}\). For instance, this is the case of local operations and shared randomness (LOSR), local operations and shared entanglement (LOSE), local operations and classical communication (LOCC), and stochastic local operations, bounded-quantum and classical communication (SLOCCQ in Def. 1). In contrast, this will not be the case for catalytic local transformations (CLO in Def. 3), central to this paper, hence the reader shall keep the more general notion of Eq. 1 in mind. Let us now formally define the later two classes of transformations.

To introduce the class of CLO along the same lines, we want to be able to bound the dimension of the catalyst system in a meaningful way. To do so we recall the definition of the Schmidt number.

The Schmidt number is a discrete measure of bipartite entanglement quantifying its dimensionality. Let us consider the following definition.

We will simply write CLO, if \(d\) is not specified. Loosely speaking, the constraint \({\rm SN}(\omega_{C_AC_B})\leq d\) guarantees that the catalyst can be encoded by a system with local quantum dimension \(d\) (plus any classical degrees of freedom). It is worth mentioning that in the case of state transformations via catalytic LOCC, restrictions on the dimension and other properties of the catalyst have also been consider, see e.g. [9], [12], [23]. We emphasise that a combination \(\{\mathcal{A},\mathcal{B},\omega_{C_AC_B}\}\) defining a CLO transformation for the input state \(\rho_{AB}\), is in general not CLO for another input state \(\rho_{AB}'\), as the marginal state of the catalyst may not be preserved. This means, in particular, that catalytic transformations cannot be performed in a black-box fashion — knowledge of the initial state is required in general. Finally, we emphasize that we require the state of the catalyst to be exactly preserved at the end of the transformation, in contrast to quantum state embezzlement [24].

3 Results↩︎

We now state our main results, which describe the nontrivial relationships among the different classes of bipartite transformations studied in this paper. See Fig. 1 for a schematic summary.

Let us start from three observations that directly follow from the way we construct the set of transformations \({\rm CLO}^{(d)}\).

The first inclusion holds by definition. Here it is implicit that in LOSE\(^{(d)}\), d also bounds the Schmidt number of the shared entangled state, which does not need to be returned. For the second inclusion note that any state with Schmidt number \(d\) can be prepared by sending a quantum system of dimension \(d\) from Alice to Bob, along with classical communication.

On the one hand, shared randomness can be viewed as a catalyst—the correlations built with the systems do not change its marginal state. On the other, any separable catalyst can be prepared from shared randomness with local operations.

This observation implies that separable states are free in CLO\(^{(1)}\). Hence, any correlation that can be prepared solely using CLO\(^{(1)}\) is, by construction, classical. At the same time, any correlations that cannot be prepared with CLO\(^{(1)}\) must be genuinely quantum. This provides CLO\(^{(1)}\) with a natural operational interpretation: in a manner analogous to LOCC, they allow one to single out classical from quantum correlations.

Consider the following classical states \(\rho_{ABR} = \sum_{x=0}^1 \frac{1}{2} [0]_A \otimes [x]_B \otimes [x]_R\) and \(\sigma_{ABR} = \sum_{x=0}^1 \frac{1}{2} [x]_A \otimes [0]_B \otimes [x]_R\). It is immediate to see that one bit of classical communication from Bob to Alice allows the transformation \[\begin{align} \nonumber \rho_{ABR}\;\xrightarrow{\;\text{LOCC}\;}\;\sigma_{ABR}. \end{align}\] Furthermore, this bit of communication is necessary for Alice and Bob to do the transformation, which is thus impossible by local operations with shared entanglement (LOSE) and hence also with CLO. To see this formally, let \(\omega_{C_A C_B}\) be the auxiliary entanglement shared between Alice and Bob and assume that the LOSE transformation is possible. Then, the marginal state \(\rho_{AC_A R}= ([0] \otimes \omega)_{AC_A} \otimes (\sum_x \frac{1}{2}[x])_R\) can be transformed to \(\sigma_{AR}= \sum_x \frac{1}{2} [x]_A \otimes[x]_R\) by means of a local operation \(\mathcal{A}:AC_A\to A\) applied by Alice. This is manifestly a contradiction, since it violates the data processing inequality for the quantum relative entropy \(H(R|AC_A)\).

A more interesting and arguably less intuitive fact is that there exist state transformations in \({\rm CLO}^{(d)}\) that cannot be realized with \({\rm SLOCCQ}^{(d-1)}\). This is the main result of this work.

To demonstrate this result, we will make use of the two following lemmas. The first shows the possibility of certain transformations within \({\rm CLO}^{(d)}\), and was established in Ref. [15] in the context of catalytic activation of Bell nonlocality, inspired by the catalytic LOCC transformation introduced in Ref. [10].

The second lemma guarantees that the Schmidt number of the initial state cannot change arbitrarily with \({\rm SLOCCQ}^{(d)}\) operations.

To show the separation between CLO\(^{(d)}\) and SLOCC\(^{(d-1)}\), we used the fact that the former class allows to increase the Schmidt number of a state by a factor \(d\). This is also the maximal possible increase saturating the bounds \(\text{SN}(\tau_{A'B'})\leq \text{SN}(\rho_{AB}\otimes\omega_{C_AC_B})\leq \text{SN}(\rho_{AB}) \times d\). This raises the natural question of which entanglement measures remain monotonic in presence of a catalyst, or otherwise how much can they increase. The former question was studied in the context of catalytic LOCC transformations. In particular, it was shown that a pure states transformation \(\ket{\Psi}_{AB}\to \ket{\Phi}_{AB}\) is possible iff the (von Neumann) entanglement entropy is non-increasing, i.e., if \(H(\tr_A \Psi)\geq H(\tr_A \Phi)\) [11]. Hence, for pure states entanglement measures inequivalent to entanglement entropy can be increased with CLOCC. In the case of mixed states it is known [11], that the squashed entanglement [25] remains a monotone, since it is additive for product states \(E_{sq}(\rho_{AB}\otimes \omega_{C_AC_B})=E_{sq}(\rho_{AB})+E_{sq}(\omega_{C_AC_B})\) and super-additive in general \(E(\tau_{A'B'C_A C_B})\geq E(\tau_{A'B'})+E( \tau_{C_AC_B}=\omega_{C_AC_B})\). Clearly, this conclusion remains true for CLO. Conversely, it is an interesting open question to understand which measures of entanglement, other than the Schmidt number, cease to be monotonic under CLO\(^{(d)}\), and how does \(d\), or other restrictions on the catalyst, limit their increase.

4 Conclusions↩︎

Our results contribute to the broader program of operational frameworks in quantum information theory. We show that catalysis can play a strong role as a resource enabling state transformations beyond what can be achieved with unlimited classical communication and limited quantum communication. Future work directions include further exploring and quantifying the trade-off between these resources, and investigating further implications of catalytic local transformations for the resource theories of quantum correlations.
Acknowledgments. We are grateful to Marco Túlio Quintino for discussions. The authors acknowledge funding from the Swiss National Science Foundation (SNSF) through NCCR SwissMAP (project 182902), project 219366 and the Swiss Postdoctoral Fellowship (project 216979). P.L.-B. also acknowledges funding from Polish National Agency for Academic Exchange (NAWA) through grant BPN/PPO/2023/1/00018/U/00001.

References↩︎