What is non-classical about quantum no-cloning?

This is a Perspective on "Contextual advantage for state-dependent cloning" by Matteo Lostaglio and Gabriel Senno, published in Quantum 4, 258 (2020).

By Ana Belén Sainz (International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland).


Contextuality is one of the features of quantum theory that have no intuitive explanation. First identified by Bell back in 1964 [1], and later posed in more general terms by Kochen and Specker [2], contextuality identifies an inconsistency between two classically ingrained ideas: (i) that the observables we measure correspond to physical properties that have a predefined value, which is merely revealed by the measurement, and (ii) that the value of a property is independent of which other observables you happen to be measuring at the same time (i.e., the $\textit{context}$ of the observation). If one is to hold on to realism for describing physical properties, i.e., condition (i), then one needs to admit a physical theory where measurement results depend on the context.

For decades, researchers have tried to make sense of how our classical intuitions fail us in the quantum world that surrounds us. We are still trying to fully comprehend the extent of this form of nonclassicality identified by Kochen-Specker, and different perspectives on the phenomenon have been put forwards to this effect [3,4,5]. Meanwhile, contextuality has already been identified as having a key role in quantum information processing [6,7].

A conceptual breakthrough came years after Kochen-Specker’s seminal result, by looking at the phenomenon from an operational lens. This gave rise to $\textit{contextuality in the generalised sense}$, discovered by Spekkens [8], and which subsumes Kochen-Specker’s notion of contextuality. In a nutshell, Spekkens’ contextuality argues that if two objects behave exactly the same when we observe and probe them in any possible way, then the underlying fine-grained description of these objects (should it exist) must be exactly the same. Such a revolutionary new perspective, grounded on Leibniz’s principle of $\textit{identity of indiscernibles}$ [9], has made us reconsider not only the very foundations of quantum physics, but also the impact that this nonclassical feature could have on technological developments.

State-dependent cloning

The no-cloning theorem [10,11,12] says that quantum information cannot be copied (i.e., cloned) in a deterministic and reliable fashion. This comes in contrast to the well-known fact that classical information can be and is perfectly copied with no problem in our daily lives.

The task of $\textit{state-dependent cloning}$ [13] refers to the situation where we wish to clone the unknown state of a quantum system, under the premise that the corresponding state belongs to a known set. The question that is asked is “how well can we succeed in such a task”, namely, what is the maximum likelihood that we can successfully clone the unknown state of the system. The advantage here is that, given that we know the possible states that the system can be in, we can potentially optimise our cloning device with respect to that set of states. An extreme example is when we are given the promise that the state belongs to a fixed orthonormal basis — in this case quantum cloning reduces to classical copying, as we can simply measure the state and prepare several copies according to the outcome. Most interesting cases lay somewhere in between, i.e., with more vague promises.

“No-cloning” as a quantum signature: an obstacle

Given that classical theory allows for information to be copied perfectly, while quantum theory may only copy quantum information with some probability of success, one could be tempted to see “the fundamental impossibility of perfect cloning” (i.e., no-cloning) as a defining non-classical feature. Namely, if we are in a situation where fundamentally we cannot copy the information reliably, we could take that as a signature that a non-classical process is taking place. But to our surprise, there are a few steps in this logical reasoning that don’t quite work, as we see next.

The first key issue is the meaning of $\textit{classicality}$ — when can something be considered classical? This is where Spekkens’ contextuality [8] comes in. It provides a broad notion of classicality which, unlike Bell nonlocality, makes sense even for experimental setups within a single lab — such as the task of cloning. More precisely, the observed experimental data is deemed to have a `classical’ explanation if it does not feature the aforementioned Spekkens’ contextuality. Hence, one could take the next step and conjecture that “if our experiment features no-cloning, then the experimental data must be contextual (i.e., non-classical)”. However, there are examples of toy theories that fundamentally cannot clone arbitrary information despite being non-contextual [14,15] — therefore, no-cloning alone cannot be considered a signature of non-classicality.

This realisation, however unfortunate, opens the door to asking a more fundamental question: what is non-classical about the no-cloning theorem featured by quantum theory? A surprising answer to this relies on considering the operational constraints present in quantum state-dependent cloning, as developed in the work of Lostaglio and Senno [16].

Operational constraints come to the aid

Lostaglio and Senno [16] study how capable classical systems are at the task of cloning when they are subjected to certain $\textit{operational constraints}$ — that is, how well classical systems can be cloned provided that the cloning apparatuses feature certain statistical properties. Because of these particular constraints, one cannot necessarily expect perfect cloning anymore, as would be usually the case. This is how Ref. [16] sets the problem on fair grounds, and by so compares (classical) apples with (quantum) apples. Such an approach was used in the past by Schmid and Spekkens [17] for studying the contextual advantage in the task of state discrimination — it all boils down to understanding the phenomena operationally [18] and identifying what is `strongly nonclassical’ about them [19].

So, what are these operational constraints? For the case of state-dependent cloning, it means $\textit{two conditions that are}$ $\textit{satisfied by quantum theory}$, and which are formulated for the states that play a relevant role on the experiment. These states are:

– the states we could be asked to clone, for example $\{ |0\rangle_A, |1\rangle_A, |+\rangle_A \}$,
– the attempted clones (i.e., the state we actually prepare when trying to clone our initial state),
– the states we would prepare if perfect cloning was possible, for example $|++\rangle_{AA’}$.

For each such state $s$, let $P_s$ denote the preparation procedure that yields it, and let $p P_s +(1-p) P_{s’}$ denote a process by which we flip a coin and with probability $p$ prepare state $s$, otherwise prepare $s’$.

Condition 1:

For each $P_s$, there exists a dichotomic measurement procedure $M_s$ that can certify that such state has been prepared — that is, if we measure $M_s$ on the system and obtain, say, outcome 1, we are certain that the system was prepared according to $P_s$. For example, for the state $|+\rangle$, the outcome 1 of a measurement of the Pauli $X$ observable certifies the state.

Condition 2:

For each $P_s$, there exists a complementary state preparation of the system, call it $P_{s^\perp}$, such that:
(2a) the measurement $M_s$ never yields outcome $1$ when performed on a state prepared by $P_{s^\perp}$,
(2b) $P_s + P_{s^\perp} = P_{s’} + P_{s’^\perp}$ for certain pairs of states $(s,s’)$, which conveys the following intuition: suppose you prepare the system by flipping an unbiased coin which decides whether to implement $P_s$ or its complement $P_{s^\perp}$; then, the resulting state of the system should be for all practical purposes independent of which particular state $s$ was chosen from the pair to implement the protocol.

As a quantum example, consider the initial states $\{ |0\rangle, |+\rangle \}$. Then, by choosing the Pauli $X$ and Pauli $Z$ measurements, respectively, as the $M_s$, and $\{ |1\rangle, |-\rangle \}$, respectively, as the complementary states, Conditions (2a) and (2b) are met. Indeed, the preparation of a system by flipping an unbiased coin and preparing either $|0\rangle$ or $|1\rangle$ yields always the maximally mixed state, which is also the case when choosing between $|+\rangle$ and $|-\rangle$.

Conditions 1 and 2 are deemed $\textit{operational}$, since they can be checked by studying the outcome statistics of the preparation and measurement devices, in a way that is independent of the underlying physical theory describing those apparatuses.

The figure of merit that Lostaglio and Senno [16] use to assess what is non-classical about quantum no-cloning is the $\textit{success probability}$ with which a state can be cloned in the task of state-dependent cloning. That is, the average probability (over all the states we could be asked to clone) that our imperfect clone gives the outcome 1 when the certificate measurement $M_s$ (with $s$ being the state of the perfect clone) is performed on it. The question Ref. [16] asks is what is the maximum success probability that a classical version of the experiment can achieve, provided that it complies with the operational constraints? The answer is “well, some number which depends on the particular case, but that number will always be smaller than what quantum theory lets you achieve”. In other words, when asked to perform the same task under the same constraints, quantum theory provides a larger success probability than what any classical model (including the aforementioned toy theories) would allow.

Finally, Lostaglio and Senno [16] also show that this fundamental difference between quantum and non-contextual theories still exists in the more realistic state-dependent cloning experiment where the measurement procedures $M_s$ are allowed to be noisy.

What do we learn?

A straightforward lesson that we learn from the work of Lostaglio and Senno [16] is that quantum theory does provide an advantage over classical theories on the task of state-dependent cloning. That is, contextuality is required to reproduce the cloning fidelities featured by quantum theory when the classical models are under the same operational constraints, even in non-idealised setups. This is quite surprising and counter-intuitive, given our ubiquitous perception that classical information can be perfectly copied.

But an important fundamental lesson here is that such a claim pertaining to `contextuality as a resource’ can be stated with rigor and clarity once a proper account of the operational constraints is made. For this `proper account’ to be feasible, the experiment needs to be reconceptualised in a manner that makes no reference at all to quantum theory. Such an approach, endorsed also by the previous work of Ref. [17], can spur further conceptual advances in quantum foundations and quantum information.

► BibTeX data

► References

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