Rényi entropies and negative central charges in non-Hermitian quantum systems

This paper proposes the definition of a family of entanglement measures for Hermitian and non-Hermitian systems. Their definition is a variant of the entanglement measures found in the literature. The authors claim that, in 1+1d critical systems, the entanglement measures follows the logarithmic scaling expected from conformal field theory (CFT), with a multiplicative constant given by the central charge. In support of this claim, they report some numerical results for three examples of 1+1d critical models, on the entanglement measures of a single interval. They do not give analytical arguments for their claim.

The numerical results reported in this manuscript are convincing. They show that the one-interval entanglement entropies defined by the author do follow the logarithmic scaling expected from CFT in the three examples they have chosen to study. For this reason, I think it is worth publishing this manuscript as a SciPost Physics article.

However, some inaccuracy in the short review of literature given in the introduction should be corrected. Indeed, refs [16-20] claim that the ordinary Renyi entropies of non-Hermitian models are governed by the "effective central charge" c-24h (where h is the lowest conformal weight of the model), whereas ref. [21] argues that this result relies on an incorrect determination of the corresponding partition function, and that, for example, the entanglement entropy scales as (c-12h)/3 log(L_A).

Moreover, when dealing with a quantum model defined by a 1d Hamiltonian, the notion of a "true central charge" (as referred to below eq. 3) is undefined. Indeed, fundamentally, the central charge is a feature of the stress-energy tensor, which specifies the response of the action to a local change of metrics. Hence, the central charge of a given model is well-defined only if the action (or the lattice Boltzmann weights) is given for a family of surfaces with varying metrics. In particular, for the XXZ model of eq. (3), by introducing "gauge" phase factors, one can construct distinct six-vertex models, with the same Hamiltonian limit given by eq. (3), but corresponding to different CFTs in the scaling limit. The definition of the modified trace eq. (15), taken from ref. [20] corresponds to the "gauge" associated to the CFT with central charge given below eq. (3), whereas the usual trace corresponds to the CFT with central charge c=1.

Finally, it is interesting to remark that the "generic" Renyi entropy defined by the authors reads

$1/(1-n) \log Tr[ (\rho_A^\dagger)^{(n-1)/2} (\rho_A)^{(n+1)/2} ]$

and thus, if n is an odd integer, it is given by a partition function on an n-sheeted surface (the same as for the usual Renyi entropy), with time-reversal applied along the subsystem A on (n-1)/2 copies. The numerical results reported in the manuscript show that, on the specific examples under study, this time-reversal has no effect on the scaling of the Renyi entropies.

1. Change the introduction to reflect correctly the debate in the literature, about entanglement entropies of non-Hermitian systems.
2. Correct the discussion of central charges in section 2.2

*The paper presents an interesting new definition of standard entanglement measures for non-hermitian systems. The definitions are a priori general, only dependent on the eigenvalues of the reduced density matrix.
*The authors back up their proposal with numerical results for several non-hermitian models.

*I find that the authors slightly misrepresent their contribution to the subject (namely, defining a consistent entanglement measure for non-Hermitian quantum systems).
*Both in their abstract and in other places in the paper they give the impression that they are the first team to address the problem and also that previous proposals are somehow wrong (at least they repeatedly refer to their proposal as the "correct" or "proper" definition).
*I think there is scope to present a more nuanced discussion of what has been done before.

In this paper the authors present a general proposal for the description of the entanglement entropies of non-Hermitian quantum systems. Their proposal is distinct from some of the proposals that have come before and so provides a new viewpoint on the generic question of how to define consistent measures of entanglement in non-Hermitian quantum systems.

In my view the paper is of good quality and provides a good numerical analysis of the formulae that are proposed, by considering several interesting models. Overall I think the work should be published in SciPost.

However, I do think that there should be a slightly more balanced discussion of contributions by other groups, as I feel that, perhaps inadvertently, those proposals are presented in an unfairly negative light.

I also think that the authors do not really address some of the issues their proposal has, which are still rather fundamental questions about what an entanglement entropy should be. Let me summarise my main points below:

1) In the introduction the authors write "The negative entanglement entropy seems problematic because the reduced density matrix is positive semi-definite which cannot give rise to a negative value of the entanglement entropy. "

I think this sentence is a bit unclear. The point is that by defining the reduced density matrix as the authors do in the first line of Section 2, that is in terms of right and left eigenvectors which can be distinct for non-hermitian systems, automatically such density matrix is not positive-definite in general (as the authors say, it can even have complex eigenvalues). So, I think the correct statement would be that "Defining a reduced density matrix which involves both right and left eigenvectors one naturally finds an entropy which is no longer guaranteed to be positive (or even real!)"

2) From the sentence quoted in 1) it sounds as if the authors recognise that getting a negative entropy is problematic. Then they write that
"To reconcile this issue, Refs. [17–19] suggest that the entanglement entropy is still positive and the true central charge is replaced by an effective central charge ceff which is positive [17]." and this is correct. In these papers, a different reduced density matrix is chosen so that the entropy is again guaranteed to be positive.

3) The next sentence in the introduction says "However, a proper redefinition of the entanglement measures should be considered in the non-Hermitian systems, and the correct properties such as the negative central charges can be obtained."

I find this sentence a bit inconsistence with what came earlier. First of all, in point 1) the authors seemed to recognise that there is something a bit unphysical about having negative entropies. In this sentence however they now say that such negative entropies are the "proper" entropies obtained from a "proper definition", which also suggests the definitions of previous works are all "not proper".

In the context of the entanglement of non-hermitian quantum systems, I think that it is not clear, even after the present work, what a "proper definition" of entanglement should be and in fact the existing proposals in the literature precisely differ in what they consider to be a "proper definition". Whereas the works [17]-[19] aim for a measure that is positive-definite and that still encodes universal information about the CFT, the authors of [21] instead argue that the "proper definition" of the entanglement entropy is the one following from the reduced density matrix in the first sentence of Section 2 of this paper. As a consequence their entropies are often negative.

I would argue that having negative entropies poses at least a question of interpretation, given the usual probabilistic meaning of entropy, so it would be worth mentioning this.

Following my report above I recommend the following:

1) In the discussion of previous contributions I suggest the authors mention that previous proposals were aiming for measures of entanglement with different properties from what the authors prioritise in this paper. While the works [17-19] defined a measure that is guaranteed to be positive-definite for all non-unitary CFTs, the authors of [21] proposed a measure that could be negative but which was based on the "true" reduced density matrix of the non-hermitian system. The current proposal is an addition to this body of work. It is a new measure which guarantees scaling with the true central charge, even when it is negative.

2) I think it should be mentioned somewhere that, while this definition of entanglement entropy, leads to quantities that can be measured in numerical experiments, so it is numerically useful, it still raises the usual issue of interpretation. What is the physical meaning of a negative entropy?

3) The abstract of the paper strongly suggests that this paper is the first attempt at dealing with non-hermitian systems. I suggest that the authors add a sentence in the abstract right after "Here, we propose a natural extension of entanglement and Rényi entropies to non-Hermitian quantum systems." which says something like "There have been other proposals for the computation of these quantities, which are distinct from what is proposed in the current paper."

4) In the first paragraph of page 3 the authors write "all the proposed entanglement measures in non-Hermitian systems are restricted to their specific models". I agree that what the authors propose in this paper is more general than previous treatments, however those treatments were also quite generic. I would replace "specific models" by "non-hermitian conformal field theories" which is what [17-19] and [21] really looked at.

5) One additional comment is that even if the proposal of [17] might look quite different from the authors do here, it is actually easy to see from [17] how one could get an entropy that scales with c instead of ceff. You would just need to set Delta=0 in equation (11) which means choosing not the field with smallest dimension of the CFT by the identity field. That would immediately produce scaling with c.

6) Finally, a questions about the main equations in the paper (1). I am actually a bit confused about these definitions, so maybe the authors can clarify this. They say that the eigenvalues w can be complex in general and so can be written in the Gauss form as modulus times a phase. Given that, it seems to me that the definitions (1) can produce complex results. Isn't that the case, since there is always a factor w which does not appear with a modulus? And is this a desired feature of the proposal? Or is there some assumption that the eigenvalues appear in complex conjugated pairs and so their imaginary parts will cancel in the sum? I am asking because the numerics all seem to produce negative entropies but never complex ones.

7) One additional comment that the authors might want to consider is that complex eigenvalues do not only occur for open quantum systems as a consequence of gain and loss (which seem to be most of the examples considered here). One can have complex eigenvalues even in closed systems. A well-known example is the complex periodic Ising chain that was studied by von Gehlen. https://iopscience.iop.org/article/10.1088/0305-4470/24/22/021