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Coherence requirements for quantum communication from hybrid circuit dynamics
by Shane P. Kelly, Ulrich Poschinger, Ferdinand Schmidt-Kaler, Matthew P. A. Fisher, Jamir Marino
This is not the latest submitted version.
|Authors (as registered SciPost users):||Shane Kelly|
|Preprint Link:||https://arxiv.org/abs/2210.11547v2 (pdf)|
|Date submitted:||2023-07-29 20:43|
|Submitted by:||Kelly, Shane|
|Submitted to:||SciPost Physics|
The coherent superposition of quantum states is an important resource for quantum information processing which distinguishes quantum dynamics and information from their classical counterparts. In this article we determine the coherence requirements to communicate quantum information in a broad setting encompassing monitored quantum dynamics and quantum error correction codes. We determine these requirements by considering hybrid circuits that are generated by a quantum information game played between two opponents, Alice and Eve, who compete by applying unitaries and measurements on a fixed number of qubits. Alice applies unitaries in an attempt to maintain quantum channel capacity, while Eve applies measurements in an attempt to destroy it. By limiting the coherence generating or destroying operations available to each opponent, we determine Alice's coherence requirements. When Alice plays a random strategy aimed at mimicking generic monitored quantum dynamics, we discover a coherence-tuned phase transitions in entanglement and quantum channel capacity. We then derive a theorem giving the minimum coherence required by Alice in any successful strategy, and conclude by proving that coherence sets an upper bound on the code distance in any stabelizer quantum error correction codes. Such bounds provide a rigorous quantification of the coherence resource requirements for quantum communication and error correction.
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Well-written manuscript discussing timely topics (error-correcting codes, monitored dynamics) from an unusual, but important perspective (coherence resource theory).
Some technical aspects deserve further comments, as well as the generality claims in the manuscript conclusion
The work by S. Kelly and collaborators discusses coherence (as a resource theory) in a class of stabilizer circuits, encapsulating aspects of measurement-induced dynamics and error-correcting codes.
In particular, the Authors consider three variant of a quantum game (with adaptive Clifford circuits) from a classical setup to a classical setup with quantum initial states to a fully quantum setting to corroborate their results, namely:
1. The determination of the coherence requirements for quantum communications in stabilizer circuits.
2. Coherences shed light on what is genuinely quantum about measurement-induced transitions, corroborating previous discussions in [75-77]. (Remark: the Authors should add Rev. Lett. 129, 260603 to the references [75-77] as this manuscript is relevant to the topic).
Overall, the manuscript is well written and presented, and the pedagogical structure may help inexperienced readers. Therefore, the manuscript is worthy of publication in Scipost Physics once the Authors discuss the following comments.
As highlighted by the Authors, basis dependence is intrinsically related to the resource theory of coherence. Indeed, the observable of choice, i.e., the relative entropy of coherence, includes the participation entropy.
It is, therefore, a natural question: are there basis independent results? For instance, it is known that participation entropy captures the dynamical criticality, e.g., Ref.  and [SciPost Phys. 15, 045 (2023)]. While the leading term is volume, the subleading term captures the universal basis-independent content of the phase transition. Given the close analogy between participation entropy and the relative entropy of coherence, it would be interesting to clarify this point. Can the Authors comment on this point?
Relatedly, I'm not sure that going beyond stabilizer setups will be a mild variation (referring to Sec. VI A). Indeed, suppose I consider a non-Pauli basis, like a random (local or Global) Hilbert space basis. In that case, any well-defined stabilizer state will be fully delocalized; hence, the coherence relative entropy may be uninformative.
Therefore, it is non-trivial to generalize the ideas and results of this paper to more general setups, including Haar circuits or measurements on non-Pauli strings: magic, for instance, can play a critical role.
Therefore, I suggest the Authors highlight these non-trivial interplay (e.g., beyond Pauli basis dependence, global stabilizer basis-dependence, etc.) in their conclusions and soften the generality section of their results (if they don't have quantitative data supporting these conjectures).
(Ultimately, this is good news: coherences in more generic setups are an interesting venue for future investigations!)
1 - new connection between measurement induced transitions and coherence properties, with applications to quantum channels
2 - extensive discussions and carefully written
the only minor weakness I could spot is that the discussion is at time too qualitative, and one has to resort to consulting appendices often
In their manuscript, Kelly and coworkers discuss the dynamics of (random) Clifford circuits in relation to coherence. The main goal is to achieve an understanding of channel properties under a number of scenarios, where Alice and Eve are allowed only certain types of operations.
Relevant cases discussed are those in which case Alice protects a classical diary (without inputing coherence/entanglement into the system), the competition between quantum phases in case both Alice and Eve can perform coherence-enhancing operations, and quantum communication channels. While the results have overall only somewhat distant implications for concrete quantum communication tasks, their conceptual insights are important. In addition, the work sheds light on the overall connection between measurement induced transitions and quantum communication protocols in the context of coherence, something that could easily see further applications.
Overall, the paper is well written, and its conclusions are sound and interesting.
Presentation-wise, I found the authors' choice of deferring to the appendices most technical details somewhat unfortunate, as I had to stop reading to understand what was really done at least a few time. This is of course a matter of personal taste, and I am not insisting on substantial changes: just, I'd recommend the authors to consider moving some of the material back to the text (maybe, shortening the discussion paragraphs at the beginning of some subsections, like V.B, etc.)
I have append below some more detailed comments:
1) why is 'random' crossed in the caption of Fig. 1?
2) Sec. III.a has a somewhat bizzare order. I'd anticipate the content of III.a.1 before conclusions are drawn (in fact, I was unable to understand how Fig. 5 was obtained when it is discussed, this is only discussed in III.a.1);
3) I have missed the discussion about error bar averaging. While the number of realizations considered is large (as are system sizes), I still find hard to believe that late time entropies (see, e.g., Fig. 4) have negligible error bars.
4) Frankly, I did not find particularly surprising the fact that "Alice can not encode quantum information without being able to generate X coherence". Isn't this somehow to be expected? Now, I understand that this then requires details to be nailed down properly - as the authors do in Sec. III. But still, I'd take emphasis out of this claim, and just mention this is a sanity check that illustrates the relevance of coherence in these settings.
A1) the collapse scaling in Fig. 14 is not clear. It would be informative to discuss the quality of the collapse by reporting, e.g., the sum of residuals.
A2) in appendix E, since (as correctly pointed out by the authors) the circuits considered here are quite different from , I think the sentence:
In particular, Ref.  finds the coherent information exponent as β = 0, which is inconsistent with our data.
shall be changed into something like
"In particular, Ref.  finds the coherent information exponent as β = 0, which is not compatible with the critical properties we observe in the present scenario."
please see the questions above, some of which require revisions.
The paper makes a connection between measurement induced & purification transitions, and the amount of quantum coherence provided in to the state (whether initially or during its evolution). This connection is new and interesting. It may lead to further insight into this timely subject.
The paper is very well written and presents a deep a discussion into the role of coherence. For example, it nicely distinguishes between classical and quantum information and shows that coherence is essential to protect the latter from errors, while the former can be protected even in its absence.
The paper also makes a nice connection between coherence resources and the ability to perform quantum error correction.
I don't see many weaknesses. The paper is a bit long and possibly a table of contents would be helpful. But overall it flows very nicely.
The authors investigate the role of quantum coherence in the ability to protect against attacks in a quantum channel. They show that a purification transition can be tuned by tuning the amount of coherence and thus emphasize the role of coherence in the measurement phase transition. The paper also elucidates the very nicely the difference between classical and quantum channels and what resources are needed to protect information in each case. By that it makes a connection to quantum error correction codes and their breakdown.
The paper is original, interesting and important. It may lead to further insight into the field of quantum information and the measurement phase transition in general. I therefore recommend to publish this paper.
I have a few humble question:
1. The coherence transition is tuned, eg, by tuning the ratio of Y measurements vs. Z and X at very small pm. It seems to me that if pm is large enough there is a second transition into a "disentangling" phase even if py>|px-pz|, right? Therefore, is it possible that the transition at py=|px-pz| is not the same as the MPT at large pm? (also, I understand the special role of Y is set by the basis choice for the CNOT gate?) This connects to the question whether the difference in the critical exponents compared to Ref. (Appendix E) is indeed due to the difference in the applied gates or because the coherence transition discussed here is different in nature? The first explanation sounds suspicious to me and I would recommend to check it unless I am wrong and there is no second transition.
2. The authors comment about gates drawn from the random Haar measure. A numerical verification of this statement might a nice compliment to the paper (using the amount of coherence they produce to predicts the location of the coherence transition). However, I understand this is beyond the scope of this paper?