Renyi relative entropies and renormalization group flows

Renyi relative entropies and renormalization group flows


Quantum Renyi relative entropies provide a one-parameter family of distances between density matrices, which generalizes the relative entropy and the fidelity. We study these measures for renormalization group flows in quantum field theory. We derive explicit expressions in free field theory based on the real time approach. Using monotonicity properties, we obtain new inequalities that need to be satisfied by consistent renormalization group trajectories in field theory. These inequalities play the role of a second law of thermodynamics, in the context of renormalization group flows. Finally, we apply these results to a tractable Kondo model, where we evaluate the Renyi relative entropies explicitly. An outcome of this is that Anderson’s orthogonality catastrophe can be avoided by working on a Cauchy surface that approaches the light-cone.


1 Introduction

Quantum field theory (QFT) describes the long-distance limit of many systems of interest in high energy and condensed matter physics. Novel collective phenomena are often observed at strong coupling, and a long-term goal is to develop tools to understand strongly interacting QFTs. In this direction, we have witnessed important recent progress by applications of results from quantum information theory (QIT). By studying how degrees of freedom are entangled, and how this changes from microscopic to macroscopic scales, new results on the nonperturbative behavior of QFTs have been obtained. These range from ansatze for ground state wavefunctions, to irreversibility of the renormalization group and insights into quantum gravity.

In this work we will focus on certain nonperturbative aspects of the renormalization group (RG). The RG gives flows or trajectories in the space of couplings as a function of some distance or energy scale. These flows generically include fixed points, as well as relevant and irrelevant trajectories [1]. This is illustrated in Fig. 1, which shows two fixed points and , relevant flows from to (in red), and irrelevant flows (in blue).

Figure 1: Possible RG trajectories between two fixed points, following [1]. Red lines denote relevant trajectories from , while blue lines are associated to irrelevant deformations.

By now, it has been established that the RG for unitary relativistic QFTs is irreversible in two [2, 3], three [4] and four space-time dimensions [5, 6]. This means that one can associate an intrinsic quantity to a fixed point, and a necessary condition to connect two fixed points by the RG is that


This can be understood in terms of the entanglement entropy (EE),




is the vacuum reduced density matrix obtained by tracing over the degrees of freedom in the complement of a region . The irreversibility of the RG is a consequence of unitarity and strong subadditivity of the entanglement entropy [3, 4, 6].1

Here we are interested in the following question: given two fixed points that satisfy (1.1), which RG flows between them are possible? In particular, we would like to establish necessary conditions that must be satisfied by consistent RG trajectories in unitary QFTs. For this, it is natural to analyze the RG in terms of a quantum information distance between two reduced density matrices: , characterizing the UV fixed point, and , associated to the theory that undergoes the flow. One very useful notion of distance is the relative entropy,


which measures the distinguishability between the two states. The relative entropy is positive and monotonic under increasing the size of the region. Based on this, Ref. [7] proved the entropic version of irreversibility of boundary RG flows, and [8] gave an alternative proof of the -theorem. Another notion of distance is the quantum fidelity,


which reduces to the overlap between wavefunctions when the states and are pure. Such overlaps appear for instance in the evaluation of the boundary entropy in terms of boundary states [9].

In this work we will analyze a family of distance measures called quantum Renyi relative entropies,


Although other definitions of measures exist in the QIT literature (e.g. [10]), our main motivation for studying (1.6) is that the interpolate between (1.4) and (1.5) as varies between and . This fact, together with monotonicity properties satisfied by the , will allow us to derive new necessary conditions for consistent RG flows.

The nonlinear dependence of on and makes explicit calculations quite hard, and there has not been much work on understanding the physical content of these measures. Some previous works include [11], where the were evaluated in 2d CFTs using the replica trick, and [12], which studied the quantum Renyi divergences [10] for excited CFT states. Furthermore, the paper [13] studies quantum Renyi divergences to second order in perturbation theory using holography.

In this paper we compute in free field theories using real time methods (Sec. 3). We next consider in Sec. 4 general consequences of the monotonicity properties of for RG flows. By focusing on the light-cone limit studied in [7, 8], we show that the , which characterize the full RG trajectory, are bounded by quantities intrinsic to the fixed points, such as the boundary entropy or the central charge. Our main results on this, the inequalities (4.8) and (4.13), closely resemble the second law of thermodynamics , where a change in the entropy (a function of state) bounds a quantity that depends on the process. Finally, we illustrate these results in Sec. 5 with an explicit evaluation of in a tractable Kondo model. In particular, we find that Anderson’s orthogonality catastrophe [14] can be avoided by computing overlaps of ground states in the light-cone limit; this could be of interest for more general impurity problems in condensed matter physics.

2 Quantum Renyi relative entropies

Let us begin by reviewing some basic properties of the . The quantum Renyi relative entropies (QRRE in what follows) are defined as [15, 16]


They appear as a natural generalization of the Renyi relative entropies [17] that includes the quantum non-commutativity of the density matrices involved.

In this work we focus mostly on the range . In particular the edges of this interval are characterized by previously known quantum information measures. When we have the fidelity distance,


where denotes the quantum fidelity (1.5). Another interesting case is the limit


Here is the quantum relative entropy (1.4). Then, quantum Renyi relative entropies appear, when , as an interpolation between quantum fidelity and quantum relative entropy. As discussed in Sec. 1, this is one of our main motivation for considering (2.1), as opposed to other alternative forms such as [10]

For the fidelity, a useful representation is given by Uhlmann’s theorem [18], which states that


over purifications of . Given this result, properties of the fidelity can be easily proved. For instance, (2.4) makes it clear that the fidelity is symmetric in its inputs . One can also see that it is bounded . If , , while if and only if and have support on orthogonal subspaces.

The Renyi relative entropies also admit representations in terms of extremizing quantities. For instance, in [19] it was shown that


for ; the minimum should be replaced by the maximum for . A similar representation is derived in [15]. These representations are at the basis of the monotonicity properties that we will now review.

The are monotonically increasing in  [15, 20, 19]


Both the fidelity distance and the relative entropy are positive, and equal to zero only when . Eq. (2.6) then gives the same properties for the ,


Another important property is monotonicity when increasing the size of the algebra. If we consider two regions , then


This result uses (2.5); see e.g. [19]. This property is intuitive in QFT: the information-theoretic distance decreases for smaller regions, because there are less operators localized in the region that can be used to distinguish the states.

In the following sections we will study the consequences of these equations for the RG.

3 Renyi relative entropies in free field theory

In this section we will calculate the QRREs in free QFT. These are the simplest possible models in field theory, and hence provide a natural place to start understanding the . Notwithstanding their simplicity, free models provide an interesting setup for QIT measures, where properties of more general QFTs may be recognized.2 Gaussian states also play a prominent role in quantum information theory, quantum optics and atomic physics –see e.g. [24] for a review on Gaussian quantum information. In fact, our approach will make some contact with previous works in these areas, such as [25, 26, 27].

We will work in real time, relating the Gaussian correlators on a fixed Cauchy slice to the density matrix; this procedure is reviewed in [28]. This approach is also useful for lattice calculations, and the results will be applied to a Kondo model in Sec. 5 below. Our results are valid for Gaussian states, which have broader applicability than free QFTs. This approach is presented in Appendix A. In this section, however, we frame the discussion in terms of free theories.

3.1 Renyi relative entropies for free fermions

Consider two field theories of fermions, with the same field content, but with different Hamiltonians. We will restrict to free theories (quadratic Hamiltonians), which lead to Gaussian ground states. In the present derivation we work at zero temperature and vanishing chemical potential, but do not require Poincaré invariance. Let us denote the two different Hamiltonians on the lattice by


In the standard case, these Hamiltonians arise from discretizing the theories of interest on a constant time Cauchy surface. But let us point out from the start that we will also be interested in more general Cauchy surfaces. In particular, in relativistic theories below, the appropriate Cauchy surfaces will approach the light-cone limit.

The reduced density matrices are denoted by and , respectively; as in (1.3), these are obtained by tracing over the fermions on the sites in the complement of the set . Our goal is to compute .3

The fermion modes obey . The non-vanishing two-point correlators on the Cauchy surface are given by the zero-temperature Fermi-Dirac distribution,


with . Similar expressions hold for the other theory, with .

Consistently with Wick’s theorem, the reduced density matrix is given by a Gaussian state [29, 28]


where , which is known as the modular Hamiltonian, is fixed in terms of the correlator by requiring . The result is


The normalization constant .

The QRREs can be calculated explicitly because of two key properties. First, for a Gaussian state , the power is again a Gaussian state, with modular Hamiltonian . Second, because of the algebra of creation and annihilation operators, the product of two different Gaussian states is again a Gaussian state, whose modular Hamiltonian matrix can be obtained in terms of the Baker-Campbell-Hausdorff (BCH) formula.

In order to see this, it is convenient to introduce Majorana fermions , and rewrite the reduced density matrix as


where is real and antisymmetric. Then using obtains [30]


This allows to compute products of Gaussian density matrices, which is what we need to evaluate the . This method was used in [26] to compute the fidelity, and a similar approach is presented in the Appendix for the calculation of . The final result is

Recall that is the correlator associated to , while is the one associated to .

3.2 Resolvent method

Eq. (3.1) gives a closed expression for in terms of the fermion correlators y . In order to compute the previous nontrivial powers of operators, it is often convenient to use their resolvents.

Let us introduce the resolvent of an operator ,


We have added the term compared to the standard definition in order to achieve convergence at large . We can do this because, in all our expressions below, this term will be multiplied by functions that vanish at .

The correlators and have eigenvalues between 0 and 1. For a given eigenvalue ,


and thus


Next, we focus on the more complicated matrix


and we need to compute . The matrix has positive eigenvalues. For a single eigenvalue , we have


where is a contour that runs anti-clockwise around (it does not contain ). The term proportional to has vanishing integral, but is added in order to have an integrable integrand at large . Let us choose the branch cut of to be at . We can then deform the contour to run between , with the result




Using (3.10) and (3.14), the Renyi relative entropies (3.1) become

In Sec. 5 we will apply these results to the case of a Dirac fermion coupled to a Kondo impurity, which undergoes an RG flow.

3.3 Free bosons

Let us now focus on free bosons. The lattice Hamiltonian is of the form


where and obey the canonical commutation relations . We will consider two different Hamiltonians, with quadratic kernels and , and evaluate for their corresponding reduced density matrices.

The two-point functions on the Cauchy surface are parametrized as




See e.g. [28] for a review of these points. The equations in (3.17) imply the matrices and are real Hermitian and positive. Furthermore, introducing


the eigenvalues of are greater than .

Consistently with Wick’s theorem, the reduced density matrix is given by a Gaussian state of the general form


This density matrix can be diagonalized by a Bogoliubov transformation, which allows to relate and to the correlators (3.17). Explicit expressions may be found in [28].

In order to compute the QRREs, it is convenient to introduce the variables , which satisfy , where is the symplectic matrix. In this parametrization, the reduced density matrix is of the form


where is real and symmetric, and its blocks are determined by and in (3.20). Products of Gaussian states are then given by [30]


This method was used in [27] to compute the fidelity for the bosonic case.

Using these properties, in the Appendix we evaluate , obtaining


where is given by the following expression


(Recall that is the correlator associated to , while is the one associated to .) The remaining nontrivial powers of and can be computed explicitly, but the final expressions in terms of the original correlators are rather complicated and will not be presented here. These complications are due to the fact that, unlike the fermionic case, here the matrices are not block-diagonal, and depend on (or ) as well as on .

4 Bounds on renormalization group flows

In this section we analyze some general implications of the monotonicity properties of the for RG flows.

For reduced density matrices on a region of typical size , and for a relevant flow with energy scale , the Renyi relative entropies will depend on the dimensionless combination . The then define a distance that characterizes the RG. The limit corresponds to the UV, where the relevant deformation flows to zero and ; in this case, . From (2.8), increases with , signaling an increased distinguishability between the two states. The limit parametrizes the IR, where approaches the density matrix of another fixed point. The RG flow will generically be nonperturbative in nature, and will be sensitive to the full trajectory. The other property that will play an important role is (2.6), which implies that the QRREs with are bounded above by the relative entropy,


4.1 Boundary RG flows

Boundary RG flows occur when a 2D boundary CFT is perturbed by a relevant operator at the boundary ,4


This triggers a nontrivial RG flow, which we assume ends at a different infrared boundary CFT, . A boundary CFT is characterized by an intrinsic quantity known as the boundary entropy . It can be obtained as the part of the thermal entropy that is independent of the size of the system [31],


This quantity decreases along boundary RG flows [31, 32], , a statement known as the -theorem. A physical realization of this setup occurs for instance in the Kondo problem, where measures the impurity entropy. The boundary entropy can also be obtained from the entanglement entropy on an interval  [33],


with a short distance cutoff and a bulk constant contribution that drops out from the differences we are interested in.

Let be the density matrix of reduced to the interval , and the corresponding quantity for the theory (4.2) with nontrivial RG flow. Introducing the modular Hamiltonian , the relative entropy can be written as


where , and . Since the relative entropy contains a piece that is the difference between entanglement entropies of the two theories, it is sensitive to the change in boundary entropy . However, in general the change in the modular Hamiltonian dominates in the relative entropy, with .

Figure 2: Choice of different Cauchy surfaces inside the interval . As the Cauchy surface approaches the light-cone, the contribution .

A direct connection between the relative entropy and the change in boundary entropy obtains by quantizing on a Cauchy surface that approaches the light-cone [7]. This is illustrated in Fig. 2. is independent of the choice of Cauchy surface, but depends on because the two density matrices evolve with different unitary operators. In the light-cone limit, , and then


Positivity of the relative entropy then implies that decreases monotonically under boundary RG flows, thus establishing the entropic -theorem [7],


In the present setup of boundary RG flows with Cauchy surface on the light-cone, the inequality (4.1) gives


for a region with one endpoint at the boundary. In particular, in the IR limit , with the mass scale associated to the RG,


The right hand side of this inequality is finite and depends only on the UV and IR fixed points, and not on the specific RG trajectory that connects them. On the other hand, we expect the left hand side to depend on the RG trajectory.

Eq. (4.9) is our main result for boundary RG flows; it provides a bound on all possible RG trajectories connecting two BCFTs. The upper bound depends only on intrinsic quantities of the fixed points. From a thermodynamic perspective, (4.9) resembles the second law . The thermal entropy is a function of state, while heat transfer depends on the process. In our present context, the “function of state” corresponds to the boundary entropy, which depends only on properties of the fixed point, while the QRRE, like heat transfer, is sensitive to the specific trajectory in coupling space. One interesting difference between the thermodynamic and quantum cases is that, while in the former case there exists nontrivial adiabatic processes with no exchange of heat, in the quantum setup “adiabatic RG flows” are not possible. In other words, is always strictly positive. This follows from the fact that this difference is a relative entropy, which vanishes only for . But if the two density matrices agree, then all correlators are the same, and there is no RG flow.5

Finally, we note that as , we expect the fidelity to approach the overlap of the corresponding vacuum wavefunctionals. The inequality (4.9) for then yields


The wavefunctionals here are defined on the past (or future) null infinity. The finite right hand side implies a nonzero overlap . This is an interesting outcome, which provides a way of avoiding Anderson’s orthogonality catastrophe in relativistic systems. Anderson’s result [14] states that, under mild assumptions, a many-body fermion ground-state wavefunction in the presence of a local perturbation is orthogonal to the unperturbed ground state, . In fact, if we work on a Cauchy surface at constant time, we expect the same result for in the more general boundary RG flows we are considering –we will see an example of this in Sec. 5. However, the orthogonality is avoided by taking the light-cone limit. The finite overlap is guaranteed by the relative entropy becoming finite in this limit, and corresponds to both theories being less distinguishable on the light-cone. It would be interesting to understand other consequences of this result.

4.2 RG flows in dimensions

We will now consider RG flows in spacetime dimensions, where the fixed points –denoted by and – are Poincare invariant unitary CFTs. These flows can be produced by turning on relevant deformations in ,


with the scaling dimension of at the UV fixed point. The light-cone construction summarized in the previous section has been extended to this case in [8], and we will now examine the implications of (4.1).

We again introduce two reduced density matrices and , associated to the sphere ; corresponds to , while arises in (4.11). Each of them is obtained by starting from the vacuum state of the corresponding theory, and tracing over the degrees of freedom in the complement of sphere. The theories have the same operator content, but evolve with different hamiltonians. The QRREs provide distance measures for the RG. We can evaluate them on different Cauchy surfaces inside the causal domain of dependence on (recall Fig. 2), and we will focus on surfaces that approach the light-cone.

Let us focus first on spacetime dimensions. Ref. [8] showed that the modular Hamiltonian contribution to the relative entropy vanishes in the light-cone limit, as in the case of boundary RG flows. Denoting the characteristic mass scale of (4.11) by , the relative entropy in the large distance limit becomes [8]


where is the CFT central charge. Positivity of the relative entropy then provides an alternative proof of the -theorem, .6 Combining this with (4.1), we find the following restriction on RG trajectories:


As in (4.9), we have here a function of the trajectory being bounded above by a quantity that is intrinsic to the fixed points. We conclude that the distances can grow at most logarithmically at long distances, and with a coefficient that is smaller than .

In higher dimensions, the relative entropy in the light-cone limit is dominated by the area term in the entanglement entropy,


(Note that in , the central charge also appears as the leading area term.) For relevant deformations with dimension , the modular Hamiltonian contribution to the relative entropy vanishes, and [8]


in the limit . Therefore, we arrive at the constraint


Unlike the previous cases, is not a combination of intrinsic quantities. However, it is still an interesting object in QFT. It is given by the integral of the stress tensor correlator [34, 35]


where is the trace of the stress tensor. This quantity is finite when , and depends on the RG trajectory.7 It is also proportional to the renormalization of Newton’s constant due to the field-theoretic degrees of freedom in flat space.

This ends our general discussion on constraints for RG flows. In the next section we will study the QRRE in a concrete Kondo model with a nontrivial flow.

5 Application to the free Kondo model

Finally, we will study the distances in the Kondo problem introduced in [36]. This model is free but it supports a nontrivial boundary RG flow, providing an interesting setup where the can be evaluated nonperturbatively in the relevant deformation.

5.1 The free Kondo model

The model consists of a free Dirac fermion living in half-space . This ‘bulk’ fermion is coupled quadratically to a fermionic degree of freedom that lives at the boundary ,


In the UV, we choose the boundary condition that relates the two chiralities,


In the IR, , the mass term dominates over the impurity kinetic term, and extremizing over sets


Hence we obtain a boundary RG flow between ’’ and ’’ boundary conditions for the Dirac fermion.

The lattice version of the theory contains a single fermion hopping in a one-dimensional lattice, with just one special site corresponding to the impurity,


Here is the lattice spacing, and is the impurity fermion. The spectrum of contains left and right moving low energy modes, as expected from the usual fermion doubling. Furthermore, constructing a Majorana fermion out of the lattice fermion , the last term in (5.4) produces the quadratic coupling of (5.1). In this way, the continuum limit of this lattice model reproduces (5.1).

As a first step, let us evaluate the fidelity distance on a Cauchy surface at constant time. In the next section we will analyze the light-cone limit.

For this, we need to calculate the equal-time fermion two-point functions and . is the correlator for arbitrary mass , while arises for the particular case . This calculation was described in detail in [7], and for completeness here we summarize the main points.

The quadratic kernel for the Hamiltonian in (5.4) can be diagonalized in terms of momentum modes ; the energies in units of read . The reflection coefficient follows from the first two equations in the diagonalization of , and becomes


Taking the continuum limit, in the UV, and in the IR. This reproduces the boundary RG flow in (5.2) and (5.3). The equal time correlator is simply given by


Using then in (5.6) the explicit expressions for the wavefunctions we can compute the fermionic correlators for the theories we want to compare and, with the help of (3.1), we finally evaluate numerically .

In Fig. 3 we show as a function of the interval size, for intervals with one end at the boundary. The increase of with is rather slow, but our analysis shows that as . In other words, the fidelity in this limit. This is consistent with the bound (4.1), since grows more slowly than the relative entropy, which scales like for large intervals [7]. Our numerical findings indicate then that the whole family of diverges when , since the QRREs are monotonically increasing in .

Figure 3: Quantum fidelity for the RG flow in the Kondo model, formulated on the line.

As stated by Uhlmann’s theorem (2.4), the fidelity can be defined as a maximization over purifications. For , the system containing the impurity is becoming pure, and then the fidelity is given by the overlap of vacuum wavefunctions of the theory with and without mass perturbation. We can then understand the fact that as an expression of Anderson’s orthogonality catastrophe [14], to the effect that the ground state in the presence of a local perturbation should become orthogonal to the original ground state in the thermodynamic limit.

5.2 The Kondo model on the null line

Given the previous results on a Cauchy surface at constant time, we will now analyze the measures on regions lying on the null line with one end on the boundary. From the general discussion in Sec. 4.1, this should give finite QRREs as . The reason for this change in behavior as we modify the Cauchy surface is that we are comparing two density matrices and that evolve with different Hamiltonians.

The fermion correlator on the null line takes the form [7]






Here Ei is the exponential integral function


In these expressions, .8

In order to compute the QRRE, we evaluate the resolvents defined in (3.2). This requires calculating a few inverses and powers of correlators. The density matrix corresponds to a correlator (5.7) with (this is the UV fixed point), while for we need to take an arbitrary . This computation is quite lengthy but straightforward, and we detail the steps in Appendix B. The resulting expression for the QRRE is


In Fig. 4 we show the result of evaluating (5.11) numerically for several values of . As predicted from the general properties of QRRE, the curves are monotonically increasing as we increase the region size . Also, we observe the monotonicity in , with for and for all values . In the limit , all the collapse to zero, consistent with . These curves provide measures of distances between the states and along the RG, with corresponding to the UV (high energies), while approaches the IR limit.

Figure 4: QRREs for the RG flow in the Kondo model, formulated on the light-cone. We show different values of , and the limiting result for and .

Finally, let us evaluate the limit . In this limit, the integral for approximates to


The prefactor drops out from all the expressions, since only enters. The integral can now be performed analytically, giving


Fig. 5 shows the results for when . In particular, we find that these distances asymptote to different values which depend on . For we recover the result for the change in the impurity entropy [7]. On the other hand, for we expect to depend on the RG flow in between the fixed points. Finally, for , measures the overlap between the wavefunctionals with and without perturbation. We obtain a finite result on the null Cauchy surface, providing a way to avoid Anderson’s orthogonality catastrophe for this system.

Figure 5: in the limit, for different values of .

6 Conclusions and future directions

In this work we studied the quantum Renyi relative entropies for RG flows in quantum field theory. These measures provide interesting distances that characterize new aspects of the RG. We presented explicit expressions for the QRREs in free field theories, focusing in particular on fermionic systems. In this case, we studied a nontrivial RG flow in a Kondo model, and evaluated numerically the . The results on the light-cone limit are finite, show the irreversibility of the RG, and are expected to depend on the trajectory in coupling space.

Using QIT properties of the , we obtained constraints that consistent RG trajectories need to obey. These are strongest for boundary RG flows and flows in two spacetime dimensions – we showed that the measures are bounded above by differences of quantities that are intrinsic to the fixed points (impurity entropy or central charge). In higher dimensions, the bound is given by the area term in the entanglement entropy, which also measures the renormalization of Newton’s constant. For boundary RG flows, and flows in two spacetime dimensions, these constraints closely resemble the second law of thermodynamics. This hints towards the thermodynamic nature of the RG, and it would be interesting to develop this analogy further.

Our analysis suggests several directions to explore. At the level of concrete examples, one can generalize the Kondo model to include more impurities with various relevant parameters. This would provide dimensionless couplings that parametrize different RG trajectories, and we expect a nontrivial dependence of the QRREs on such parameters. More generally, it would be useful to find a more direct connection between a given RG trajectory and the , perhaps in the form of a sum rule as in (4.17). It would be interesting to study RG flows in other number of dimensions and with defects of different codimension. It would also be important to develop tools to evaluate these distances for more general CFTs and their RG flows, for instance using conformal perturbation theory. In this direction, the euclidean approach could provide a new handle on the problem. Holographic duals of these measures should also give insights into their physical properties.


This work was partially supported by CONICET (PIP grant 11220150100299), CNEA, and Universidad Nacional de Cuyo, Argentina. H.C. acknowledges an “It From Qubit” grant of the Simons Foundation. G.T. is also supported by ANPCYT PICT grant 2015-1224.

Appendix A for Gaussian states

In this Appendix we present explicit calculations of for Gaussian states. For the purpose of this paper, they arise as the vacuum-reduced density matrices of free QFTs, but they also occur in more general setups, where the Hamiltonians are not necessarily quadratic.

a.1 Fermions

In this section we focus on fermionic Gaussian states, and derive (3.1). A similar procedure was used for the fidelity in [26].

Let us consider a system of fermionic modes described by a set Majorana operators . In terms of these variables, the two point correlation function is . The complex matrix is imaginary and anti-symmetric. Now, let us consider a gaussian fermionic state written in the form


with real and antisymmetric. It is possible then, to cast G in the canonical form by an orthogonal matrix


with the eigenvalues of . Now, let be the new Majorana operators. In this new basis, we find the following expression for the state


The value of the normalization constant Z is fixed by requiring ,


where we used the fact that the eigenvalues of are . and the correlation matrix are related by


Let us now evaluate the for states of the form (A.1). Recalling (3.6), it follows that


Finally, using (A.6), (3.6) and (A.4) we find for the QRRE


In order to express the QRRE as a function of the fermionic correlators we define the following convenient parametrization


in terms of which


Lastly, we take into account that we are interested in models with charge conjugation symmetry, which fixes . The matrix becomes


Using (A.10) and (A.9) we arrive at our final result

a.2 Bosons

For bosons, we shall discuss Gaussian states of the form


with vanishing terms. This appears naturally in systems with time-reversal invariance. Free bosonic QFTs of the form discussed in Sec. 3.3 are a special case; see (3.20).

Performing a Bogoliubov transformation as in [28], obtains


where and