REVIEW
Towards measuring Entanglement Entropies in Many Body Systems
Statistical Mechanics · Submitted 28 Feb 2006 · 25 references
Pith's verdict: Correct and useful core inequality, but 'in general' overclaims; needs a refocused abstract and a cleaned-up Eq. (8).
T0 · AI review
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Referee report
The paper
The paper claims
For pure states, measuring a conserved observable gives a lower bound on entanglement entropy.
The reading
What carries the argument
The central object is the class $L$ of observables of the form $\hat O = \hat O_A\otimes I + I\otimes \hat O_B$ for which the state of interest is an eigenstate. The load-bearing identity is $S_E = S[\hat O_A] - \sum_\alpha P_\alpha \, \mathrm{tr}\,\rho_\alpha \log \rho_\alpha \ge S[\hat O_A]$, which connects the von Neumann entropy of the reduced density matrix to the Shannon entropy of the measurement outcomes of $\hat O_A$. This identity carries the argument because it converts an abstract entanglement quantifier into a distribution that can in principle be measured experimentally.
Extended reading notes
Core claim
The central claim is the identity in Eq. (1): for a pure bipartite state that is an eigenstate of $\hat O_A\otimes I + I\otimes \hat O_B$, the entanglement entropy decomposes as $S_E = S[\hat O_A] - \sum_\alpha P_\alpha \, \mathrm{tr}\,\rho_\alpha \log \rho_\alpha \ge S[\hat O_A]$, where $P_\alpha$ are the probabilities of the outcomes of $\hat O_A$ and $\rho_\alpha$ are the conditional reduced states. Therefore the Shannon (measurement) entropy $S[\hat O_A]$ of the symmetry observable always lower-bounds the entanglement entropy, and equality holds exactly when every $\rho_\alpha$ is pure, for instance when the eigenvalue $\alpha$ is non-degenerate. Moreover, the paper shows that $S_E = \max_{O\in L} S[\hat O_A]$, so the entanglement entropy is the maximum measurement entropy over the class $L$ of observables for which the state is an eigenstate. This relation is then used to obtain the entropy in examples including a BEC state, a two-mode squeezed state, the Dicke model, noninteracting fermions, and the strongly disordered XXZ spin chain.
Load-bearing premise
The method assumes the physical system has a conserved or symmetry observable of the form $\hat O_A\otimes I + I\otimes \hat O_B$ that is compatible with the state's Schmidt decomposition, and that this observable can be measured; the authors note in footnote 9 that not every Hamiltonian admits such an observable.
Editorial extensions
If this is right
- In any pure state with a conserved observable of the form $\hat O_A\otimes I + I\otimes \hat O_B$ and non-degenerate Schmidt coefficients, repeated measurement of $\hat O_A$ directly yields the entanglement entropy as the Shannon entropy of the outcome distribution.
- For noninteracting fermions, the entanglement entropy of a spatial region can be expressed as a convergent series in the cumulants of the particle number in that region, with universal coefficients that are independent of the fermionic eigenmodes.
- In the random singlet phase of the disordered XXZ chain, the entanglement entropy of a segment equals four times the variance of the total $S^z$ in that segment, so measuring spin fluctuations gives the entropy directly.
- The measurement entropy of a symmetry observable provides a general lower bound that can be used even when the system is not exactly solvable, as long as a suitable conserved or symmetry observable exists.
- For extensive observables with Gaussian outcome distributions, the measurement entropy grows at most logarithmically with the variance, so the variance itself often serves as a useful entanglement proxy.
Reading between the lines
- The identity suggests a practical experimental protocol: in cold-atom systems with a conserved particle number, the histogram of atom counts in a subregion could directly estimate the entanglement entropy without full state tomography; the authors sketch this for fermions in a trap but do not develop the protocol in detail.
- A similar logic might extend to mixed states at finite temperature, where the measured entropy would combine thermal and entanglement contributions; the paper notes this limitation but leaves the extrapolation to $T=0$ as future work.
- The universal cumulant coefficients for free fermions imply that the same data-analysis procedure could be applied across different noninteracting systems, so a single measurement protocol might work for trapped fermions, mesoscopic conductors, and optical lattices.
- The observed failure of the variance-entropy inequality in some resonating valence bond states with strong ferromagnetic correlations suggests that variance-based entanglement estimation requires conditions beyond mere conservation of the observable.
Original abstract
We explore the relation between entanglement entropy of quantum many body systems and the distribution of corresponding, properly selected, observables. Such a relation is necessary to actually measure the entanglement entropy. We show that in general, the Shannon entropy of the probability distribution of certain symmetry observables gives a lower bound to the entropy. In some cases this bound is saturated and directly gives the entropy. We also show other cases in which the probability distribution contains enough information to extract the entropy: we show how this is done in several examples including BEC wave functions, the Dicke model, XY spin chain and chains with strong randomness.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We explore the relation between entanglement entropy of quantum many body systems and the distribution of corresponding, properly selected, observables. Such a relation is necessary to actually measure the entanglement entropy. We show that in general, the Shannon entropy of the probability distribution of certain symmetry observables gives a lower bound to the entropy. In some cases this bound is saturated and directly gives the entropy. We also show other cases in which the probability distribution contains enough information to extract the entropy: we show how this is done in several examples including BEC wave functions, the Dicke model, XY spin chain and chains with strong randomness.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We explore the relation between entanglement entropy of quantum many body systems and the distribution of corresponding, properly selected, observables. | contribution 0.87 | 0.62 | 0.09 | asserted 0.59 |
| s1 | Such a relation is necessary to actually measure the entanglement entropy. | background 0.68 | 0.07 | 0.07 | asserted 1.00 |
| s2 | We show that in general, the Shannon entropy of the probability distribution of certain symmetry observables gives a lower bound to the entropy. | result 0.99 | 0.90 | 0.51 | asserted 1.00 |
| s3 | In some cases this bound is saturated and directly gives the entropy. | result 0.98 | 0.83 | 0.16 | asserted 0.83 |
| s4 | We also show other cases in which the probability distribution contains enough information to extract the entropy: we show how this is done in several examples including BEC wave functions, the Dicke model, XY spin chain and chains with strong randomness. | contribution 0.75 | 0.94 | 0.25 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
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References
How the literature uses it
4 arXiv papers cite this in 6 sentences
Papers citing it · 3 sentences without a confident label
Cites it as background 3 papers
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In the context of condensed matter, even cumulants of conserved charges—in particular the variance—have received considerable attention [ref] .
Full-counting statistics of corner charge fluctuations · Introduction · 0.98
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An interesting proposal relates entanglement in a bipartitioned system (between the subsystem $A$ and its complement) to fluctuations of a conserved $U(1)$ charge, such as the particle number, across the partition boundary [ref] , for systems that can be mapped to non-interacting fermions.
Aspects of capacity of entanglement · Bipartite particle fluctuations and entanglement fluctuations · 0.94
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We also note that in the quantum context, fluctuations have been used to characterize fractional charge [ref] , as well as to characterize the many-body entanglement of quantum states [ref] .
Random walk with horizontal and cyclic currents · Introduction · 0.94
Discussion
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