This is an expanded written version of a plenary talk delivered at ICMP 2021. We describe some rigorous results in quantum field theory that have been obtained in recent years, with particular emphasis on those results on relative entropies in the setting of conformal field theory. These results are motivated in part by recent work of physicists which, however, depends on heuristic arguments—such as introducing cutoff and using path integrals and replica trick—which are hard to justify mathematically. Our main technical tools are from the theory of operator algebras, such as modular theory and the theory of subfactors. A discussion about open problems is presented at the end the paper.
In the last few years, there has been an enormous amount of work by physicists concerning entanglement entropies in quantum field theory (QFT) motivated by the connections with condensed matter physics, black holes, etc.; see the references in Ref. 1 for a partial list of references. See Refs. 2–10 for a partial list of recent mathematical work.
For a nice introduction to various aspects of entropy, we refer the reader to Chaps. 5 and 6 of Ref. 11. However, some very basic mathematical questions remain open. For example, most of the entropies computed in the physics literature are infinite, so the singularity structures, and sometimes the cutoff independent quantities, are of most interest. Often, the mutual information is argued to be finite based on heuristic physical arguments, and one can derive the singularities of the entropies from the mutual information by taking singular limits. However, it is not clear that such mutual information, which is well defined as a special case of Araki’s relative entropy (cf. Refs. 12 and 13), is indeed finite.
In Ref. 4, we begin to address some of these fundamental mathematical questions motivated by the physicists’ work on entropy. The cutoff independent quantity, i.e., relative entropy, in particular mutual information, considered in our paper can be computed explicitly in many cases and satisfies many conditions, but not all, proposed by physicists. Our work in Ref. 4 is strongly motivated by Edward Witten’s questions, in particular his question to make physicists’ entropy computations rigorous. In Ref. 4, we focus on the chiral conformal field theory (CFT) in two dimensions, where the results we obtained were most explicit and had interesting connections to subfactor theory, even though some of our results did not depend on conformal symmetries and applied to more general QFT (cf. Ref. 10 for some results on massive free fermionic theories). The main results of Ref. 4 are as follows:
Exact computation of the mutual information (through the relative entropy as defined by Araki for general states on von Neumann algebras) for free fermions. Note that this was not even known to be finite; for example, the main quantity defined in Ref. 1 is smaller. Our proof used Lieb’s convexity and the theory of singular integrals; to the best of our knowledge, this and the related cases were the first time that such relative entropies were computed in a mathematically rigorous way. The results verified earlier computations by physicists based on (cutoff dependent) heuristic arguments, such as those found in Refs. 14 and 15.In particular, for the free chiral net associated with r fermions, and two intervals A = (a1, b1), B = (a2, b2) of the real line, where b1 < a2, the mutual information associated with A, B iswhere is the cross ratio of A, B, 0 < η < 1 (cf. Theorem 3.18 of Ref. 4).
It follows from (1) and the monotonicity of relative entropy that any chiral CFT in two dimensions that embeds into free fermions, and their finite index extensions, verified most (but not all) of the conditions discussed, for example, in Ref. 14. This included a large family (in fact all known examples, cf. p. 1480 of Ref. 10) of chiral CFTs. Much more can be obtained if the embedding has finite index. In this case, we also verified a proposal of Ref. 16 about an entropy formula related to a derivation of the c theorem. Our theorem also connected the relative entropy and index of subfactors in an interesting and unexpected way. This connection was different but related to the Pimsner–Popa result that connected the Connes–Stormer entropy to index (cf. Ref. 17).
There was also one bit of surprise: It was usually postulated that the mutual information of a pure state such as the vacuum state for complementary regions should be the same. However, in the chiral case, this is not true, and the violation is measured by the global dimension of the chiral CFT. The violation, which is in some sense proportional to the logarithm of the global index, also turns out to be what is called a topological entanglement entropy. In fact, there is a precise formula relating such violation to the relative entropy defined by conditional expectation to disjoint interval nets, cf. Ref. 6.
The rest of this article, with the exception of Secs. VI B and IX, is devoted to explaining some of the results mentioned above.4 In Sec. VI B, we consider a special case of the general results obtained in Ref. 6. We hope that this will give the reader some idea about the motivation for these results and how they are obtained. The reader is encouraged to referrer to Refs. 4 and 6 for detailed proofs. In Sec. IX, we propose a few open problems.
Even though in Ref. 4 we focus on relative entropy that is cutoff independent, we refer the reader to Ref. 5 for results on entropy for type I factors from split property. Our main technical tools are from the theory of operator algebras, such as modular theory and the theory of subfactors (cf. Refs. 18 and 19).
A. Entropy and relative entropy
The von Neumann entropy is a quantity that is associated with a density matrix ρ on a Hilbert space by
It can be viewed as a measure of the lack of information about a system to which one has ascribed the state. This interpretation is in accord, for instance, with the facts that S(ρ) ≥ 0 and that a pure state ρ = |Ψ⟩⟨Ψ| has vanishing von Neumann entropy. Note that von Neumann entropy is nonlinear in the state and in general not easy to compute even in finite dimensional cases, such as those in the statistical mechanical model on finite lattices.
A related notion is that of relative entropy. It is defined for two density matrices ρ, ρ′ by
Like S(ρ), S(ρ, ρ′) is nonnegative, and it can be infinite.
A generalization of the relative entropy in the context of von Neumann algebras of arbitrary type was found by Araki and is formulated using modular theory. Given two faithful, normal states ω, ω′ on a von Neumann algebra in standard form, we choose the vector representatives in the natural cone , called |Ω⟩, |Ω′⟩. The antilinear operator Sω,ω′a|Ω′⟩ = a*|Ω⟩, , is closable and one considers again the polar decomposition of its closure . Here, J is the modular conjugation of associated with and is the relative modular operator with respect to |Ω⟩, |Ω′⟩. Of course, if ω = ω′, then Δω = Δω,ω′ is the usual modular operator or modular Hamiltonian in physics literature.
The relative entropy with respect to ω and ω′ is defined by
S is extended to positive linear functionals that are not necessarily normalized by the formula S(λω, λ′ω′) = λS(ω, ω′) + λ log(λ/λ′), where λ, λ′ > 0 and ω, ω′ are normalized. If ω′ is not normal, one sets S(ω, ω′) = ∞.
For a type I algebra , states ω, ω′ correspond to density matrices ρ, ρ′. The square root of the relative modular operator corresponds to ρ1/2 ⊗ ρ′−1/2 in the standard representation of on ; that is, is identified with the Hilbert–Schmidt operators with the left/right multiplication of /. In this representation, ω corresponds to the vector state , and the abstract definition of relative entropy becomes
As another example, let us consider a bipartite system with Hilbert space and observable algebra . A normal state ωAB on corresponds to a density matrix ρAB. One calls the “reduced density matrix,” which defines a state ωA on (and similarly for system B). The mutual information is given in our example system by
For a tripartite system with Hilbert space and observable algebra , we have the following strong subadditivity (cf. Ref. 20):
This inequality was originally proved in Ref. 20, but it follows immediately from the monotonicity of relative entropy that will be described in the following.
In general, it is desirable to have a formula for S(ω, ω′) directly in terms of states. This is provided by Kosaki (cf. Ref. 21),
where xt is a step function valued in M that is equal to 0 when t is sufficiently large. Many properties of relative entropies follow easily from Kosaki’s formula. An example is as follows: Let ω and ϕ be two normal states on a von Neumann algebra M, and denote by ω1 and ϕ1 the restrictions of ω and ϕ to a von Neumann subalgebra M1 ⊂ M respectively. Then, S(ω1, ϕ1) ≤ S(ω, ϕ). As another example, let be Mi an increasing net of von Neumann subalgebras of M with the property . Then, S(ω1 ↾ Mi, ω2 ↾ Mi) converges to S(ω1, ω2), where ω1, ω2 are two normal states on M.
Finally, Let ω and ω1 be two normal states on a von Neumann algebra M. If ω1 ≥ μω, then S(ω, ω1) ≤ ln μ−1. Following is a property of relative entropies (cf. Theorem 5.15 of Ref. 11) that does not follow directly from Kosaki’s formula: Let M be a von Neumann algebra and M1 a von Neumann subalgebra of M. Assume that there exists a faithful normal conditional expectation E of M onto M1. If ψ and ω are states of M1 and M, respectively, then S(ω, ψ · E) = S(ω ↾ M1, ψ) + S(ω, ω · E).
For type III factors, the von Neumann entropy is always infinite, but we shall see that in many cases, mutual information is finite. By taking singular limits, we can also explore the singularities of the von Neumann entropy from mutual information, which is important from physicists’ point of view. The formal properties of von Neumann entropies are useful in proving properties of mutual information as we shall see in Sec. IV.
B. Graded nets and subnets
We shall denote by Möb the Möbius group, which is isomorphic to and acts naturally and faithfully on the circle S1.
By an interval of S1, we mean, as usual, a non-empty, non-dense, open, connected subset of S1 and we denote by the set of all intervals. If , then also where I′ is the interior of the complement of I. Intervals are disjoint if they have disjoint closure. We will denote by the set that consists of disjoint union of intervals.
This section is contained in Ref. 22 and is an adaption of Doplicher-Haag-Roberts (DHR) analysis to chiral CFT, which is most suitable for our purposes. We refer the reader to Ref. 22 for more details and proofs.
A Möbius covariant net of von Neumann algebras on the intervals of S1 is a map
from to the von Neumann algebras on a Hilbert space that verifies the following:
- Isotony: If I1, I2 are intervals and I1 ⊂ I2, then
- Möbius covariance: There is a nontrivial unitary representation U of G [the universal covering group of PSL(2, R)] on such thatThe group PSL(2, R) is identified with the Möbius group of S1, i.e., the group of conformal transformations on the complex plane that preserve the orientation and leave the unit circle globally invariant. Therefore, G has a natural action on S1.
Positivity of the energy: The generator of the rotation subgroup U(R)(·) is positive. Here, R(ϑ) denotes the (lifting to G of the) rotation by an angle ϑ.
- Locality: A -grading on is an involutive automorphism g = AdΓ of , such that Γ2 = 1, for all I. There exists a -grading g of such that, if I1 and I2 are disjoint intervals,Here, [x, y] is the graded commutator with respect to the grading automorphism g.
Existence of the vacuum: There exists a unit vector Ω (vacuum vector) that is U(G)-invariant and cyclic for .
Uniqueness of the vacuum (or irreducibility): The only U(G)-invariant vectors are the scalar multiples of Ω.
By a conformal net (or diffeomorphism covariant net) , we shall mean a Möbius covariant net such that the following holds:
- Conformal covariance: There exists a projective unitary representation U of Diff(S1) on extending the unitary representation of G such that for all , we havewhere Diff(S1) denotes the group of smooth, positively oriented diffeomorphism of S1 and Diff(I) the subgroup of diffeomorphisms g such that g(z) = z for all z ∈ I′.
we have that the unitary Z fixes Ω and
(twisted locality with respect to Z).
Let be a Möbius covariant Fermi net on S1. Then, Ω is cyclic and separating for each von Neumann algebra , .
If , we shall denote by ΛI the one parameter subgroup of Möb of “dilation associated with I.”
Let and ΔI, JI be the modular operator and the modular conjugation of . Then, we have the following:
- U extends to an (anti-)unitary representation of determined byacting covariantly on , i.e.,Here, rI: S1 → S1 is the reflection mapping I onto I′.
Part (1) of the above theorem says that the modular Hamiltonian is the boost generator or, as mathematicians would say, that the modular automorphism group is geometric, and it plays an important role in recent work on entropies in physics literature.
Now, let G be a simply connected compact Lie group. Then, the vacuum positive energy representation of the loop group LG at level k gives rise to an irreducible local net denoted by (cf. Refs. 23 and 24). Every irreducible positive energy representation of the loop group LG at level k gives rise to an irreducible covariant representation of . When no confusion arises, we will write simply as Gk. These CFTs are what is also called Wess–Zumino–Witten CFTs with gauge group G and are important building blocks of rational CFT.
Assume is a Möbius covariant net. A Möbius covariant representation π of is a family of representations πI of the von Neumann algebras , , on a Hilbert space and a unitary representation Uπ of the covering group G of PSL(2, R), with positive energy, i.e., the generator of the rotation unitary subgroup has positive generator, such that the following properties hold:
A unitary equivalence class of Möbius covariant representations of is called superselection sector.
Denote by the set of unions of disjoint 2 elements in . Let be an irreducible conformal net. For , let I3 ∪ I4 be the interior of the complement of I1 ∪ I2 in S1, where I3, I4 are disjoint intervals. Let
Note that . Recall that a net is split if is naturally isomorphic to the tensor product of von Neumann algebras for any disjoint intervals : This means the map extends to an isomorphism of the two von Neumann algebras. is strongly additive if where I1 ∪ I2 is obtained by removing an interior point from I. We note that a conformal net is automatically split (cf. Ref. 25).
Recall that a net is split if is naturally isomorphic to the tensor product of von Neumann algebras for any disjoint intervals : This means the map extends to an isomorphism of the two von Neumann algebras. Any conformal net is split. Since are type III factors, there exists a unitary operator such that Therefore, is a type I factor such that . By choosing an increasing sequence of such that , we can get an increasing sequence of type I factors Bn such that ∪nBn is strongly dense in . This allows us to approximate relative entropy by restricting to these type I approximations. This is one reason that many formal calculations in physics literature, which only applies to type I factors, under certain conditions turn out to be true also for local algebras in QFT, which is always type III. These conditions are such that the formal computations in type I case (also in the case of suitable cutoff such as in the finite lattice approximation) converge to Araki’s relative entropy, and in general, they are rather nontrivial to prove (in the physics literature, this is called free of ultraviolet divergences).
is said to be completely rational (cf. Ref. 26), or μ-rational, if the index is finite for some . The value of the index is denoted by and is called the μ-index of . is holomorphic if . is also known as Topological Entanglement Entropy by Kitaev and Preskill in Ref. 27.
The following theorem is proved in Ref. 28:
Let be an irreducible conformal net and let G be a finite group acting properly on . Suppose that is completely rational. Then,
, the fixed point conformal net under the action of G, is completely rational and , and
there are only a finite number of irreducible covariant representations of and they give rise to a unitary modular category.
Let be a graded Möbius net. By a Möbius subnet, we shall mean a map
that associates to each interval a von Neumann subalgebra of , which is isotonic,
and Möbius covariant with respect to the representation U, i.e.,
for all g ∈ Möb and , and we also require that AdΓ preserves as a set.
The case when has finite index will be most interesting. An example is the following lemma.
If is a Möbius subnet such that is finite and . Then, .
III. MUTUAL INFORMATION IN THE CASE OF FREE FERMIONS
Let H denote the Hilbert space of square-summable -valued functions on a circle. The group LUr of smooth maps S1 → Ur, with Ur being the unitary group on , acts on H multiplication operators.
Let us decompose H = H+ ⊕ H−, where
We denote by p the Hardy projection from H onto H+.
The basic representation of LUr is the representation on Fermionic Fock space Fp = Λ(pH) ⊗ Λ((1 − p)H)*.
Such a representation gives rise to a graded net as follows: Denote by , the von Neumann algebra generated by c(ξ)′s, with . Here, c(ξ) = a(ξ) + a(ξ)* and a(ξ) is the creation operator. Let Z : Fp → Fp be the Klein transformation given by multiplication by 1 on even forms and by i on odd forms. is a graded Möbius covariant net, and will be called the net of r free fermions. is strongly additive and .
Fix and let I1, I2 be disjoint, that is, and I = I1 ∪ I2. The mutual information we will compute is S(ω, ω1 ⊗2ω2). Here, ω1 ⊗2ω2 denotes the graded tensor product state on (cf. Definition 3.3 of Ref. 4). ω on is quasi-free state as studied by Araki. To describe this state, it is convenient to use Cayley transform V(x) = (x − i)/(x + i), which carries the (one point compactification of the) real line onto the circle and the upper half plane onto the unit disk. It induces a unitary map
of onto . The operator U carries the Hardy space on the circle onto the Hardy space on the real line. We will use the Cayley transform to identify intervals on the circle with one point removed to intervals on the real line. Under the unitary transformation above, the Hardy projection on is transformed to the Hardy projection on given by
where the singular integral is (proportional to) the Hilbert transform.
We write the kernel of the above integral transformation as C,
The quasi-free state ω is determined by
Slightly abusing our notations, we will identify P with its kernel C and simply write
C will be called covariance operator.
A. Computation of mutual information in finite dimensional case
Choose finite dimensional subspaces Hi of and denote by the corresponding finite dimensional factors of dimensions generated by a(f), f ∈ Hi. Let ρ12, ρ1, ρ2 be the density matrices of the restriction of ω to CAR(H1) ⊗2CAR(H2), CAR(H1), CAR(H2), respectively, and let ρ1 ⊗2ρ2 be that of the restriction of ω1 ⊗2ω2 to CAR(H1) ⊗2CAR(H2). When working carefully with graded tensor product, we have the analog of (3) in this graded local context (cf. Proposition 3.4 of Ref. 4),
This is the formula for the case of mutual information in type I factor.
Now, we turn to the computation of von Neumann entropy S(ρ1). Let p1 be the projection onto the finite dimensional subspace H1 of . ρ1 on CAR(H1) is a quasi-free state given by the covariance operator . According to Araki,
Let Pi be projections from onto , and Ci = PiCPi, i = 1, 2.
and be the same as in the definition of σC with C replaced by Cp = pCp, if p is a projection commuting with P1.
Denote by p the projection from onto H1 ⊕ H2. We have the following:
It is clear that converges strongly to σC as P converges to identity. To compute our mutual information, we like to show that this convergence is actually in trace. Unfortunately, this is much harder. Instead, we explore additional subtle properties of such operators.
B. Inequality from operator convexity
(1) For all operator convex functions f on , and all orthogonal projections p, we have pf(pAp)p ≤ pf(A)p for every self-adjoint operator A; (2) f(t) = t ln(t) is operator convex.
Part (1) of the above theorem is known as the Sherman–Davis inequality. It is instructive to review the idea of the proof of (1): Consider the self-adjoint unitary operator Up = 2p − I; by operator convexity, we have
Now notice that
where Ap = pAp and the inequality follows.
, where p → 1 strongly. The first identity follows from the Martingale property of relative entropy. To prove the inequality, we use the fact that x ln x is operator convex, and so P1C ln CP1 ≥ C1 ln C1, and similarly with C replaced by 1 − C. It follows that σ ≥ 0, σp ≥ 0. Since σp goes to σ strongly as p → 1 strongly, the inequality follows. We shall prove later that the inequality in the above lemma is actually an equality. It would follow if one can show that goes to σC in tracial norm. This is not so easy, and we note that is not trace class. This is done by applying Lieb’s joint convexity. We give a sketch of the proof and refer the reader to Secs. 3.5 and 3.6 of Ref. 4 for more details.
C. Reversed inequality from Lieb’s joint convexity
We begin with the following Lieb’s Concavity theorem (cf. Ref. 29):
(1) For all m × n matrices K, and all 0 ≤ t ≤ 1, the real-valued map given by (A, B) → Tr(K*A1−tKB) is concave where A, B are nonnegative m × m and n × n matrices, respectively.
- If A ≥ 0, B ≥ 0 and K is trace class, then,is jointly concave.
- If A ≥ ϵI, B ≥ ϵI, ϵ > 0, and K is trace class, then,is jointly convex.
To prove (3), we note that
and (3) follows from (2).
The idea of the proof is to apply Lieb’s joint convexity to A, B and unitary Up = 2P − I, with f(A, B, K) = Tr(K*A ln AK − K*AK ln B), where K is a finite rank projection, and then let K go to identity strongly.
To perform explicit computation, we also need an explicit formula for the kernel of the resolvent of C. This is related to Riemann–Hilbert problem.
D. Riemann–Hilbert problem
Recall , where I1, I2 are disjoint, that is, and I = I1 ∪ I2. We assume that I = (a1, b1) ∪ (a1, b1) ∪ ⋯ ∪ (an, bn) in increasing order. Close up I to a simple contour, and if a function ϕ is defined on I, extend its definition on the contour by simply defining it to be zero on the complement of I. We will still denote by I the simple contour containing I. Consider the following equation:
For simplicity, it will be assumed that a, b are constants and that a2 − b2 ≠ 0. As a new unknown, let us introduce the Cauchy type integral with density ϕ.
Substituting this, we obtain the following equation:
We arrive in this manner at the Riemann–Hilbert problem: to find a function F(z) from a given linear relation between its limiting values on the inside and on the outside of a contour. The solution is known (cf. Ref. 30). Using the known solution to the Riemann–Hilbert problem, the resolvent of C as restriction of an operator on ,
has the following expression:
We shall denote by . Even though both ZI(x) and are singular when x is close to the boundary of its domain, it is crucial that is a smooth function on the closure of .
if x ≠ y and , .
Then, G(t, x, y) is continuous on and
where M is a constant. The kernel for the computation of mutual information is given by
Moreover, we have the following lemma:
The kernel of
The reader is invited to do the computation in (2) of the above lemma. The number comes from using Euler’s famous solution to Basel’s problem: .
We are now ready to put things together to state the result of our computation of mutual information.
If I = (a1, b1) ∪ (a2, b2) ∪ ⋯ ∪ (an, bn) in increasing order, define
Apply Theorem 3.3 and the lemma above, we have the following theorem, which is Theorem 3.18 of Ref. 4:
IV. FORMAL PROPERTIES OF ENTROPY FOR FREE FERMION NETS AND THEIR SUBNETS
In Sec. III D, we use the Cayley transformation to identify punctured circle with real line as a tool to compute relative entropy. Now, we return to a general discussion on the formal properties of entropy, and it is now convenient to be back to intervals on the circle. Let be a disjoint union of intervals on the circle. Explicitly, we write I = (a1, b1) ∪ (a2, b2) ∪ ⋯ ∪ (an, bn) in anticlockwise order on the unit circle. We note that relative entropies are invariant under Möb transformations on the circle.
By Theorem 3.5, we have where A, B are union of disjoint intervals. When no confusion arises, we will simply write as F(A, B).
We can extend the definition mutual information to more general union of disjoint intervals by the following F(A ∪ B, A ∪ C) = F(A, B ∪ C) + F(B, C) − F(A, C) − F(A, B). The following is Theorem 4.1 of Ref. 4:
(1) F(A ∪ B, A ∪ C) ≥ 0; F(A ∪ B, A ∪ C) is continuous from inside.
F(A, B) + F(A, C) + F(A ∪ B, A ∪ C) + F(A∩C, A∩B) = F(B, C) + F(A, B ∪ C) + F(A, B∩C).
There exists a function such that F(A, B) = G(A) + G(B) − G(A ∪ B) − G(A ∩ B). Such G is uniquely determined by its value on connected open intervals.
One can choose in (3) for the r free fermion net , and such a choice determines for I = (a1, b1) ∪ (a2, b2) ∪ ⋯ ∪ (an, bn) on unit circle with anticlockwise order.
F(A ∪ B, A ∪ C) = F(A ∪ B, C) − F(A, C) = F(B, A ∪ C) − F(B, A). In particular, F(A ∪ B, A ∪ C) increases with B, C.
If is a graded subnet, then (1)–(3) is also true for the system of mutual information associated with .
We briefly describe the proof of this theorem. (1) and (5) for free fermions can be checked by using explicit formulas in Theorem 3.5, but here we present general arguments, which will also work for other cases, such as subnets of free fermions, and explain the origins of formulas that are formal manipulations of the von Neumann entropy.
Choose an increasing sequence of finite dimensional factors , invariant under the conjugate action of Γ such that , , and denote by the restrictions of ω and ω1 ⊗2ω2 to , respectively. Let and be the restrictions of ω to and , respectively. Then, we have
To simplify notations, let us write . Then, we have
It follows that
by strong subadditivity of von Neumann entropy, (1) follows and (2) also follow from the limit formula and the fact that F(A, B) is finite by Theorem 3.5.
V. STRUCTURE OF SINGULARITIES IN THE FINITE INDEX CASE
G from (3) in Theorem 4.1 can be thought as a “regularized” version of von Neumann entropy, which is always infinite in our case. From (3) of the above theorem, we see that if we only allow G to be defined on , then G is highly nonunique. Due to the continuity properties of F(A, B), we require that G(A) depends continuously only on the length rA of interval A. In addition, we require that G(A) = G(Ac) for a connected interval, and we set G(∅) = 0. Still, such G is highly nonunique. However, we shall impose further conditions coming from studying the singularities of relative entropy when we allow intervals to approach each other. Let , with |a2ϵ − a2| = ϵ > 0. We shall consider the singular limit when ϵ goes to zero while fixing a1 and C. Let B0 = (a1, a2). We will denote by , i.e., is obtained from B0 ∪ C by adding the point a2: Notice in the process that the number of components decrease by 1. The natural condition turns out to be
for some function P(ϵ) that is independent of B, C. The equation is a condition that connects the value of G for different components; as we shall see, it is a very useful condition. In general, we may take multiple singular limits. Equation (10) allows us to evaluate such limits. Let us consider such an example in detail. Let A = (a2, b2), , , . Let ϵ1 go to 0 first, we find
since the same function P(ϵ1) appears in both and with opposite signs. Then, let ϵ2 go to 0; we get by the same argument that
It is easy to see that the result is independent of the order of taking limits, and this way, we can extend the definition of F(A, B) to any F(A, B) with .
In the case of free fermions, by Theorem 3.5, we have that P(ϵ) = r/6 ln ϵ + o(ϵ), and we have
The following is Theorem 4.2 of Ref. 4:
Assume that a subnet has finite index, then the following holds:
- and this verifies Eq. (3) of Theorem 4.1; moreover,where A, B are two overlapping intervals with cross ratio 0 < ηAB < 1.
- Let B = (a1, a2ϵ), C = (a2, b2), |a2ϵ − a2| = ϵ > 0. Then,as ϵ goes to 0.
(1) Result (1) in the above theorem agrees with the postulates of Casini and Huerta in their discussion of c theorem using relative entropies in Ref. 16 . (2) It is interesting to note that the constant term in (2) of the above theorem seems to be related to the topological entropy discussed, for example, by Kitaev and Preskill (cf. Ref. 27 ), and even with the right factor: In our case, we have an additional factor 1/2 since we are discussing chiral half of CFT.
VI. FAILURE OF DUALITY IS RELATED TO NONTRIVIAL GLOBAL DIMENSION OR TOPOLOGICAL ENTANGLEMENT ENTROPY
By our theorem for the free fermion net , and two intervals A = (a1, b1), B = (a2, b2), where b1 < a2, we have
where is the cross ratio, 0 < η < 1. For simplicity, we denote by .
One checks that , which is in fact equivalent to
Similarly, for with finite index, by Theorem 5.1, is equivalent to
We note that for the free fermion net . However, here we show that with has finite index . By Lemma 2.5,
We note that is a decreasing function of η, and . So, we have
On the other hand, by Theorem 7.1,
It follows that due to the fact that .
A. What is wrong with formal manipulations
Formally, one has F(A, B) = S(A) + S(B) − S(A ∩ B) − S(A ∪ B), and for pure states, we have S(A) = S(Ac), and it follows that F(A, B) = F(Ac, Bc), but the results of Sec. VI show that this is not true (in fact, initially we tried to prove it is true). The reason is because our algebras are not type I; therefore, even though the two algebras associated with A and Ac commute with each other and generate the algebra of all bounded operators on the underlying Hilbert space, these two algebras are in general not each other’s commutant.
Moreover, formula F(A, B) = S(A) + S(B) − S(A ∩ B) − S(A ∪ B) is only true in the sense that , where An is an increasing sequence of type I factors approximating our net localized on A. Even though for pure states, we only have , and we cannot conclude that , and there is no continuity that can help because both S(An) and go to infinity as n goes to ∞.
B. More on duality
Further analysis about relative entropy and global index can be found in Ref. 6.
To give the reader an idea about the results in our paper,6 let us consider a special case of a conformal net that is chain related to a free fermion net as defined on p. 3526 of Ref. 6 or any conformal net associated with an even positive definite lattice as in Ref. 31. Then, for I = I1 ∪ I2, I′ = J1 ∪ J2, we have
where S is the relative entropy, ω is the vacuum state, c is the central charge, is the global index of , is a cross ratio, and is the conditional expectation. Previously, relations among relative entropies, central charge, and global index were given in their asymptotic form. The above relation is an identity. The duality condition as described above holds when the right-hand side is 0.
Since is monotonically increasing with respect to interval I, it follows that
and that it is monotonically increasing in I, a result which is already nontrivial.
The proof of such relations are partially based on the following result from Ref. 6:
The above result suggests that S(ω, ωE) behaves like ln IndE. Note that S(ω, ωE) ≤ ln IndE, and in cases such as those from conformal nets, where E depends on an interval I. Moreover, using the ideas presented in Sec. VII, one can show that as I approximates the whole circle,
So, S(ω, ωEI) recovers ln IndE in a limit and it seems to be a more refined invariant than ln IndE.
VII. COMPUTATION OF LIMIT OF RELATIVE ENTROPY AND ITS RELATION WITH SUBFACTORS
In this section, we determine the exact limit of relative entropies, which are necessary for analyzing the singularity structures of entropies in Theorem 5.1. The goal is to prove the following Theorem 4.4 of Ref. 4:
We remark that a more general result is stated in Proposition 2.37 of Ref. 6, whose proof is based on the same idea.
In the following, we present more details about the proof of the above result following Sec. 4.3.3 of Ref. 4 since the connection to subfactor theory can be seen clearly from the proof.
A. Basic idea from Kosaki’s formula
Recall that ϕn = ω · En. By the Pimsner–Popa inequality, En(x) ≥ λx for any positive , it follows that ϕn ≥ λω; hence,
Denote by ϕn = ω · En. By Kosaki’s formula,
where xt is a step function that is equal to 0 when t is sufficiently large.
To motivate the proof, it is instructive to see how we can get S(ω, λω) = −ln λ, 0 < λ < 1 from Kosaki’s formula. By tracing the proof of this formula, one can see that the path that leads to the approximation to −ln λ is given by the following continuous path:
with such a choice, we have
which tends to −ln λ as k goes to ∞.
This suggests that for the proof of the theorem, we need to choose a path xt, yt such that and are close to and , respectively.
Let be Jones projections for and , respectively. Let P be the projection from the vacuum representation of onto the vacuum representation of . Since Jones projections are unique up to conjugation by unitaries, there is a unitary such that ue1u* = e2. Choose an isometry such that . Note that , and Pe2P = λP. It follows that by our assumption that [Γ, P] = 0.
Since is strongly additive, we can find a sequence of bounded operators such that un → u strongly. Let . Then, e2n → e2 strongly.
For any ϵ > 0, one can find n ≥ 2 and such that
In fact, . This is proved by careful evaluation of states and using subfactor theory.
B. The proof
By Kosaki’s formula,
where xt is a step function equal to 0 when t is sufficiently large. Since we can approximate any continuous function with step functions in the strong topology and vice versa, we can assume that xt is continuous and is equal to 0 when t is sufficiently large. Given ϵ > 0, for fixed , choose e as above, and
We can choose n large enough such that
It follows that
Let k, m go to ∞ and ϵ go to 0, we have proved theorem. ■
VIII. MORE EXAMPLES THAT VERIFY THEOREM 5.1
Take . This is orbifold of U(1)1. Theorem 5.1 applies to the net . Another special case is when k = 1, we can take a further orbifold of U(1)4 corresponding to complex conjugation on U(1) to get a tensor product of two chiral Ising models with central charge . It follows that the chiral Ising model with central charge verifies Theorem 5.1 and, in particular, violates duality.
More generally, we can take any finite subgroup of U(n) that commutes with AdΓ and obtain an orbifold subnet of U(n)1. This provides a large family of examples that verify our theorem.
The second class comes from the following inclusions with finite index:
So, Theorem 5.1 applies to the net . If we take m = n, then since verifies our theorem by the example in the beginning of Sec. VIII, it follows that the net associated with SU(n)n × SU(n)n and, hence, the net associated with SU(n)n also verify our Theorem 5.1.
IX. DISCUSSIONS ON OPEN PROBLEMS
There are many interesting well-defined mathematical problems about entropy in quantum field theory. We select a few problems in this section.
A. Computation problems
While most of the computations of entropy by physicists use path integrals and replica trick, and also depend on cutoff, these computations are hard to justify mathematically. For a mathematically rigorous treatment of the replica trick in statistical mechanical models, we refer the reader to a very interesting book.32 On the other hand, relative entropy is cutoff independent and well defined in quantum field theory and so it will be interesting to compute as many examples as possible. The free fermion case is a rare case where all mutual information for the vacuum state is exactly known (cf. Ref. 33 for an interesting computation in the free boson case). Cardy computed some relative entropies in 3d free theory in Ref. 34 based on operator product expansions, and it will be interesting to justify these computations mathematically. Another interesting computation is that of the entropy for the type I factor in the split case as in Remark 3.9 of Ref. 5.
B. C Theorem and higher dimensional generalizations
In Ref. 16, a very interesting physical derivation of c theorem in two dimensions is presented using the properties of relative entropy. Our results in Sec. IV are partially motivated by this. However, to obtain an interesting c function, one must go beyond the conformal case. However, even in the free massive theory case, it is not clear how to justify the cutoff dependent computations in Ref. 35. The same comment applies to the higher dimensional considerations in Ref. 36. We expect that the results of Sec. 3 of Ref. 10 will be useful in approaching such problems.
Holographic entropy as in the context of AdS/CFT correspondence (cf. Ref. 37 and references therein) has been a driven force for the recent interest in entropy in QFT in general. In the context of AdS/CFT correspondence, there is also the Ryu–Takayanagi (RT) formula relating the entropy of QFT on a d-dimensional spacetime to the area of certain RT surfaces in d+1 dimensional spacetime. The slogan in the physics community seems to be that “Entanglement glues spacetime together” or “Entanglement builds geometry” and “gravity is the hydrodynamics of entanglement.”
Unfortunately, AdS/CFT correspondence is not on a solid mathematical ground at present. A toy model to consider may be the three dimensional AdS and two dimensional CFT case. For an instance, if we take a symmetric orbifold of a conformal net , i.e., take the orbifold of tensor with itself n times and take the orbifold to be the permutation group on n letters, then when n is large, this theory may have a holographic dual (cf. Ref. 38 and references therein). Note that such an orbifold theory is on solid mathematical ground thanks to Theorem 2.4.
Now, if one can manage to compute the mutual information of vacuum in these orbifolds, then one can try to extract leading large n terms and compare to some length of geodesic in the three dimensional AdS metric as predicted in the RT formula as described for an example on p. 178 of Ref. 37.
It will be desirable to see geometry from entanglement (cf. Ref. 39) in a rigorous mathematical framework. A natural mathematical framework may be Connes’ Noncommutative Geometry (cf. Ref. 40). So far, relative entropy does not seem to appear naturally in the usual noncommutative geometry framework, but see Ref. 41 for a recent result on von Neumann entropy in the free fermion case.
I would like to thank R. Longo for helpful discussions. This work was partially supported by NSF Grant No. DMS-1764157.
Conflict of Interest
The author has no conflicts to disclose.
Feng Xu: Writing – original draft (equal).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.