Assessing the performance of quantum repeaters for all phase-insensitive Gaussian bosonic channels

One of the most sought-after goals in experimental quantum communication is the implementation of a quantum repeater. The performance of quantum repeaters can be assessed by comparing the attained rate with the quantum and private capacity of direct transmission, assisted by unlimited classical two-way communication. However, these quantities are hard to compute, motivating the search for upper bounds. Takeoka, Guha and Wilde found the squashed entanglement of a quantum channel to be an upper bound on both these capacities. In general it is still hard to find the exact value of the squashed entanglement of a quantum channel, but clever sub-optimal squashing channels allow one to upper bound this quantity, and thus also the corresponding capacities. Here, we exploit this idea to obtain bounds for any phase-insensitive Gaussian bosonic channel. This bound allows one to benchmark the implementation of quantum repeaters for a large class of channels used to model communication across fibers. In particular, our bound is applicable to the realistic scenario when there is a restriction on the mean photon number on the input. Furthermore, we show that the squashed entanglement of a channel is convex in the set of channels, and we use a connection between the squashed entanglement of a quantum channel and its entanglement assisted classical capacity. Building on this connection, we obtain the exact squashed entanglement and two-way assisted capacities of the $d$-dimensional erasure channel and bounds on the amplitude-damping channel and all qubit Pauli channels. In particular, our bound improves on the previous best known squashed entanglement upper bound of the depolarizing channel.


Introduction
Optical quantum communication over long distances suffers from innate losses [1][2][3][4][5]. While in a classical setting the signal can be amplified at intermediate nodes to counteract this loss, this is prohibited in a quantum setting due to the no-cloning theorem [6]. This problem can be overcome by implementing a quantum repeater, allowing entanglement over larger distances [7,8]. The successful implementation of a quantum repeater will form an important milestone in the development of a quantum network [9]. At this stage however, physical implementations perform worse than direct transmission [10,11]. As the experimental results improve it will be necessary to evaluate whether or not an implementation has achieved a rate not possible via direct communications. This can be done by comparing the attainable rate with a quantum repeater [12][13][14][15][16][17][18][19] to the capacity of the associated quantum channel (i.e. direct transmission) for that task. For future quantum networks, arguably the two most relevant tasks are the transmission of quantum information and private classical communication. The capacity of a quantum channel for these two tasks, assuming that we allow the communicating parties to freely exchange classical communication, is given by the two-way assisted quantum and private capacity. We denote these quantities by ( )  Q 2 and ( )  P 2 , respectively.
Finding exact values for ( )  Q 2 and ( )  P 2 , however, is highly non-trivial thus motivating the search for upper bounds for them [20]. After having shown that the squashed entanglement of a channel is a quantity that is such an upper bound [21], Takeoka, Guha and Wilde showed that there is a fundamental rate-loss trade-off in quantum key distribution and entanglement distillation over practical channels [22]. which was introduced by Christandl and Winter [23] as an entanglement measure for a bipartite state. The squashed entanglement can be interpreted as the environment E holding some purifying system of r AB , and then squashing the correlations between A and B as much as possible by applying a channel  ¢  E E that minimizes the conditional mutual information ¢ ( | ) I A B E ; . Extending this idea from states to channels, Takeoka, Guha and Wilde [21,22] defined the squashed entanglement ( )  E sq of a quantum channel as the maximum squashed entanglement that can be achieved between A and B, AA is the state shared between Alice and Bob after the ¢ A system is sent through the channel is an upper bound on the two-way assisted capacities. Unfortunately, there is no known algorithm for computing the squashed entanglement of a channel. This is partially due to the fact that the dimension of ¢ E is a priori unbounded and that computing the squashed entanglement of a state is already an NP-hard problem [24] and thus might even be uncomputable. However, fixing the channel in (1) in general yields an upper bound on ( )  E sq . Exploiting this idea of fixing a specific 'squashing channel'  ¢  E E , Takeoka et al derived upper bounds on the squashed entanglement of several channels. Notably, they used this technique to find an upper bound for the pure-loss bosonic channel.
The main contribution of this paper is an upper bound applicable to all phase-insensitive Gaussian bosonic channels. We apply this bound to the pure-loss channel, the additive noise channel and the thermal channel.
Additionally, we obtain results for finite-dimensional channels by using tools that we develop here. The first of these consists of a concrete squashing channel that we call the trivial squashing channel which can be connected with the entanglement-assisted capacity. This connection, first observed by Takeoka et al (see [25]), allows us to compute the exact two-way assisted capacities of the d-dimensional erasure channel, and bounds on the amplitude damping channel and general Pauli channels. Second, the squashed entanglement of entanglement breaking channels is equal to zero. Third, for channels that can be written as a convex sum of channels the convex sum of the squashed entanglement of each channel is an upper bound, i.e. ( )  E sq is convex on the set of channels. We combine all three of these tools to obtain bounds for the qubit depolarizing channel.

Notation
In this section we lay out the notation and conventions that we follow in this paper.
For a quantum state r A the von Neumann entropy of r A is defined as For convenience we take all logarithms in base two and set º (·) (·) log log 2 . For a quantum state r r ≔ AB the conditional entropy of system A given B is defined as = - A AB , where we denote the partial trace over system A of a state r AB by tr r ( ) A AB . For a tripartite state r ABE the conditional mutual information is defined as . Whenever there is potentially confusion regarding the state over which we are computing an entropic quantity we will add the state as a subscript.
A quantum channel ¢  A B is a completely positive and trace preserving map [26] between linear operators on Hilbert spaces ¢  A and  B . A quantum channel  can always be embedded into an isometry ¢  V A BE that takes the input to the output system B together with an auxiliary system E that we call the environment. This isometry is called the Stinespring dilation of the channel. The action of the channel is recovered by tracing out the environment: We denote the d-dimensional maximally mixed state by p. The dimension of p is implicit and should be clear from the context. Let  be a channel with input and output dimension d.
3. Some properties of ( )  E sq In this section we prove several properties of ( )  E sq that will be of general use for obtaining upper bounds on the squashed entanglement of concrete channels. First we define a squashing channel that we call the trivial squashing channel and connect it to the entanglement assisted capacity of that channel, an observation previously made in [25] by Takeoka et al. Second, we prove that the squashed entanglement of entanglement breaking channels equals zero. The third property is that ( )  E sq is convex in the set of channels.

The trivial squashing channel
One possible squashing channel  ¢  E E is the identity channel, which we will call the trivial squashing channel. The state on ¢ ABE is pure, from which it can easily be calculated that The maximization in the right hand of (7), up to the 1/2 factor, characterizes the capacity of a quantum channel for transmitting classical information assisted by unlimited entanglement [27]. In other words, the squashed entanglement is bounded from above by one half the entanglement assisted capacity of the channel which we denote by ( )  C E . This connection, which was first observed by Takeoka et al (see [25]), allows us to bound the squashed entanglement for all channels for which ( )  C E is known.

Entanglement breaking channels
Entanglement breaking channels have zero private and quantum capacities assisted by two-way communications. We show that the squashed entanglement of these channels is also zero, following a similar approach as was done for the squashed entanglement of separable states in [28]. In order to see this note that if an entanglement breaking channel  EB is applied to half of a bipartite state, the output is always separable and can be written as a convex combination of product states which has zero conditional mutual information.

Convexity of
in the set of channels The squashed entanglement of the channel is convex in the set of channels. We prove this in the appendix following similar ideas to the ones used in [23] to prove that the squashed entanglement is convex in the set of states.

Finite-dimensional channels
To build intuition before moving to bosonic channels, let us first bound the squashed entanglement of finitedimensional channels, i.e. channels where both the input and output dimensions are finite. An illustrative example of the effectiveness of the trivial squashing channel is the d-dimensional erasure channel , where ρ is ad dimensional state and ñ |e is an erasure flag orthogonal to the support of any ρ on the input [26]. It is well known that [26] and that = - [29]. In general we have where the first inequality holds since the squashed entanglement of a channel is an upper bound on ( )  Q 2 and the second inequality follows from applying the trivial squashing channel. In the specific case of the erasure channel, we then must have that That is, the trivial squashing channel is the optimal squashing channel, yielding both two-way assisted capacities and the squashed entanglement of the d-dimensional erasure channel. Independently of our work, in [31] the two-way assisted capacities of the d-dimensional erasure channel are established by computing the relative entropy of entanglement of the channel, which is also an upper bound on P 2 . A second channel we can apply the trivial isometry to is the qubit damping channel g  AD , a channel that models energy dissipation in two-level systems. The qubit amplitude damping channel is defined as with amplitude damping parameter g Î [ ] 0, 1 . Since the entanglement assisted classical capacity of the amplitude damping channel is known [26] to be equal to is the binary entropy, we immediately find the bound s q A comparison of this bound with the best known lower bound, given by the reverse coherent information (RCI) , and an upper bound g - found by Pirandola et al [30] using a relative entropy of entanglement approach, can be seen in figure 1.
A third interesting example are d-dimensional unital channels for which the maximally entangled state on ¢ AA maximizes the mutual information ( ) I A B ; . For these channels the trivial squashing channel gives the following compact upper bound  Hence, we also obtain that -( ) H p 2 is the entanglement assisted classical capacity of a Pauli channel . Let us now apply the bound for Pauli channels to a concrete channel, the qubit depolarizing channel  p . The action of this channel is 0, 1 . This corresponds with the Pauli channel given by = - The depolarizing channel can also be written as a convex combination of two other depolarizing channels, allowing us to use the convexity of ( )  E sq in the set of channels to improve on the upper bound in equation (25). We can compute the squashed entanglement of each individual channel and multiply it by the appropriate weight. Using this idea (see appendix B), we obtain the following stronger upper bound . This bound is equal to (25) for   p 0 1 3 , after which it linearly goes to zero at = p 2 3 . See figure 2 for a comparison of this new bound, the bound by Takeoka et al [21,34], the bound by Pirandola et al [30], and the RCI [32,33].

An upper bound on phase-insensitive channels
In this section we discuss our main result, an upper bound on the squashed entanglement of any phaseinsensitive Gaussian bosonic channel. Gaussian bosonic channels are of interest because they are used to model a large class of relevant operations on bosonic systems [35]. Phase-insensitive channels are those Gaussian bosonic channels which add equal noise in each quadrature of the bosonic systems. Imperfections in experimental setups  [21], the dashed blue line is the optimized squashed entanglement bound in this paper, the solid green line is the relative entropy of entanglement upper bound by Pirandola et al [30,31] and the magenta line is a lower bound given by the reverse coherent information [32,33].
for quantum communication with photons are modeled by phase-insensitive channels, motivating us to upper bound the squashed entanglement of all such channels. In particular this motivates the search for bounds where the input of the channel has a constraint on the mean photon number N.
Any phase-insensitive channel  PI is completely characterized by its loss/gain parameter τ and noise parameter ν. The Stinespring dilation of such a channel consists of a beamsplitter with transmissivity = t t n + + T 2 1 interacting with the vacuum on E 1 , and a two-mode squeezer with squeezing parameter figure 3 and the appendix for a detailed definition of the channel). T and G also completely characterize any phaseinsensitive channel. Takeoka et al [21,22,34] found bounds for such channels by only considering the beamsplitter part of the Stinespring dilation. To be a valid channel, we must have that n t - . We further have that phase-insensitive channels are entanglement breaking whenever n t +  1 [37], or equivalently, 1 . Hence, the squashed entanglement must be zero for channels with such parameters as discussed in the tools section.
Since we are interested in phase-insensitive Gaussian channels, we make the ansatz that a good squashing map will be a phase-insensitive channel. Numerical work suggests that, if only phase-insensitive isometries are considered, the pure-loss channel and the amplification channel separately have as optimal squashing isometry the balanced beamsplitter interacting with the vacuum. This motivates us to use the isometry consisting of two balanced beamsplitters at the outputs of the first beamsplitter and the two-mode squeezer (see figure 3). Using this isometry we obtain a bound for all phase-insensitive channels with restricted mean photon number N (see appendix for a derivation and a proof that the equation is monotonically non-decreasing as a function of N). This equation equals n n n n + --  where we have set As  ¥ N , the bound above converges to its maximum value of Figure 3. A squashing isometry for any phase-insensitive Gaussian channel  PI taking ¢ A to B. The beamsplitter B 1 and the twomode squeezer S form the Stinespring dilation, while the balanced beamsplitters B 2 and B 3 form the squashing map. The beamsplitter B 1 interacts with the vacuum on E 1 and A, and the two-mode squeezer S interacts with the output of B 2 and the vacuum on E 2 . The squashing isometry consists of two balanced beamsplitters B 2 and B 3 interacting with the vacuum on F 1 and F 2 and the output of the beamsplitter B 1 and the two-mode squeezer S.
Rewriting the upper bound as function of the channel parameters τ and ν [35] we obtain the upper bound z n t n t tz t n t n n t t 5.2. Application to concrete phase-insensitive Gaussian channels with unconstrained photon input 5.2.1. Quantum-limited phase-insensitive channels A pure-loss channel has G=1. As a consequence, for pure-loss channels the bound in equation (29) reduces to . This bound coincides with the bound found by Takeoka et al. In the opposite extreme we find quantum-limited amplifying channels, that is channels with T=1 and > G 1. For these channels, the bound by Takeoka is equal to infinity while (29) is non-trivial. Concretely, it reduces to the finite value of we show in the appendix that the upper bound becomes which is a lower bound on ( )  P 2 [38]. See figure 4 for a comparison of these bounds.

Thermal channel
A thermal channel is similar to the pure-loss channel, but instead of the input interacting with a vacuum state on a beamsplitter of transmissivity τ, it interacts with a thermal state with mean photon number N B . For a thermal channel we have that In figure 5 the upper bound is plotted for N B =1 together with two other bounds and the RCI, which is a lower bound on ( )  P 2 [32,39].

Non-quantum limited noise for lossy channels
In experimental setups one does not measure ν, but the additional noise c  0. We have the relation is the minimum amount of noise that will be introduced for a loss τ (the quantum-limited noise) [35]. The upper bound from (30) can then be rewritten as

Finite-energy bounds
For low mean photon number and certain parameter ranges the finite-energy bound in equation (27) is tighter than previous upper bounds on the two-way assisted capacities. For any energy the pure-loss bound from Takeoka et al [21,34] and equation (86) coincide. In figure 6 the bound from Takeoka et al [21,34], is shown for an average photon number of N=0.1 [40,41] and the two-way assisted private capacity of the pure-loss channel [30,31]. The loss-parameter runs from 0 to´-2 10 20 , which is the expected range of losses for fiber lengths of around 1000 km. In figure 7 we plot the upper bound by Pirandola et al [30,31], the finite-energy bounds of Takeoka et al [21,34], and equation (86) for the thermal channel with N B =1. For h the thermal channel becomes entanglement breaking, so that the squashed entanglement bounds are equal to zero for those regimes. This implies that the squashing isometry in figure 3 is not optimal.

Conclusion
In this paper we have obtained bounds on the two-way assisted capacities of several relevant channels using the squashed entanglement of a quantum channel. For practical purposes, the most relevant of the channels considered are phase-insensitive Gaussian channels. Our bound for these channels is always non-zero, even when the corresponding channel is entanglement-breaking. This points to the existence of an even better squashing channel for phase-insensitive Gaussian channels. Future work could investigate this intriguing avenue, especially due to its relevance to the squashed entanglement of a bipartite state as an entanglement measure.  [21,34], the dashed blue line the new bound reported in this paper, in solid green the bound by Pirandola et al [30,31], and the dashed-dotted line shows the reverse coherent information [32,39] which is a lower bound. Furthermore, we have proven the exact two-way assisted capacities and the squashed entanglement of the ddimensional erasure channel, improved the previous best known upper bound on the amplitude-damping channel and derived a squashed entanglement bound for general qubit Pauli channels. In particular, our bound applies to the depolarizing channel and improves on the previous best known squashed entanglement upper bound.
The only credible way to claim whether an implementation of a quantum repeater is good enough is by achieving a rate not possible by direct communication. Our bounds take special relevance in this context, especially for realistic energy constraints.

Acknowledgments
KG, DE and SW acknowledge support from STW, Netherlands, an ERC Starting Grant and an NWO VIDI Grant. We would like to thank Mark M Wilde and Stefano Pirandola for discussions regarding this project. We also thank Marius van Eck, Jonas Helsen, Corsin Pfister, Andreas Reiserer and Eddie Schoute for helpful comments regarding an earlier version of this paper.

Appendix A. Bounds for convex decomposition of channels
One way of obtaining bounds on the squashed entanglement is based on decomposing the channel action as a mixture of other channels actions and bounding each of them individually. Let ¢  A B be a channel such that its action can be written as the convex combination of the action of two other channels  0 and  1 Then we can always purify r AB in the following way r r r stand for the state that we obtain after applying the channel isometry to the pure input state fñ ¢ | AA . Let us apply the following channel to rñ | ABEF F Now, let us apply the rest of the channel. We obtain å r r r r That is, r ¢ ABE F 1 is a quantum-classical system. For states of this form the conditional mutual information can be simplified to The first inequality holds by restricting the squashing channels to those channels of the form in (38). Equality (43) follows since for channels of the form (38) the resulting state is a quantum-classical state as indicated in (40), and for classical quantum states the conditional mutual information of the whole state is a convex combination of the individual conditional mutual informations as shown in (41). The last inequality follows because the state that achieves the maximum squashed entanglement might be different for each channel. This method generalizes easily to any number of channels, from which it follows that if

Appendix B. Improved bound for the depolarizing channel
It is well known that the depolarizing channel becomes entanglement breaking for  p 2 3 [42], which implies that P 2 is zero in that range. For    p 2 3 , we can write the output of the channel as a convex combination of the output of  2 3 and   . That is, there exists some a   0 1 such that r a r a r = - By expanding both sides of (46) and identifying the coefficients, we obtain . Using the decomposition of the depolarizing from (46) the action of  p on half of a pure entangled state takes the following form y y y = Ä ñá Since y ¢ ABE is a valid extension of r AB , this means that there exists some squashing channel  ¢  E E that takes the environment of the depolarizing channel to this particular ¢ E . This is easy to see, first we can find a state yñ ¢ | ABE T that purifies y ¢ ABE . Next, since all purifications are related by an isometry there exists some purification  ¢ V E E T that takes the environment of the channel to ¢ E T . After this we trace out the system T and obtain y ¢ ABE . Now, y ¢ ABE is a quantum-classical system. Hence, we can decompose the conditional mutual information ¢ (

| ) I A B E
; into the sum of the mutual information conditioned on each value of E å a a l : Furthermore the input state that maximizes (52) is the maximally entangled state on ¢ AA . Hence, the following bound upper bound on

Appendix C. Squashed entanglement upper bound for any phase-insensitive Gaussian channel
In this section we discuss a proof of an upper bound for the squashed entanglement of any phase-insensitive bosonic Gaussian channel  PI . Here we use the fact that any such channel can be decomposed as a beamsplitter with transmissivity T concatenated with a two-mode squeezer with squeezing parameter = ( ) r G acosh . We first show that we can restrict the input states to the class of thermal states with mean photon number N, after which the entropic quantity of interest is written as a function of N. We then show that this function is monotonically increasing, after which we take the asymptotic limit  ¥ N of the entropic quantity yielding To show this is true, we first use a different form of ( )  E sq , which was proven by Takeoka et al [21], There are two differences between the characterization in (55) and the one in (2). First, the maximization runs over density operators on ¢ A instead of running over pure states on ¢ AA . Second, instead of taking the infimum over the squashing maps, it is taken over their dilations: squashing isometries  ¢ V E E F that take the system E to ¢ E and an auxiliary system F. The entropies are then taken on the state ω on systems ¢ BE F. The total operation, which we denote by D, consists of the Stinespring dilation of the channel (B 1 and S) and the squashing isometry consisting of two balanced beamsplitters (B 2 and B 3 ), see figure C1 . We now write C.1. Bound for finite N A Mathematica file is included in the supplementary material to guide the reader through the calculations performed in this section. For the proof we first need to be able to calculate the entropy of a Gaussian state as a Figure C1. A squashing isometry for any phase-insensitive Gaussian channel  PI taking ¢ A to B. The beamsplitter B 1 and the twomode squeezer S form the Stinespring dilation, while the balanced beamsplitters B 2 and B 3 form the squashing map. The beamsplitter B 1 interacts with the vacuum on E 1 and A, and the two-mode squeezer S interacts with the output of B 2 and the vacuum on E 2 . The squashing isometry consists of two balanced beamsplitters B 2 and B 3 interacting with the vacuum on F 1 and F 2 and the output of the beamsplitter B 1 and the two-mode squeezer S. function of its covariance matrix. The entropy of an M -mode Gaussian state ρ can be calculated by finding the M symplectic eigenvalues n  1 k of the covariance matrix G of ρ [43]. It turns out that the M 2 eigenvalues of the matrix WG are of the form n i k [44], where The entropy of the state is then n To obtain the state at the end of the isometry we determine first the optimal state for a specified mean photon number N, after which we apply the Gaussian transformations of the Stinespring dilation of the channel and the isometry, shown in figure C1. To find the maximizing input state on ¢ A , we follow the same approach [21,34] as Takeoka et al. Since the concatenation of multiple Gaussian transformations is still a Gaussian transformation, having a Gaussian state as input will always give a Gaussian state on any of the outputs. From the extremality of Gaussian states for conditional entropy [45], we get that the optimal input state is a Gaussian state.
To find the optimal Gaussian state, we note that the covariance matrix of all single-mode Gaussian states can be written as [46] q q q q cosh 2 cos sinh 2 sin sinh 2 sin sinh 2 cosh 2 cos sinh 2 57 for some  r 0 and q Î . Since the channel from ¢ A to ¢ ¢ ¢ ¢ BE E F F 1 2 1 2 is covariant with displacements and all unitariesŨ such that the corresponding symplectic matricesS U act on the thermal state as q q q q to be the representative of that equivalence class, and we only have to consider thermal states for the optimization.
The total system G ¢ ¢ ¢ A E F E F 1 1 2 2 consists then of a thermal state G ¢ A with mean photon number N on ¢ A and vacuum states on all the other inputs: the two-mode squeezer S on ¢ A and E 2 with the relation = ( )  We then have that the total symplectic transformation matrix D is = ( ) D B SB B 65 The covariance matrix on the subsystems E E 1 2 is then Multiplying by W gives Taking the covariance matrix corresponding to ¢ ¢ E E 1 2 we find using Mathematica the symplectic eigenvalues to be The covariance matrix corresponding to From this the symplectic eigenvalues can be calculated to be where we used that = ( ) g 1 0.

C.2. Monotonicity of the bound
For this section we restrict ourselves to the picture of calculating the squashed entanglement on the systems is the total isometry (see figure C1). In this picture the optimization is over the purification of the thermal state, the two-mode squeezed vacuum state Y N . To show monotonicity of equation (86) in N, we use that, up to a displacement on B (conditioned on a measurement outcome k at ¢ A ), it is possible to transform the state Y ¢ AA N to Y ¢ ¢ AA N , using a local operation L A on Alice (where ¢ < N N ) [47], see figure C2. Suppose now that A performs the operation L A on the state after the isometry, 1 2 Here we used that displacement operations can always be removed by local operations [48], so that for fixed outcome k the state r ¢ ¢  are flagged on the classical outcome k, and that the conditional mutual information of the whole state can not be smaller than the sum of the values of the conditional mutual information of the individual states [23]. In equations (91) and (92) we use the fact that all the r ¢ ¢ ¢ ABE E N k , 1 2 states are related to r ¢ ¢ ¢ ABE E N 1 2 by local unitaries on B and ¢ ¢ E E 1 2 and that the conditional mutual information of those states thus must be equal.
That is, the conditional mutual information computed over the isometry V with input state Y N is always greater than the conditional mutual information computed over the isometry V with input state Y ¢ N if ¢ < N N . This thus implies that equation (86) is a bound for all phase-insensitive Gaussian bosonic channels and all energy restrictions.

C.3. Expression as  ¥
N To obtain an explicit form for the expression in (86) as  ¥ N , we expand the eigenvalues around = ¥ N for three different regimes of G and T using Mathematica. For = G T 1 we have . Furthermore, this also means that if we have . We will call these relations the asymptotic entropic relations for short.
Using these asymptotic entropic relations, we find n n n n ¢ ¢ - Here we used the asymptotic entropic relations in equations (98) and (99). Equation (100) is basic rewriting, equation (101) follows directly from the definition of (·) g , and equation (102) follows from rewriting the terms In equation (103) we collect the terms proportional to ( ) G log 4 , from which we can see that these terms sum up to zero. In equation (105) we expand the quadratic terms, collect corresponding terms in equation (106) and write the upper bound both as a function of G and T in the last equality.
For > G T 1 we get in the asymptotic limit that equations (78), (79), (82) and (83)  For both regimes, the eigenvalues and in particular their leading terms are always positive. We see that for both > G T 1 and < G T 1 the absolute value of the eigenvalues are the same up to ordering, so that where we have used the definition of (·) g in the first equality and simplified the terms in the second step. We can expand the logarithms and collect the different terms and simplify to rewrite equation (122). Let us consider one by one the terms proportional to each logarithmic term. The terms proportional to + ( ) G log 1 are + + + -+ --