CConstraining Quantum Fields using Modular Theory
Nima Lashkari
School of Natural Sciences, Institute for Advanced Study,Einstein Drive, Princeton, NJ, 08540, USA
Abstract
Tomita-Takesaki modular theory provides a set of algebraic tools in quantumfield theory that is suitable for the study of the information-theoretic propertiesof states. For every open set in spacetime and choice of two states, the mod-ular theory defines a positive operator known as the relative modular operatorthat decreases monotonically under restriction to subregions. We study the con-sequences of this operator monotonicity inequality for correlation functions inquantum field theory. We do so by constructing a one-parameter R´enyi family ofinformation-theoretic measures from the relative modular operator that inheritmonotonicity by construction and reduce to correlation functions in special cases.In the case of finite quantum systems, this R´enyi family is the sandwiched R´enyidivergence and we obtain a new simple proof of its monotonicity. Its monotonic-ity implies a class of constraints on correlation functions in quantum field theory,only a small set of which were known to us. We explore these inequalities for freefields and conformal field theory. We conjecture that the second null derivativeof R´enyi divergence is non-negative which is a generalization of the quantum nullenergy condition to the R´enyi family. lashkari@ias . edu a r X i v : . [ h e p - t h ] N ov Introduction
In recent years, the tools and techniques of quantum information theory have foundmany applications in many-body local quantum systems from condensed matter tofield theory and gravity. In relativistic quantum field theory, studying the constraintsof unitarity and causality on information-theoretic measures has led to the discoveryof new inequalities that hold universally in all theories. The strong sub-additivity ofentanglement entropy has been used to derive c-theorems in various dimensions [1, 2, 3].Strong sub-additivity is a special case of the monotonicity of relative entropy whichhas been used to prove the average null energy condition (ANEC) in flat space andthe generalized second law in curved spacetimes [4, 5]. In this work, we explore a largeclass of monotonicity constraints of information-theoretic measures in quantum fieldtheory.In quantum field theory, there are no density matrices associated to finite regionsof spacetime. This is an obstacle on the way of applying the entanglement theory toquantum fields which reflects itself as ultraviolet divergences that appear in entangle-ment entropy. In some cases, one can use scaling arguments to isolate universal piecesin the calculations. These ultraviolet divergences point to the fact that entanglementin quantum field theory is a property of the algebra and not the states [6].A more satisfactory approach is provided by the Tomita-Takesaki modular theorythat studied the algebra of local operators in quantum field theory; see [7, 6] for areview. Modular theory connects with information theory by defining a positive opera-tor known as the relative modular operator. Araki realized that the expectation valueof the logarithm of this operator generalizes the density matrix definition of relativeentropy to arbitrary quantum systems including quantum field theory [8]. Relativeentropy is a distance measure central to quantum information theory from which mostother entanglement measures can be derived; see [9] for a review.In modular theory, the relative modular operator ∆ A Ω | Φ is defined for every space-time region A and choice of two states | Ω (cid:105) and | Φ (cid:105) . The locality of quantum fieldtheory imposes a strong constraint on how the relative modular operator changes asone changes the region A . Most importantly for us, the relative modular operator ismonotonic, that is to say for any region A ⊂ B we have∆ A Ω | Φ ≥ ∆ B Ω | Φ (1)as an operator inequality. It implies the monotonicity of relative entropy, however, asan operator inequality, it is much stronger. In this work, we study the consequences ofthis inequality in quantum field theory.In section 2, we review the modular theory emphasizing those properties of therelative modular operator that play an important role in our work. In section 3,we introduce the Petz divergence as an information-theoretic measure that monoton-ically increases as one enlarges the region, and show that it satisfies all the desirableinformation-theoretic properties one expects from a R´enyi generalization of relativeentropy. In section 4, motivated by the work of Araki and Masuda in [10] we define analternative R´enyi family for relative entropy and show that it is the generalization of1he sandwiched R´enyi divergence to quantum field theory. Similar discussions appearin [11, 12, 13]. Our approach has the advantage that we do not need to resort to theproofs in [14, 15, 13] to show the monotonicity of sandwiched R´enyi divergences. Itis implied by the monotonicity of relative modular operator. For completeness, wereview of the proof of the monotonicity of relative modular operator with an emphasison finite dimensional Hilbert spaces in section 5.In [16] a Euclidean path-integral was constructed that computes sandwiched R´enyidivergences, and it was shown that for certain values of the R´enyi index they are givenby correlation functions. Section 6 starts with this path-integral construction andexplores the consequences of monotonicity for these correlation functions. We find newinequalities that the 2 n -point correlation functions have to satisfy as a consequenceof monotonicity. They can be interpreted as the non-negativity of certain off-diagonalelements of null momentum. See the inequality in (84) for a concrete example.Studying the dependence of these 2 n -point correlation functions on the region sizewe are naturally led to the conjecture that the second null (or spacelike) derivativesof the sandwiched R´enyi divergence is non-negative. This can be thought of as ageneralization of quantum null energy condition to the R´enyi family. We provideevidence for the conjecture using reflection positivity of Euclidean correlators. Weperform sample calculations of the sandwiched R´enyi divergence and its derivatives inshape deformations in free two-dimensional bosons and higher-dimensional conformalfield theories in a local quench limit.Finally, we conclude in section 7 by discussing generalizations and comment on po-tential implications of our work for constraining bulk effective field theories in holog-raphy. A general quantum system is described by an algebra of bounded operators with a † operation and a notion of a norm. In this abstract language, states are linear mapsfrom the algebra to correlation functions. Given a state, there is a well-defined pre-scription to construct an irreducible representation of such an algebra in the Hilbertspace. If the algebra is generated by operators that satisfy the canonical commuta-tion relation Lie algebra then such an irreducible representation is unique up to unitaryequivalence. This is not the case in quantum field theory. There are many inequiva-lent representations of the algebra of operators in the Hilbert space that correspond todifferent superselection sectors. This is one of the main motivations to study the localalgebra of operators in quantum field theory. The algebra of local operators in quantum field theory differs from finite dimensionalquantum systems in essential ways. Given a state living on a Cauchy slice, there is nonotion of density matrix operator corresponding to a subregion. This causes a problem This prescription is called the Gelfand–Naimark–Segal construction. See [7] for a review. This is called the Stone-von Neumann theorem. By the local algebra we mean the algebra associated with a region of spacetime. D spacetimedimensions. For every open set A in spacetime, there exists a von Neumann algebra ofbounded operators A A that is generated by bounded functions of fundamental fieldsintegrated against test functions supported on A . Locality requires that A B ⊂ A A if B ⊂ A . The closure of the union of algebras associated with all open sets formsthe global algebra of quantum field theory. In a relativistic quantum field theory, aPoincar´e transformation that sends A → Λ( A ) corresponds to an automorphism of theglobal algebra that transforms A → A Λ( A ) . We think of states on A as linear maps ω A from the local algebra A A to expectation values, and denote the states of the globalalgebra with vectors | Ω (cid:105) in the Hilbert space.Every von Neumann algebra A A has a notion of † . Therefore, for any two vectors | Ω (cid:105) and | Φ (cid:105) and a region A one can define the relative Tomita operator by its action S A Φ | Ω a | Ω (cid:105) = a † | Φ (cid:105) , (2)where a is an operators in the algebra A A of region A , and | Ω (cid:105) is carefully chosen suchthat a | Ω (cid:105) and a (cid:48) | Ω (cid:105) are dense in the Hilbert space. We further assume that thereexists no a ∈ A such that a | Ω (cid:105) = 0. Such states are called cyclic and separating.The definition in (2) can be extended to arbitrary vectors, however, to simplify thepresentation we restrict our discussion to vectors that are cyclic and separating.The relative modular operator ∆ Φ | Ω is the norm of the relative Tomita operator:∆ Φ | Ω ≡ S † Φ | Ω S Φ | Ω . (3)It is unbounded and manifestly non-negative. From the definition of the Tomita oper-ator it is clear that S Φ | Ω S Ω | Φ = 1 , S † Φ | Ω S † Ω | Φ = 1 , (4)hence the action of ∆ − | Φ = S Φ | Ω S † Φ | Ω . Furthermore, the Tomita operator for region A is the Hermitian conjugate of that of A (cid:48) (its causal complement): ( S A Φ | Ω ) † = S A (cid:48) Φ | Ω [17].As a result, the relative modular operators (∆ A Ω | Φ ) − = ∆ A (cid:48) Φ | Ω . This operator playsan important role in our discussion. Most importantly for us, the relative modularoperator for any two vectors | Φ (cid:105) and | Ω (cid:105) satisfies the monotonicity inequality∆ B Ω | Φ ≤ ∆ A Ω | Φ (5) This is often called the Isotony condition. The relative Tomita operator is the closure of the operator defined in (2). In this work, we usethe notation S Φ | Ω to mean the closed operator. We have assumed that the algebra of A and A (cid:48) commute and have no operators in commonexcept for the identity operator. Such a von Neumann algebra is called a “factor”. A A ⊂ A B . This is a consequence of the fact that the domain of S B Φ | Ω is anextension of that of S A Φ | Ω . That is to say for any a ∈ A A we have S A Ω | Φ a | Φ (cid:105) = S B Ω | Φ a | Φ (cid:105) = a † | Ω (cid:105) . (6)This implies that for vectors a | Φ (cid:105) with a ∈ A A the monotonicity of the relative modularoperator is saturated: (cid:104) Φ | a † ∆ A Ω | Φ a | Φ (cid:105) = (cid:104) Φ | a † ∆ B Ω | Φ a | Φ (cid:105) . (7)To obtain an inequality we consider non-linear functions of ∆ Ω | Φ that inherit mono-tonicity by construction. A function f : (0 , ∞ ) → (0 , ∞ ) is called operator monotoneif for any self-adjoint operators X ≥ Y ≥ f ( X ) ≥ f ( Y ) ≥
0. An importantexample is f ( z ) = z α which is operator monotone for for 0 < α < X > Y > Y − > X − . The relative modular operator isinvertible, therefore it satisfies the monotonicity operator inequalitysign( α )(∆ A Ω | Φ ) − α ≤ sign( α )(∆ B Ω | Φ ) − α (8)for 1 ≥ α ≥ −
1. Another important operator monotone function is the principal branchlogarithm on positive definite operators which implies that Araki’s relative entropy S (Φ (cid:107) Ω) = −(cid:104) Φ | log ∆ Ω | Φ | Φ (cid:105) (9)is monotonic. In this work, we mostly focus on f ( z ) = z α and briefly comment onother operator monotone functions until section 7.To study the entanglement between region A and A (cid:48) it is natural to focus on quan-tities that are independent of unitaries in the region A and A (cid:48) . Hence, we shouldunderstand how the relative modular operator transforms when the vectors are ro-tated by unitaries in A and A (cid:48) . One can check directly from the definition in (2) that S UU (cid:48) Ω | V V (cid:48) Ψ = V U (cid:48) S Ω | Ψ V (cid:48)† U † satisfies S UU (cid:48) Ω | V V (cid:48) Ψ aV V (cid:48) | Ψ (cid:105) = a † U U (cid:48) | Ω (cid:105) (10)for unitaries U, V ∈ A A and U (cid:48) , V (cid:48) ∈ A A (cid:48) . As a result, the relative modular operatortransforms according to ∆ UU (cid:48) Ω | V V (cid:48) Ψ = U V (cid:48) ∆ Ω | Ψ V (cid:48)† U † . (11)This transformation rule plays an important in section 3 when we engineer information-theoretic quantities from the relative modular operator that are invariant under localunitaries. See [6] for a proof based on the ideas presented in [18]. For completeness, in section 5 we reviewa variant of this proof adapted for finite-dimensional systems presented in [19]. The inequality
X > Y > > X − / Y X − / . The operator X − / Y X − / is non-negative and has a spectral decomposition, therefore X / Y − X / > Y − > X − .
4n a quantum field theory with a stress tensor, shape deformations are generatedby unitaries that are exponentiated integrals of the stress tensor. Consider a diffeo-morphism that deforms region A continuously to its subregion B and represent it byunitary U . In this work, we consider only the shape deformations that leave the vacuuminvariant: A A = U A B U † , U | Ω (cid:105) = | Ω (cid:105) . (12)Examples of such unitaries are Poincare transformations and arbitrary deformationson null hypersurfaces. The reason is that for such unitaries, we can explicitly relatethe relative modular operator of B with that of A using the definition of the relativeTomita operator: S B Ψ | Ω a B | Ω (cid:105) = ( U † S AU Ψ | Ω U ) a B | Ψ (cid:105) (13)and ∆ B Ψ | Ω = U † ∆ AU Ψ | Ω U . (14)In the limit the size of region A shrinks to zero the state | Φ (cid:105) and | Ω (cid:105) are indistin-guishable, therefore one expects the matrix elements of f (∆ Ω | Φ ) to converge to thoseof f (∆ Φ ). In particular, this implies that for any operator monotone function f andany region A we have (cid:104) Φ | f (∆ A Ω | Φ ) | Φ (cid:105) − f (1) ≥ . (15)We are now ready to construct information-theoretic measures out of the relative mod-ular operator. The class of measures that we will show to be related to the quantumfield theory correlation functions is called the sandwiched R´enyi divergence. In thenext section, we start by a closely related but simpler class of measures called Petzdivergences. The monotonicity of relative entropy in (8) is an operator equation which impliessign( α ) (cid:104) Ψ | (∆ A Ω | Φ ) − α | Ψ (cid:105) ≤ sign( α ) (cid:104) Ψ | (∆ B Ω | Φ ) − α | Ψ (cid:105) (16)for any A ⊂ B , α ∈ [ − ,
1] and any vector | Ψ (cid:105) in the Hilbert space. The matrix element (cid:104) Ψ | (∆ A Ω | Φ ) − α | Ψ (cid:105) depends on the information in both A and the complementary region A (cid:48) . In entanglement theory we are interested in “semilocal” quantities, meaning thosethat are independent of the information content in A (cid:48) . If | Φ (cid:105) is cyclic and separating We thank Edward Witten for pointing this out to us. a | Φ (cid:105) with a ∈ A is dense in the Hilbert space, therefore it is natural toconsider Petz quasi-entropies introduced in [20] D Aα,a (Φ (cid:107) Ω) = 1 α log (cid:104) Φ | a † (∆ A Ω | Φ ) − α a | Φ (cid:105) . (17)where the operator a is normalized such that (cid:104) Ω | a † a | Ω (cid:105) = 1 and α ∈ [ − ,
1] . Thisnormalization makes sure quasi-entropies vanish at α = − (cid:104) Φ | a † (∆ A Ω | Φ ) a | Φ (cid:105) = (cid:104) Ω | a † a | Ω (cid:105) = 1 . (18)Under unitary rotations in A (cid:48) the relative modular operator transforms accordingto (11), and since [ a, U (cid:48) ] = 0 the Petz quasi-entropy remains invariant.In addition, this R´enyi family has the following properties:1. If A ⊂ B and a ∈ A A then it increases monotonically with system size: D Aα,a (Φ (cid:107) Ω) ≤ D Bα,a (Φ (cid:107) Ω) .
2. If U is a unitary in A A then D Aα,a ( U Φ (cid:107) V Ω) = D Aα,V † aU (Φ (cid:107) Ω).3. It increases monotonically in α . If α ≤ β then D Aα,a (Φ (cid:107) Ω) ≤ D Aβ,a (Φ (cid:107) Ω).See appendix A for a proof. In the remainder of this work, we focus on quasi-entropies in the case a = 1, where the subregion monotonicity property extends to allsubregions. For a = 1, we have the Petz divergence: D α (Φ (cid:107) Ω) = 1 α log (cid:104) Φ | ∆ − α Ω | Φ | Φ (cid:105) (19)with the following additional properties:1. It is non-negative and vanishes when A shrinks to zero.2. It is invariant under the rotation of both vectors by the same unitary in A .3. At α = 0 it is smooth and equal to the relative entropy.4. Under swapping vectors | Φ (cid:105) and | Ω (cid:105) it satisfies D − α (Φ (cid:107) Ω) = 1 − αα D α − (Ω (cid:107) Φ) . (20)See appendix B for proofs.Petz divergences vanish at α = − α . They acquiretheir maximum at α = 1: D (Φ (cid:107) Ω) = log (cid:104) Φ | ∆ − | Φ | Φ (cid:105) = log (cid:104) Ω | S † Φ | Ω ( S Φ | Ω S † Φ | Ω ) S Φ | Ω | Ω (cid:105) = log (cid:104) Ω | ∆ | Ω | Ω (cid:105) . (21)If | Φ (cid:105) ∼ Φ (cid:48) | Ω (cid:105) with Φ (cid:48) ∈ A (cid:48) since ( S A Φ | Ω ) † is an extension of S A (cid:48) Φ | Ω we have D (Φ (cid:107) Ω) = log (cid:18) (cid:104) Ω || Φ (cid:48) | | Ω (cid:105)(cid:104) Ω || Φ (cid:48) | | Ω (cid:105) (cid:19) . (22) In [20] Petz defines the quantity in (17) for a general operator monotone function f . Sandwiched R´enyi divergences
The Petz divergence satisfies all the desired properties an information-theorist wouldwant, however it cannot be written in terms of the correlation functions of quantumfield theory in a simple way. A closely related monotonic R´enyi family called thesandwiched R´enyi divergence was introduced in [21, 22] for finite quantum systems. In[16] it was shown that for special values of α , in quantum field theory, the sandwichedR´enyi divergences can be written in terms of correlation functions. We focus on thisnew family in this section.To motivate an algebraic definition of the sandwiched R´enyi divergence we followthe approach in [10]. Consider a vector | Φ (cid:105) that is in the intersection of the domainof ∆ − | Ψ for all | Ψ (cid:105) ∈ H . We define the sandwiched R´enyi divergence to be S Aα (Φ (cid:107) Ω) ≡ α sup | Ψ (cid:105)∈H log (cid:104) Φ | (∆ A Ω | Ψ ) − α | Φ (cid:105) S A − α (Φ (cid:107) Ω) ≡ − α inf | Ψ (cid:105)∈H log (cid:104) Φ | (∆ A Ω | Ψ ) α | Φ (cid:105) (23)for 0 ≤ α < If the intersection of the domains of ∆ − | Ψ for all | Ψ (cid:105) does notinclude | Φ (cid:105) the Sandwiched R´enyi divergence is defined to be infinite. However, wewill show that it is finite for a dense set of states Φ (cid:48) | Ω (cid:105) where Φ (cid:48) is an operator in thecommutant. Sandwiched R´enyi divergences satisfy all the desired properties that led usto Petz divergences. The subregion monotonicity and non-negativity are inherited from(16) by definition. The monotonicity in α follows from Holder’s inequality similar tothe argument presented in appendix B. It is also invariant under rotation by untiariesin A (cid:48) , and under unitary rotations in A , it transforms according to S α ( V Φ (cid:107) U Ω) = S α ( U † V Φ (cid:107) Ω) = S α (Φ (cid:107) V † U Ω). Similar to the Petz entropy it vanishes when the twovectors are the same, but as opposed to the Petz entropy, it does not vanish at α = − α = − S α (Φ (cid:107) Ω) ≥ D α (Φ (cid:107) Ω) . (24)An analogy with finite dimensional quantum systems helps to unpack the definitionin (23). This definition is based on a generalization of the notion of the p -norm of amatrix to unbounded operators [10]. Remember that for a matrix X the p -norm is A similar discussion appears in a recent work [13]. We assume all vectors are normalized (cid:104) Ψ | Ψ (cid:105) = 1 unless mentioned otherwise. There is a potential for confusion here due to the notation we have chosen. It is sometimes saidthat Petz divergences are larger than sandwiched ones. That statement uses a different definition ofPetz divergences which in our notation becomes the inequality (32) shown in appendix B. See section5. (cid:107) X (cid:107) p = tr ( | X | p ) /p . It is invariant under unitary rotations, (cid:107) X (cid:107) p = (cid:107) U † XU (cid:107) p , and it satisfies the Holder inequality (cid:107) XY (cid:107) ≤ (cid:107) X (cid:107) p (cid:107) Y (cid:107) q for any p, q > p − + q − = 1. By the definition of the norm | tr ( XC ) | ≤ (cid:107) XC (cid:107) for any operator C . As a result,sup (cid:107) C (cid:107) q =1 | tr ( XC ) | ≤ sup (cid:107) C (cid:107) q =1 (cid:107) XC (cid:107) ≤ (cid:107) X (cid:107) p (25)where in the second step we have used Holder’s inequality. If we pick C proportionalto | X | p/q U † where X = U | X | is the polar decomposition of X and the overall factor of C is fixed such that (cid:107) C (cid:107) q = 1, we find that the inequality above is saturated. In otherwords, we have the norm duality relation (cid:107) X (cid:107) p = sup (cid:107) C (cid:107) q =1 | tr ( XC ) | . (26)Our operators of interest are not trace class, however, choosing a reference state | Ω (cid:105) and an algebra of region A following Araki and Masuda [10] one can still define the p -norm of a vector | Φ (cid:105) using the definition p ∈ [2 , ∞ ] (cid:107)| Φ (cid:105)(cid:107) Ap, Ω = sup | Ψ (cid:105) (cid:107) ∆ / − /p Ψ | Ω | Φ (cid:105)(cid:107) p ∈ [1 , (cid:107)| Φ (cid:105)(cid:107) Ap, Ω = inf | Ψ (cid:105) (cid:107) ∆ / − /p Ψ | Ω | Φ (cid:105)(cid:107) . (27)As we discussed in section 2, we know that (∆ A Ω | Ψ ) − = ∆ A (cid:48) Ψ | Ω , therefore the sandwichedR´enyi divergence can be written as S Aα (Φ (cid:107) Ω) = 2 α log (cid:107)| Φ (cid:105)(cid:107) A (cid:48) − α , Ω . (28)The reason for switching from supremum to infimum as we go from positive to negativevalues of α is a generalization of (26) to the norms defined in (27) that relates the p and q norms when p + q = 1. To see this, consider the Cauchy-Schwarz inequality |(cid:104) Φ | χ (cid:105)|(cid:104) Φ | ∆ − α Ω | Ψ | Φ (cid:105) / ≤ (cid:104) χ | ∆ α Ω | Ψ | χ (cid:105) / . (29)Taking the infimum over | Ψ (cid:105) we haveinf | Ψ (cid:105) |(cid:104) Φ | χ (cid:105)|(cid:104) Φ | ∆ − α Ω | Ψ | Φ (cid:105) / ≤ inf | Ψ (cid:105) (cid:104) χ | ∆ α Ω | Ψ | χ (cid:105) / . In fact, the supremum of the left-hand-side over all vectors | Φ (cid:105) saturates the inequality[10] sup | Φ (cid:105) |(cid:104) Φ | χ (cid:105)| sup | Ψ (cid:105) (cid:104) Φ | ∆ − α Ω | Ψ | Φ (cid:105) / = inf | Ψ (cid:105) (cid:104) χ | ∆ α Ω | Ψ | χ (cid:105) / . (30)8s a result, we have the norm duality relationsup (cid:107)| Φ (cid:105)(cid:107) p, Ω =1 |(cid:104) Φ | χ (cid:105)| = (cid:107)| χ (cid:105)(cid:107) q, Ω . (31)This is a generalization of (26) to von Neumann algebras.For special values of α the sandwiched R´enyi divergences in (23) is of particularinterest in information theory: • α = 1: Due to the monotonicity in α , this is the maximum of the sandwichedR´enyi divergence in α and it is given by the norm of the operator that createsthe state. If | Φ (cid:105) = Φ (cid:48) | Ω (cid:105) we have S (Φ (cid:107) Ω) = sup | Ψ (cid:105) log (cid:104) Φ | ∆ − | Ψ | Φ (cid:105) = sup | Ψ (cid:105) log (cid:104) Φ | S Ψ | Ω S † Ψ | Ω | Φ (cid:105) = sup | Ψ (cid:105) log (cid:104) Ψ |(cid:107) Φ (cid:48) (cid:107) | Ψ (cid:105) = 2 log (cid:107) Φ (cid:48) (cid:107) where we have used ( S A Ψ | Ω ) † = S A (cid:48) Ψ | Ω . An important implication of this result isthat for states Φ (cid:48) | Ω (cid:105) with Φ (cid:48) bounded all the sandwiched Renyi divergences arefinite and free of ultraviolet divergences. This is a dense set of states. Haag andSwieca have argued that the norm of the operator that creates the state shouldbe interpreted as a measure of how localized the state is [23]. Smaller S (Φ (cid:107) Ω)corresponds to a state more localized in A (cid:48) . A unitary U (cid:48) creates an excitationcompletely localized in A (cid:48) . • α = : This value corresponds to a generalization of “collision entropy” to vonNeumann algebras: S (Φ (cid:107) Ω) = 2 sup Ψ (cid:104) Φ | ∆ − / | Ψ | Φ (cid:105) . As we will see in the nextsection, this value of α can be related to Euclidean four point functions. • α = 0: Similar to the Petz divergence the sandwiched divergences is equal tothe relative entropy at α = 0. To see this, we show the following inequality inappendix C: D α − α (Φ (cid:107) Ω) ≥ S α (Φ (cid:107) Ω) ≥ D α (Φ (cid:107) Ω) . (32)In the limit α → • α = −
1: This value of α provides a generalization of “quantum Fidelity” to vonNeumann algebras: S − (Φ (cid:107) Ω) = − F (Φ , Ω) F (Φ , Ω) ≡ (cid:107)| Φ (cid:105)(cid:107) A (cid:48) , Ω (33) Note that the R´enyi divergences including the relative entropy are not continuous. Therefore,in general, there exist states for which our measure is infinite. This is clear from the fact that thedomain of ∆ − | Ψ is not the whole Hilbert space for all | Ψ (cid:105) .
9n the next section, we will show that for density matrices F (Φ , Ω) reduces tothe standard expression for quantum Fidelity. From norm duality we know thatFidelity as defined above is the supremum over overlaps (cid:104) Ψ | Φ (cid:105) for | Ψ (cid:105) = Ψ (cid:48) | Ω (cid:105) and (cid:107) Ψ (cid:48) (cid:107) = 1: F (Φ , Ω) = sup (cid:107)| χ (cid:105)(cid:107) A (cid:48)∞ , Ω =1 |(cid:104) χ | Φ (cid:105)| . (34)This is the analogue of Uhlemann’s theorem in quantum field theory [24]. Up to here, the discussion of the relative modular operator and our information-theoretic measures applied to any quantum system from qubits to quantum field theory.To come in contact with the information theory literature, in this section, we write ourmeasures for finite quantum systems in terms of density matrices, and provide a simpleproof of the monotonicity of the Petz divergence and the sandwiched R´enyi divergence.We closely follow the arguments presented in [19] and [6]. Consider a bipartite quan-tum system described by a density matrix σ and its reduced density matrix on thefirst subsystem σ . It is convenient to purify σ in a pure state of a four-partite system | Ω (cid:105) = d (cid:88) i,j =1 c ij | ij, ij (cid:105) ∈ H (35)where | ij (cid:105) are the eigenvectors of σ ij and we have assumed that the local Hilbert at eachsite is d -dimensional. In our notation, we separate the indices corresponding to theHilbert space of the system from those of the purification by a comma. The coefficients c ij are the Schmidt coefficients of | Ω (cid:105) and the square root of eigenvalues of σ . Forsimplicity, we assume that σ is full-rank which implies that all c ij >
0. Similarly, wepurify σ in a bi-partite pure state | ω (cid:105) = (cid:88) i d i | i, i (cid:105) ∈ K d i = (cid:88) j c ij . (36)The local algebra on each site is the algebra of d × d complex matrices with itsnatural notion of † and norm. We consider the algebras A = A and B = A ⊗ A thatare generated by a ik = | i (cid:105)(cid:104) k | and b ijkl = | ij (cid:105)(cid:104) kl | , respectively. Acting on the state | Ω (cid:105) with b ∈ B we can obtain arbitrary states H . Similarly, any state in K can be writtenas a | ω (cid:105) with a ∈ A .The state | ω (cid:105) can be though of as | ω (cid:105) = σ / | E ω (cid:105)| E ω (cid:105) = (cid:88) j | j, j (cid:105) (37)10here | E ω (cid:105) is an unnormalized maximally entangled state in the basis spanned by | j (cid:105) ,the eigenvectors of σ [25]. Note that for any full rank density matrix σ we have (cid:104) ω | X ⊗ Y | ω (cid:105) = (cid:104) E ω | σ / Xσ / ⊗ Y | E ω (cid:105) = tr ( σ / Xσ / Y ) . (38)Consider the state | ω (cid:105) on systems 13 tensored with another copy of it on 24: | ω (cid:105) ⊗| ω (cid:105) . If we act on it by the operator˜ W = σ / (cid:16) σ − / ⊗ σ − / (cid:17) (39)we obtain the four-partite state | Ω (cid:105) : | Ω (cid:105) = ˜ W ( | ω (cid:105) ⊗ | ω (cid:105) ) (40)where by σ we mean σ as a density matrix of system 34. Furthermore, we find thatfor a ∈ A : a | Ω (cid:105) = ˜ W ( a | ω (cid:105) ⊗ | ω (cid:105) ) (41)Since a | ω (cid:105) is dense in K we have constructed a linear linear embedding W : K → H that sends a ik | ω (cid:105) → a ik | Ω (cid:105) . More explicitly, this map is W = (cid:88) ijk c kj d k | ij, kj (cid:105)(cid:104) i, k | . (42)Consider the normalized pure states of subsystem 2 and 4: | Ψ k (cid:105) = 1 d k (cid:88) j c kj | j, j (cid:105) . (43)It is straightforward to check that W † W = I K , W W † = I ⊗ (cid:88) k | k (cid:105) (cid:104) k | ⊗ | Ψ k (cid:105) (cid:104) Ψ k | = P where P is a projection in H . Therefore, W is an isometry.The anti-linear Tomita operator for algebras A and B satisfies S Aω a ik | ω (cid:105) = a † ik | ω (cid:105) S B Ω b ijkl | Ω (cid:105) = b † ijkl | Ω (cid:105) which are solved by S Aω | k, i (cid:105) = d k d i | i, k (cid:105) S B Ω | kl, ij (cid:105) = c kl c ij | ij, kl (cid:105) . (44)11n this basis, the Tomita operator is the anti-linear operator that swaps the role of thesystem and its purifier. One can compute the modular operators∆ Aω = (cid:88) ik d k d i | k, i (cid:105)(cid:104) k, i | = σ ⊗ σ − ∆ B Ω = (cid:88) ijkl c kl c ij | kl, ij (cid:105)(cid:104) kl, ij | = σ ⊗ σ − (45)and check directly that W † | αβ, kj (cid:105)(cid:104) αβ, kj | W = c kj d k | α, k (cid:105)(cid:104) α, k | (46)and therefore ∆ Aω = W † ∆ B Ω W . (47)Since projectors are positive operators, for positive λ and β , we have (cid:0) ∆ B Ω + β (cid:1) − ≥ (cid:0) ∆ B Ω + β + λ (1 − P ) (cid:1) − (48)At large λ the right-hand-side can be expanded as (cid:0) ∆ B Ω + β + λ (1 − P ) (cid:1) − = ( P (∆ B Ω + β ) P ) − + O (1 /λ )= (cid:0) W (∆ Aω + β ) W † (cid:1) − + O (1 /λ ) (49)where we have used W W † = P . Taking the λ → ∞ limit we find W † ( β + ∆ B Ω ) − W ≥ ( β + ∆ Aω ) − (50)as an operator inequality in K . For a positive operator ∆ and 0 < α < α = sin( πα ) π (cid:90) ∞ dβ β α (cid:18) β − β + ∆ (cid:19) ∆ − α = sin( πα ) π (cid:90) ∞ dβ β − α (cid:18) β + ∆ (cid:19) . (51)Therefore, for − < α < α ) (cid:0) ∆ Aω (cid:1) − α ≤ sign( α ) (cid:0) ∆ B Ω (cid:1) − α . (52)Now, consider a second density matrix φ and its purification φ = (cid:88) αβ f αβ | αβ (cid:105)(cid:104) αβ || Φ (cid:105) = (cid:88) αβ f αβ | αβ, αβ (cid:105) ∈ H . (53)12imilarly, the density matrix φ can be purified with | ϕ (cid:105) = (cid:88) α g α | α, α (cid:105) ∈ K g α = (cid:88) β f αβ . (54)The relative Tomita operator now solves S Aϕ | ω a αk | ω (cid:105) = a † αk | ϕ (cid:105) S B Φ | Ω b αβkl | Ω (cid:105) = b † αβkl | Φ (cid:105) (55)where b αβkl = | αβ (cid:105)(cid:104) kl | and a αk = | α (cid:105)(cid:104) k | . These equations imply S Aϕ | ω | α, i (cid:105) = g α d i | i, α (cid:105) S B Φ | Ω | αβ, ij (cid:105) = f αβ c ij | ij, αβ (cid:105) . (56)The relative modular operators can be worked out explicitly:∆ Aϕ | ω = (cid:88) iα g α d i | α, i (cid:105)(cid:104) α, i | = φ ⊗ σ − ∆ B Φ | Ω = (cid:88) ijαβ f αβ c ij | αβ, ij (cid:105)(cid:104) αβ, ij | = φ ⊗ σ − (57)From (38) and (57) we find that the Petz divergence written in terms of density matricesis D α (Φ (cid:107) Ω) = 1 α log tr (cid:0) φ α σ − α (cid:1) . (58)Once again, we can check explicitly that W † ∆ B Φ | Ω W = ∆ Aϕ | ω . (59)By the same argument we used for the modular operator, we find the monotonicityinequality W † (cid:0) ∆ B Φ | Ω + β (cid:1) − W ≥ (cid:0) ∆ Aϕ | ω + β (cid:1) − . (60)Therefore, in the range α ∈ ( − ,
1) we havesign( α )(∆ Aϕ | ω ) − α ≤ sign( α ) W † (∆ B Φ | Ω ) − α W (61) Often the Petz divergence is defined in the range α ∈ (0 ,
2) using ¯ D n = n − log tr ( φ n ω − n ). Inour notation, this is D n − . | ω (cid:105) , and using the fact that W | ω (cid:105) = | Ω (cid:105) we get sign( α ) (cid:104) ω | (∆ Aϕ | ω ) − α | ω (cid:105) ≤ sign( α ) (cid:104) Ω | (∆ B Φ | Ω ) − α | Ω (cid:105) . (62)As a result, we find the monotonicity of the Petz divergence D α ( ω (cid:107) φ ) ≤ D α (Ω (cid:107) Φ) . (63)Next, we would like to find the expression for the sandwiched R´enyi divergence interms of density matrices and prove its monotonicity. Consider a third vector | Ψ (cid:105) andits corresponding density matrix ψ , and ψ with its purification | ψ (cid:105) . Let us fix theregion for the moment. It follows from the definition in (23) and the expressions in(57) that S α (Φ (cid:107) Ω) = 1 α sup tr ( ψ )=1 log tr (cid:0) ψ α φ / σ − α φ / (cid:1) , (64)where we have used the cyclicity of the trace, and the supremum is over all densitymatrices ψ . Defining the positive matrix η = ψ α and X = σ − α/ φ / we can write thesandwiched R´enyi divergence in (58) in terms of p -norms as S α (Φ (cid:107) Ω) = 1 α log (cid:32) sup (cid:107) η (cid:107) /α =1 tr (cid:0) ηX † X (cid:1)(cid:33) (65)From (26) the supremum can be evaluated explicitly to give S α (Φ (cid:107) Ω) = 1 α log (cid:107) X † X (cid:107) − α = 1 α log (cid:107) XX † (cid:107) − α = 1 − αα log tr (cid:104)(cid:0) σ − α/ φσ − α/ (cid:1) − α (cid:105) (66)where we have used the polar decomposition of X = U | X | and the fact that p -norm isunitary-invariant. In the literature, the sandwiched R´enyi divergence is often definedfor n ∈ (1 / , ∞ ) to be ¯ S n = n − log tr (cid:104)(cid:16) σ − n n ρσ − n n (cid:17) n (cid:105) . In our notation, this is S n − n .The statement that ¯ D n ≥ ¯ S n in our notation becomes D n − ≥ S n − n which is the sameas (32). The expression in (66) continues to hold when α <
0. This can be seen bycomputing the generalized p -norm in (27) in terms of density matrices: (cid:107)| Φ (cid:105)(cid:107) A (cid:48) − α , Ω = sup tr ( χ )=1 tr (cid:0) ψ α φ / σ − α φ / (cid:1) = tr (cid:107) η (cid:107) /α =1 (cid:0) ηX † X (cid:1) = (cid:107) X † X (cid:107) − α = (cid:107) X (cid:107) − α (67)where X = ω − α/ φ / and η = ψ α . Therefore, S − α (Φ (cid:107) Ω) = − α log sup (cid:107) σ − α/ ψ / (cid:107) − α =1 | tr (cid:0) ψ / φ / (cid:1) | = − α log sup (cid:107) ν (cid:107) − α =1 | tr (cid:0) νφ / σ α/ (cid:1) | = − α log (cid:107) φ / σ α/ (cid:107) α = − αα log tr (cid:104)(cid:0) σ α/ φσ α/ (cid:1) α (cid:105) (68)14hich is the same as (66).Now, we are ready to prove the monotonicity of the sandwiched Renyi divergence.Assume | ϕ (cid:105) is cyclic and separating. Consider the subset of vectors in H : | Ψ a (cid:105) = a | Φ (cid:105) with a an invertible element of A , the algebra of the first subsystem. Theircorresponding states on K are | ψ a (cid:105) = a | ϕ (cid:105) that form a dense set in K [25]. Thefunction (cid:104) ϕ | ∆ − αω | ψ | ϕ (cid:105) is real-valued and continuous in finite-dimensional Hilbert spaces.Therefore, since a | ψ (cid:105) with invertible a is dense in K we findsup ψ (cid:104) ϕ | (cid:0) ∆ Aω | ψ (cid:1) − α | ϕ (cid:105) = sup ψ a (cid:104) ϕ | (cid:0) ∆ Aω | ψ a (cid:1) − α | ϕ (cid:105) . (69)From the operator inequality in (61) we havesign( α ) sup ψ a (cid:104) ϕ | (cid:0) ∆ Aω | ψ a (cid:1) − α | ϕ (cid:105) ≤ sign( α ) sup Ψ a (cid:104) ϕ | W † Ψ a (cid:0) ∆ A Ω | Ψ a (cid:1) − α W Ψ a | ϕ (cid:105) (70)We observe that W Ψ a | ϕ (cid:105) = W Ψ a a − | ψ a (cid:105) = a − | Ψ a (cid:105) = | Φ (cid:105) . (71)Therefore, sign( α ) sup ψ (cid:104) ϕ | (cid:0) ∆ Aω | ψ (cid:1) − α | ϕ (cid:105) ≤ sign( α ) sup Ψ a (cid:104) Φ | (cid:0) ∆ A Ω | Ψ a (cid:1) − α | Φ (cid:105)≤ sign( α ) sup Ψ (cid:104) Φ | (cid:0) ∆ A Ω | Ψ (cid:1) − α | Φ (cid:105) (72)Hence, we have established a simple proof of the monotonicity of the sandwiched R´enyidivergence: S α ( ϕ (cid:107) ω ) ≤ S α (Φ (cid:107) Ω) . (73) In this section, we start with the expression for the sandwiched R´enyi divergence interms of density matrices in (66) and use the Euclidean path-integral formalism towrite them in terms of correlation functions. At the first look, this appears to beagainst our philosophy to avoid the use of density matrices in quantum field theory.However, since we defined our R´enyi families directly within the modular theory weare guaranteed that they are ultraviolet finite for states Φ | Ω (cid:105) with Φ an operatorin the algebra or the commutant. This is manifestly the case in [16] that writes thesandwiched R´enyi divergence for values of α = 1 − n and positive integer n > n -point correlation functions. We review this constructionbelow. It is to be contrasted with the standard expression for R´enyi entropies thatis an n -sheeted partition function and ultraviolet divergent. Unfortunately, the Petzdivergence involves fractional powers of both density matrices for all values of α and wecannot have a simple Euclidean path-integral representation for it, except for the casethat the excited state is the vacuum of the theory deformed with a relevant operator,e.g. see [26]. For the remainder of this section, we focus on the sandwiched R´enyidivergence. 15 ath-integral approach Consider the vacuum of a quantum field theory | Ω (cid:105) , an operator Φ supported in A (cid:48) andexcited states | Φ (cid:105) = ∆ − θ/π Φ | Ω (cid:105) with 0 ≤ θ ≤ / − ( π − θ ); see figure 1. The vacuum density matrix ω A for half-space x > φ A with two operator insertions Φ and Φ † at angle − ( π − θ ) and ( π − θ ),respectively. If θ + πα ≤ π there is a path-integral representation for the operator ω − α/ φω − α/ as a wedge of angle 2 π (1 − α ) with two operator insertions; figure 1. Ifwe choose α such that 1 − α = n with n > n such wedges can besewn together to obtain tr (cid:16) ( ω − n n φω − n n ) n (cid:17) as a 2 n -point function of Φ and Φ † withoperators inserted at z ( ± ) k = re i ( kπn ± θ ) for k = 0 , · · · , n − tr (cid:16) ( ω − n n φω − n n ) n (cid:17) = (cid:42) n − (cid:89) k =0 Φ † ( z + k )Φ( z − k ) (cid:43) (74)where we have suppressed the D − x ⊥ . For integer n > S A − n (Φ (cid:107) Ω) = 1 n − (cid:32) (cid:10)(cid:81) n − k =0 Φ † ( z + k )Φ( z − k ) (cid:11) (cid:104) Φ † ( z +0 )Φ( z − ) (cid:105) n (cid:33) . (75)The allows us to relate the monotonicity of sandwiched R´enyi divergences to constraintson correlation functions. It is worth noting that the non-negativity of sandwiched R´enyidivergence is already a non-trivial constraint implying that for all positive integer n > A . As we discussed in section 2, for a one-parameter family of smoothdeformations given by U λ with U = 1 that are symmetries of the vacuum state, therelative modular operator transforms according to∆ A ( λ )Ψ | Ω = U † λ ∆ AU λ Ψ | Ω U λ . (76)Plugging this into the definition of the sandwiched R´enyi divergence we find S A ( λ ) α (Φ (cid:107) Ω) = S Aα ( U λ Φ (cid:107) Ω) . (77)The unitary U λ acts geometrically on local operators: Φ( z ) → Φ ( λ ) ( z λ ). The sand-wiched R´enyi divergence of the deformed region A ( λ ) is given by the expression in (75)with Φ( z ) replaced with Φ ( λ ) ( z λ ).To be specific, we consider the Rindler space parameterized by ( z, x ⊥ ) where z = x + iτ and x ⊥ are the perpendicular directions. We focus on two classes of shapedeformations and assume that operator Φ is a scalar:16
The Euclidean path integrals that prepare (a) excited state | Φ (cid:105) , (b) vacuumdensity matrix on region A , (c) the density matrix of A in the excited state. (d) The operator ω − n n φω − n n corresponds to a path-integral on a wedge of angular size 2 π/n . (e) The correlatorthat appears in the definition of the sandwiched R´enyi divergence. Translations:
The unitary U λ ( ζ ) = e iλP ζ with P ζ = ζ P + ζ P corresponds toa translation in ζ direction.2. Null deformations:
The unitary U λ ( f, x ⊥ ) = e iλQ f with Q f = (cid:82) duf ( x ⊥ ) T uu ( u, x ⊥ )a conserved charge associated with deformations on a null hypersurface parame-terized by u = t − r and x ⊥ .The second class includes the first one as a special case. However, it is harder towork with Q f because, as opposed to P ζ , it is not a topological charge. If the stateswe consider are created by the action of only one operator acting on the vacuum, then U λ ( f, x ⊥ ) acts as: U λ ( f, x ⊥ )Φ( z, x ⊥ ) | Ω (cid:105) = U λ ( ζ )Φ( z, x ⊥ ) | Ω (cid:105) , (78)where ζ = f ( x ⊥ ) ∂ u is a null translation. We will use this simplifying assumption andpostpone the study of the consequences of monotonicity under null deformations forgeneral states to future work.Consider the Euclidean translation ( τ, x ) → ( τ + λζ , x + λ ) thenΦ( − τ , x ) → e − λ ( ζ P + P ) Φ( − τ , x ) e λ ( ζ P + P ) = Φ( − τ − λζ , x − λ ) , Φ † ( τ , x ) → e − λ ( − ¯ ζ P + P ) Φ † ( τ , x ) e λ ( − ¯ ζ P + P ) = Φ † ( τ + λ ¯ ζ , x − λ ) , (79)where we have taken a to be complex so that we can analytically continue it toreal time. If we take a to be pure imaginary the pair Φ † Φ move together and thedenominator in (75) stay the same. The other pairs of operators Φ † Φ in the numeratorof (75) are the same as Φ † Φ above but rotated by 2 πk/n . To simplify our notation, werefer to Φ( z k ) as Φ k .Changing the region corresponds to moving Φ in the path-integral. The first deriva-tives in shape deformation corresponds to the first λ derivatives in (79). The m th λ derivative is a nested commutator of the pair Φ † Φ with the momentum that generatesthe translation: d m dλ m (Φ † Φ) = 1 m ! [ P ζ , [ P ζ , · · · , [ P ζ , Φ † Φ]]] . (80)The momentum P ζ is a topological charge, so it can be written as P ζ = (cid:90) C d Σ i ζ j T ij , (81)where C is any codimension one surface that encircles the pair Φ † Φ and no otheroperators. We choose this surface to be x = c with τ sin( π/n ) < c < x ; see figure 2.The momentum P ζ written on this surface is P ζ = (cid:90) dτ ( T x x + ζ T τx ) = P + ζ P , (82) The statement that the second class of transformation are symmetries of the vacuum is theso-called Markov property of the vacuum in quantum field theory [27, 28]. P and P are momenta in τ and x directions, respectively. We are interestedin deformations of region A that send ( x , x ) → ( x + it, x + λ ) with t <
1, so we set ζ = it . This makes sure A ( λ ) ⊂ A ( λ (cid:48) ) for any λ < λ (cid:48) .To the first order in λ by rotation symmetry we have ∂ λ S A ( λ )1 − n (Φ (cid:107) Ω) λ =0 = n (cid:104) [ P ζ , Φ † Φ ]Φ † Φ · · · Φ † n − Φ n − (cid:105)(cid:104) Φ † Φ · · · Φ † n − Φ n − (cid:105) (83)where the factor of n comes the rotation invariance of correlators. It is convenient tointerpret the correlator above by choosing ∂ x as the generator of time-translations.In this quantization frame, P becomes the Hamiltonian and P L = iP becomes theLorentzian momentum along a spacelike direction; see figure 2. Therefore, the mono-tonicity of R´enyi divergences can be understood as the non-negativity of the followingoff-diagonal matrix elements of H − tP L with 0 < t < ∀ n ≥ (cid:104) Φ † Φ | H − tP L | Φ † Φ · · · Φ † n − Φ n − (cid:105) ≥ . (84)Note that the denominator in (83) is manifestly positive from reflection positivityaround τ = 0. Since H − tP L = tP u + (1 − t ) H with P u the null momentum in u direction, we only need to consider the constraint at t = 0 and t = 1. When n = 2the inequality above follows from the positivity of P u . To our knowledge, for n > The physical interpretation of this result is that not only H and P u are positiveoperators but also they have the following non-zero off-diagonal matrix elements: (cid:104)O| P u |O n − (cid:105) ≥ , (cid:104)O| H |O n − (cid:105) ≥ |O(cid:105) = | Φ † Φ (cid:105) and |O n − (cid:105) = | Φ † Φ · · · Φ † n − Φ n − (cid:105) .Consider the quantization frame that chooses τ as the Euclidean time. Now the 2 n -point function without momentum commutators can be thought of as the the norm ofa state | χ (cid:105) where | χ (cid:105) = | Φ Φ † Φ · · · Φ † n/ (cid:105) if n is even and | χ (cid:105) = | Φ · · · Φ † ( n − / Φ ( n − / (cid:105) if n is odd.In the special case Φ = Φ † the correlator is symmetric under the reflection τ → − τ .Since this reflection sends P to − P but preserves P we find that ∂ t S − n (Φ (cid:107) Ω) | t =0 = 0 ∂ x S − n (Φ (cid:107) Ω) | t =0 ≥ . (86)Clearly, this holds only for the first derivative and just in the special case of a real fieldΦ = Φ † . According to the reconstruction theorems the reflection positivity and the analyticity of Euclideancorrelators are sufficient to insure unitarity and causality in Lorentzian signature [29]. This suggeststhat there has to be a way to derive this constraint from reflection positivity. However, we could notfind such an argument for n > It we tune initial state to have be a real field, as the operator evolves in time it becomes a complexfield.
2. Note that P (1) is the same as P but rotatedby π/ A conjecture
The sandwiched R´enyi divergence is smooth at α = 0 and equal to the relative entropy.It was conjectured and recently argued that, in addition to the first derivative of relativeentropy in deformation parameter that is positive due to monotonicity, the second nullderivative of relative entropy is also non-negative in field theory [30, 31]. This iscalled the quantum null energy condition. In this subsection, we investigate the higherderivatives of the sandwiched R´enyi divergence.Let us start with n = 2. From (80) we know that the m th derivatives in nulldeformations is the expectation value of the positive operator P mu in the state |O(cid:105) : ∂ mλ S / (Φ (cid:107) Ω) = 1 m ! (cid:104)O| P mu |O(cid:105) ≥ . (87)Next, consider the second space-like derivative ( t = 0) of R´enyi divergences with n > † . Then, ∂ x S A ( λ )1 − n (Φ (cid:107) Ω) λ =0 = n (cid:104) [ P , [ P , Φ Φ ]] · · · Φ n − Φ n − (cid:105)(cid:104) Φ Φ · · · Φ n − Φ n − (cid:105) + n n − (cid:88) k =2 (cid:104) [ P , Φ Φ ] · · · [ P (1 ,k ) , Φ k − Φ k − ] · · · Φ n − Φ n − (cid:105)(cid:104) Φ Φ · · · Φ n − Φ n − (cid:105) where P (1 ,k ) is the same operator as P (1) rotated by 2 πk/n ; see figure 2. The firstterm above is symmetric under reflection around θ = 0, and the k th term in the sum20s symmetric under the reflection around θ = πk/n ; see figure 2. Therefore, for thedeformation generated by ∂ x the second derivative of R´enyi relative entropy is non-zerofor all n if Φ = Φ † : ∂ x S (Φ (cid:107) Ω) ≥ . (88)Repeating the argument above for a null deformation and the special state Φ = Φ † wefind that the second null derivative is also positive. Therefore, for this special class ofstates, we have proved our conjecture ∂ u S (Φ (cid:107) Ω) ≥ . (89) Examples
It is instructive to explicitly compute the sandwiched R´enyi divergence and its deriva-tives in some simple theories. The theory has to be simple enough that we cancompute the 2 n -point correlation functions. Two instances when we can access thesecorrelators are free fields and small θ limit in conformal field theory. Consider a two dimensional massless boson and a coherent operator e iβφ with real β .Since this operator is a conformal primary, and there are the same number of operatorsin the numerator and the denominator of (75), the sandwiched R´enyi divergence isindependent of r in z ( ± ) k = re i ( πkn ± θ ) . The n -point function of the coherent operator isgiven by [33] (cid:42) n − (cid:89) k =0 e − iβ k φ ( z k ) (cid:43) = (cid:89) k>i ( z k − z i ) β i β k . (90)For the initial configuration at λ = 0 we have (cid:10)(cid:81) n − k =0 e − iβφ ( z + k ) e iβφ ( z − k ) (cid:11)(cid:10) e − iβφ ( z +0 ) e iβφ ( z − ) (cid:11) n = n − m (cid:89) m =1 (cid:32) sin (cid:0) πmn (cid:1) sin (cid:0) πmn + θ (cid:1) sin (cid:0) πmn − θ (cid:1) (cid:33) ( n − m ) β = (cid:18) n sin θ sin( nθ ) (cid:19) nβ The sandwiched R´enyi divergence is [16] S A (0) n ( e iβφ (cid:107) Ω) = nβ n − (cid:18) n sin θ sin( nθ ) (cid:19) . (91)Initially starting at ( τ = r sin θ, x = r sin θ ) and deforming the region, the operator Φ moves to ( − τ (1 + λζ ) , x (1 − λ )) and Φ † goes to ( τ (1 + λ ¯ ζ ) , x (1 − λ )), where ¯ ζ is In a recent work the R´enyi divergences were computed in free field theory using real-time tech-niques [32]. Our Euclidean approach has the advantage that it makes the monotonicity constraintsmanifest. It would be interesting to study these constraints in large N theories. ζ . Choosing ζ = it we find that (cid:104) Φ † k Φ k (cid:105) remains unchanged.For displaced operators from the n -point function formula we have (cid:10)(cid:81) n − k =0 e − iβφ ( z + k ( λ )) e iβφ ( z − k ( λ )) (cid:11)(cid:10) e − iβφ ( z +0 ) e iβφ ( z − ) (cid:11) n = n − (cid:89) m =1 (cid:32) c ( θ ) c ( − θ ) sin (cid:0) mπn (cid:1) X ( θ ) X ( − θ ) (cid:33) ( n − m ) β X ( θ ) = 1 − cos (cid:18) πmn + 2 θ (cid:19) + 2 λ (cid:18) cos (cid:18) πmn + θ (cid:19) − cos( θ ) (cid:19) + λ (1 − t ) (cid:18) − cos (cid:18) πmn (cid:19)(cid:19) c ( θ ) = 1 − λ (cos θ + it sin θ ) + λ (1 − t ) . (92)First consider the case n = 2. Then, X ( θ ) = 1 + cos(2 θ ) − λ cos θ + 2 λ (1 − t ) (93)and S A ( λ )1 / ( e iβφ (cid:107) Ω) = β (cid:16) λ t sin ( θ ) + (1 − λ cos( θ ) + λ (1 − t )) (cid:17) (1 + cos(2 θ ) − λ cos( θ ) − λ ( t − . (94)At t = 0 we have S A ( λ )1 / ( e iβφ (cid:107) Ω) = β log (cid:18) (1 + λ − λ cos θ )( λ − cos θ ) (cid:19) (95)According to Bernstein’s theorem a function f ( λ ) has ∂ kλ f ( λ ) ≥ k if and onlyif f ( − λ ) is the Laplace transform of a probability distribution µ ( s ) [34]: f ( − λ ) = (cid:90) ∞ ds e − sλ µ ( s ) . (96)In many physics applications, the probability distribution µ ( s ) corresponds to thedensity of states that is a probability measure, and f ( − λ ) is the partition functionwhose odd (even) derivatives in inverse temperature are negative (positive). The secondR´enyi divergence has an inverse Laplace transform S A ( − λ )1 / ( e iβφ (cid:107) Ω) = (cid:90) ∞ ds e − λs µ ( s ) µ ( s ) = 2 β e − s cos θ (1 − cos( s sin θ )) s ≥ (cid:90) ∞ ds µ ( s ) = 2 β log sec θ < ∞ (97)which is indeed a probability measure. Therefore, we learn that all spatial derivatives ofthe second R´enyi divergence are non-negative. Next, we consider t = 1 that correspondsto a null deformation: S / ( e iβφ (cid:107) Ω) = β (cid:18) λ − λ cos θ (2 λ − cos θ ) (cid:19) + β log sec θ (98)22he λ dependent piece in (98) is the same as (95) with λ → λ , hence all of its λ derivatives as was proved before.Same logic generalizes to n >
2. Writing sandwiched R´enyi for n > t = 0 asa Laplace transform we find S A ( − λ )1 − n ( e iβφ (cid:107) Ω) = 2 β n − n − (cid:88) m =1 ( n − m ) (cid:90) ∞ dse − λs µ ( s ) µ ( s ) = e − s cos θ s (cid:16) cosh (cid:16) s sin θ cot (cid:16) mπn (cid:17)(cid:17) − cos( s sin θ ) (cid:17) ≥ ≤ θ ≤ π/n and integer n . In the null case, it is easier to consider S A ( λ )1 − n ( e iβφ (cid:107) Ω) − S A (0)1 − n ( e iβφ (cid:107) Ω) = β ( n − n − (cid:88) m =1 ( n − m ) (cid:90) ∞ ds e − sλ µ ( s ) µ ( s ) = e − s cos θ s (cid:16) cosh (cid:16) s θ cot (cid:16) mπn (cid:17)(cid:17) − cos (cid:16) s θ (cid:17)(cid:17) ≥ . (100)To summarize, we established that all spatial and null derivatives of R´enyi divergencesof coherent states with respect to vacuum are positive. The fact this holds for higherthan two derivatives in deformations is special to coherent states. To show this, we con-sider the chiral primary operator ∂φ of the same theory. This operator has dimension(1 ,
0) and its 2 n -point functions were computed in [35]: G n = (cid:42) n (cid:89) i =0 ( ∂φ ) † ( ∂φ ) (cid:43) = 14 n det (cid:34) (cid:0) z i − z j (cid:1) (cid:35) i,j ∈ [1 , n ] = Γ (cid:16) n + n csc( nθ )2 (cid:17) Γ (cid:16) − n + n csc( nθ )2 (cid:17) . (101)Therefore, S − n ( ∂φ (cid:107) Ω) = 1 n − (cid:0) (2 sin θ ) n G n (cid:1) (102)A spatial deformation correponds to cos θ → cos θ − λ in the formula above. Then, thefirst spatial derivative is given by ∂ λ S A ( λ )1 − n ( i∂φ (cid:107) Ω) λ =0 = 2 n cos θ ( n − (cid:18) − n tan θ tan( nθ ) sin( nθ ) (cid:18) ψ (cid:18) n + n csc( nθ )2 (cid:19) − ψ (cid:18) − n + n csc( nθ )2 (cid:19)(cid:19)(cid:19) . Note that this operator is real, so our proof of the non-negativity of the second deriva-tive in spatial deformations applies to it. Figure 3 is a plot of the first four derivativesin spatial deformations. The first and the second derivatives of sandwiched R´enyi di-vergence are non-negative as expected, however the fourth derivative becomes negativefor large enough n . 23 .02 0.04 0.06 0.08 0.10 0.12 θ First spatial derivative θ Second spatial derivative θ Third spatial derivative θ - - Fourth spatial derivative
Figure 3: (top) The first and the second spatial derivatives of the sandwiched R´enyi diver-gence for n = 2 to n = 26 are non-negative and consistent with monotonicity and our proof.(bottom) The third and the fourth spatial derivatives are plotted for n = 2 to n = 16. Forlarge enough n , the fourth derivative becomes negative.
24s the second example, we consider the states of an interacting conformal fieldtheory with θ (cid:28)
1. We expand each pair of operators in small θ using the operatorproduct expansion:Φ † k Φ k (cid:104) Φ † k Φ k (cid:105) = (cid:88) p (2 sin θ ) h p C pφφ O p ( z k , z k ) z k = (cos θ − λ ) sin (cid:18) πmn (cid:19) − (sin θ + itλ ) cos (cid:18) πmn (cid:19) z k = (cos θ − λ ) cos (cid:18) πmn (cid:19) − (sin θ + itλ ) sin (cid:18) πmn (cid:19) , (103)where O p represent primaries and their descendants. The 2 n -point function in thislimit islog (cid:32) (cid:10)(cid:81) n − k =0 Φ † ( z + k ( λ ))Φ( z − k ( λ )) (cid:11)(cid:10)(cid:81) n − k =0 Φ † ( z + k )Φ( z − k ) (cid:11) (cid:33) = log (cid:32) C ∆ φφ ) n − (cid:88) m =1 ( θ ) (cid:0) c ( θ = 0) sin (cid:0) mπn (cid:1)(cid:1) ∆ + · · · (cid:33) = (cid:18) θ − λ + λ (1 − t ) (cid:19) ∆ n − (cid:88) m =1 (cid:0) πmn (cid:1) + · · · (104)where ∆ is the dimension of the lightest scalar primary. The inverse Laplace transformof the first term with λ → − λ is: (cid:18) θ λ + λ (1 − t ) (cid:19) ∆ = (cid:90) ∞ ds e − sλ µ t ( s ) µ ( s ) = e − s s − Γ(2∆) µ ( s ) = e − s/ s ∆ − ∆ Γ(∆) , (105)which implies that all derivatives of the sandwiched R´enyi divergence are non-negativefor spatial and null deformations at this order in perturbation theory.One can go to higher orders in perturbation theory. If we we focus on the case t = 0and assume that the second light primary has dimension ∆ > ∆ , at the next orderin perturbation theory we havelog (cid:32) C ∆ φφ ) (cid:88) m =1 (2 sin θ ) (cid:0) c ( θ ) sin (cid:0) mπ (cid:1)(cid:1) ∆ + C ∆∆∆ (cid:18) (2 sin θ )3(1 − λ ) (cid:19) + · · · (cid:33) (106)Since the first order term is non-negative it is hard to use the monotonicity constraintto derive an inequality regarding only the OPE coefficients. We thank Shu-Heng Shao for pointing this out to us. Discussion and Generalizations
In summary, we defined the sandwiched R´enyi divergence as a R´enyi generalizationof relative entropy in any von Neumann algebra and explored the consequences of itsmonotonicity for correlation functions of field theory. We found new inequalities forcorrelation functions and conjectured a constraint on the second in null derivativesof the R´enyi family that is a generalization of quantum null energy condition. It isinteresting to explore the implications of these inequalities in large N and holographictheories. The holographic dual of sandwiched R´enyi divergences was constructed in[36]. It is natural to ask whether monotonicity can be used to constrain the effectivefield theory in the bulk. We postpone this to future work.The physics interpretation of the R´enyi divergences comes from their connectionwith the resource theory of thermodynamics. In an out-of-equilibrium quantum systemwith long-range interactions, there are many independent second laws of thermody-namics that constrain state transformations, each corresponding to a generalized freeenergy [37]. These free energies are precisely the sandwiched R´enyi divergences andPetz divergences we studied here. This suggests that in conformal field theory andgravity the monotonicity of R´enyi divergences for n > f ( z ) = z α . An arbitraryoperator monotone function can be characterized in the following way: A function isoperator monotone if and only if it has the representation [39] f ( z ) = az + b + (cid:90) ∞ dβµ ( β ) (cid:18) β − z + β (cid:19) (107)for a, b ≥ µ ( β ) a measure that satisfies (cid:90) dββ µ ( β ) + (cid:90) ∞ dββ µ ( β ) < ∞ . (108)For instance, if we take µ ( β ) = sin( πα ) π β α the condition above is achieved if 0 < α < f ( z ) = z α . It would be interesting to see if one can relatethe monotonicity of other f (∆ Ω | Ψ ) to correlation functions and obtain new constraints.26 Acknowledgments
I am greatly indebted to Edward Witten whose suggestion to consider the non-commutative L p spaces initiated this work. I would also like to thank Nima Arkani-Hamed, HongLiu, Raghu Mahajan, Srivatsan Rajagopal, Shu-Heng Shao, Douglas Stanford and YohTanimoto for discussions at various stages of this project. This work was supported bya grant-in-aid (PHY-1606531) from the National Science Foundation. A Proof of Properties Listed for Quasi-entropy
In this appendix we prove the the follwoing properties of the Petz quasi-entropy:1. If A ⊂ B and a ∈ A A then it increases monotonically with system size: D Aα,a (Φ (cid:107) Ω) ≤ D Bα,a (Φ (cid:107) Ω) .
2. If U is a unitary in A A then D Aα,a ( U Φ (cid:107) V Ω) = D Aα,V † aU (Φ (cid:107) Ω).3. It increases monotonically in α . If α ≤ β then D Aα,a (Φ (cid:107) Ω) ≤ D Aβ,a (Φ (cid:107) Ω).Statements (1) and (2) follows respectively from the monotonicity relation in (16)and the transformation rule (11) for the relative modular operator under unitaries inthe algebra. To show the monotonicity in α we consider the spectral decomposition ofthe relative modular operator: ∆ − | Φ = (cid:90) ∞ λ P ( dλ ) , (109)where P ( dλ ) is a projection-valued measure. According to Holder’s inequality, if µ ( λ )is a normalized probability measure, α > p > p + q = 1 we have: (cid:90) λ α dµ ( λ ) ≤ (cid:18)(cid:90) λ pα dµ ( λ ) (cid:19) /p (cid:18)(cid:90) dµ ( λ ) (cid:19) /q = (cid:18)(cid:90) λ pα dµ ( λ ) (cid:19) /p . (110)Choosing p = β/α > (cid:104) Ψ | ∆ − α Ω | Φ | Ψ (cid:105) = (cid:90) λ α (cid:104) Ψ | P ( dλ ) | Ψ (cid:105) ≤ (cid:18)(cid:90) λ β (cid:104) Ψ | P ( dλ ) | Ψ (cid:105) (cid:19) α/β = (cid:104) Ψ | ∆ − β Ω | Φ | Ψ (cid:105) α/β , where | Ψ (cid:105) is an arbitrary state. This establishes monotonicity in α . That is for0 ≤ α ≤ β ≤ D α,a (Φ (cid:107) Ω) = 1 α log (cid:104) Φ | a † ∆ − α Ω | Φ a | Φ (cid:105) ≤ β (cid:104) Φ | a † ∆ − β Ω | Φ a | Φ (cid:105) = D β,a (Φ (cid:107) Ω) . (111)A similar argument using the spectral decomposition of ∆ Ω | Φ shows that the aboveequation holds for any − ≤ α ≤ β ≤
1. 27
Proof of Properties Listed for Petz divergence
In this appendix, we prove of the properties of the Petz divergence listed below (19).1. It is non-negative and vanishes when A shrinks to zero.2. It is invariant under the rotation of both vectors by the same unitary in A .3. At α = 0 it is smooth and equal to the relative entropy.4. Under swapping vectors | Φ (cid:105) and | Ω (cid:105) it satisfies D − α (Φ (cid:107) Ω) = 1 − αα D α − (Ω (cid:107) Φ) . (112)Statement (1) follows from the inequality in (15). Since Petz divergence is mono-tonic under the restriction to subregions and it vanishes as the region size shrinks tozero, it follows that it is non-negative. The second statement follows trivially by setting a = 1 and U = V in property (2) of quasi-entropies. The limit α → α → D α (Φ (cid:107) Ω) = − log (cid:104) Φ | log ∆ Ω | Φ | Φ (cid:105) = S (Φ (cid:107) Ω) . (113)Statement (4) says that under swapping vectors | Φ (cid:105) and | Ω (cid:105) the Petz divergencesatisfies D − α (Φ (cid:107) Ω) = 1 − αα D α − (Ω (cid:107) Φ) . (114)Remember that the Petz divergence is invariant under | Φ (cid:105) → U (cid:48) | Φ (cid:105) with U (cid:48) a unitaryin the A (cid:48) . This freedom can be used intelligently to satisfy | Φ (cid:105) = ∆ / | Ω | Ω (cid:105) . Such achoice is called the vector representative of the state in the “natural cone”. See [7] formore detail. Equipped with this fact, the claim follows: D − α (Φ (cid:107) Ω) = − α log (cid:104) Φ | ∆ α Ω | Φ | Φ (cid:105) = 1 − αα (cid:18) α − (cid:104) Ω | ∆ / | Ω ∆ α Ω | Φ ∆ / | Ω | Ω (cid:105) (cid:19) = 1 − αα (cid:18) α − (cid:104) Ω | ∆ − α Φ | Ω | Ω (cid:105) (cid:19) = 1 − αα D α − (Ω (cid:107) Φ) . (115)At the symmetric point D − / (Φ (cid:107) Ω) = D − / (Ω (cid:107) Φ) = − (cid:104) Φ | Ω (cid:105) . The non-negativityof the Petz divergence implies that 0 ≤ (cid:104) Φ | Ω (cid:105) ≤ Proof of Inequality (32)
The claimed inequality in (32) is: D α − α (Φ (cid:107) Ω) ≥ S α (Φ (cid:107) Ω) ≥ D α (Φ (cid:107) Ω) . (116)The lower bound follows from the definition of the sandwiched R´enyi divergences.The upper bound can be written as (cid:104) Φ | ∆ − α − α Ω | Φ | Ω (cid:105) (1 − α ) α ≥ sup Ψ (cid:104) Φ | ∆ − α Ω | Ψ | Φ (cid:105) α . (117)Using | Φ (cid:105) = ∆ − / | Φ | Ω (cid:105) it simplifies to (cid:107) ∆ − − α ) Ω | Φ | Φ (cid:105)(cid:107) (1 − α ) ≥ sup Ψ (cid:107) ∆ − α Ω | Ψ | Φ (cid:105)(cid:107) . (118)Then, we use (∆ A Ω | Ψ ) − = ∆ A (cid:48) Φ | Ω and α = 1 − p to we write it as (cid:107) (∆ A (cid:48) Φ | Ω ) p | Ω (cid:105)(cid:107) p ≥ sup Ψ (cid:107) (∆ A (cid:48) Ψ | Ω ) − p | Φ (cid:105)(cid:107) . (119)This inequality is a generalization of the Araki-Lieb-Thirring inequality [40] to von Neu-mann algebras that can be proved using the interpolation theory of non-commutative L p spaces. However, this goes beyond the scope of this work and we refer the interestedreader to the proof presented in Theorem 12 of [13]. References [1] H. Casini and M. Huerta,
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