EXACT CLAIM

The exactly-vacuum-port one-mode setting is already settled by De Palma–Trevisan–Giovannetti (2017), Theorem 4, Eq. (24), with environment energy E=0; this includes every physical finite-rank coherent perturbation of a thermal first input with finite energy.

A joint low-temperature boundary lemma holds as follows. Fix 0<eta<1, q=1-eta, a,b>0 and z_A,z_B in C. On the span of |0>,|1>, define rho_A(t)=[[1-at,z_A t],[conj(z_A)t,at]] and rho_B(t)=[[1-bt,z_B t],[conj(z_B)t,bt]], and set both states to zero on the orthogonal complement. Take 0<t<min{a/(a^2+|z_A|^2),b/(b^2+|z_B|^2)}. These are independent physical finite-energy inputs. Let rho_C(t) be the retained output of c=sqrt(eta)a_op+sqrt(q)b_op. Put s=eta a+qb and K=2eta qab. Then, as t decreases to zero with all these parameters fixed,

 g^{-1}(S(rho_C(t)))-eta g^{-1}(S(rho_A(t)))-q g^{-1}(S(rho_B(t)))
 = [eta q(a-b)^2+K(1-log(K/s^2))] t^2/log(1/t)+o(t^2/log(1/t)).

The displayed coefficient is at least K(1+log 2)>0. Consequently EPnI is strict for all sufficiently small positive t in this family, including when both fixed coherences are nonzero. This is not a resolution of the arbitrary-input conjecture.

FULL FROZEN PROOF

First, the supplied primary-source excerpt of DTG17 states in Theorem 4, Eq. (24), that for every quantum state rho, 0<=lambda<=1 and E>=0,
 S(E_{lambda,E}(rho)) >= g(lambda g^{-1}(S(rho))+(1-lambda)E).
Here E_{lambda,E} is the one-mode thermal attenuator. Mixing with vacuum is precisely E_{eta,0}. Since g is increasing, this gives EPnI with a vacuum second input. If the first input is vacuum, exchange the ports and use lambda=1-eta. Finite mean energy ensures finite input and output entropies. No smallness or coherence restriction is required. Thus the proposed vacuum-port perturbation setting needs no new proof.

We now prove the joint boundary assertion independently. Throughout, all O and o constants may depend on the fixed parameters. Write L=log(1/t) and T=eta|z_A|^2+q|z_B|^2.

1. Physical inputs. For a positive parameter d and a complex parameter z, the matrix [[1-dt,zt],[conj(z)t,dt]] has trace one and determinant dt-(d^2+|z|^2)t^2. The stated upper bound on t makes its determinant and both diagonal entries positive. Its mean photon number is exactly dt. Thus both factors are normalized positive states with finite energy. Their tensor product has at most two photons in total. Photon-number conservation by the beam splitter implies that the retained output has exact support in span{|0>,|1>,|2>}. No truncation approximation is involved.

2. Input entropies. The smaller eigenvalue of the matrix with parameters d,z is
 r(t)=dt-|z|^2t^2+O(t^3).
Expanding the binary entropy gives
 S(rho_{d,z}(t))=dt(L+1-log d)-|z|^2t^2(L-log d)-d^2t^2/2+O(t^3 L).
Also g(dt)=dt(L+1-log d)+d^2t^2/2+O(t^3). Therefore
 S(rho_{d,z}(t))=g(dt)-|z|^2t^2(L-log d)-d^2t^2+O(t^3 L).    (I)

3. Output eigenvalues. Define H=2sqrt(eta q) Re(z_A conjugate(z_B)) and h=sqrt(eta)z_A+sqrt(q)z_B, so |h|^2=T+H. Denote output matrix entries in the number basis by C_jk. The probability of two retained photons is exactly
 C_22=2eta q ab t^2=Kt^2:
only the input |1,1> population can contribute, and its probability to put both photons in the retained port is 2eta q. The retained mean photon number is exactly st+Ht^2, by the beam-splitter transformation of c^dagger c and independence. Hence
 C_11=st+(H-2K)t^2,
 C_00=1-st+(K-H)t^2.
The first-order input expansion and the transformation of a single photon give
 C_01=ht+O(t^2), C_02=O(t^2), C_12=O(t^2).
Indeed all first-order input terms have ket and bra total photon numbers at most one. All entries are polynomials in t of degree at most two.

The eigenvalue near one is consequently
 lambda_0=1-st+(K-H+|h|^2)t^2+O(t^3)
          =1-st+(K+T)t^2+O(t^3).
For completeness, the usual second-order correction here follows by eliminating the lower two coordinates in the eigenvector equation: the leading vacuum-to-one-photon entry is ht, the eigenvalue denominator tends to one, and the vacuum-to-two-photon entry is O(t^2).
The smallest eigenvalue is
 lambda_2=Kt^2+O(t^3).
This follows by eliminating the vacuum coordinate and then the one-photon coordinate. The latter diagonal is st+O(t^2), with s>0, whereas its coupling to the two-photon coordinate is O(t^2); its correction is therefore O(t^3). Eliminating the vacuum changes the two-photon diagonal only by O(t^4). The separation of the two small eigenvalues follows from their distinct first-order slopes s and zero. Finally trace one gives
 lambda_1=st-(2K+T)t^2+O(t^3).
These expansions can equivalently be obtained from the three-dimensional characteristic polynomial.

Since s,K>0, entropy expansion at these eigenvalues is valid with remainder O(t^3 L). Specifically,
 -lambda_0 log lambda_0=st-(K+T+s^2/2)t^2+O(t^3),
 -lambda_1 log lambda_1=st(L-log s)-(2K+T)t^2(L-log s-1)+O(t^3 L),
 -lambda_2 log lambda_2=Kt^2(2L-log K)+O(t^3 L).
Adding and comparing with g(st) yields
 S(rho_C(t))=g(st)-Tt^2(L-log s)
             +[K(1-log(K/s^2))-s^2]t^2+O(t^3 L).    (II)

4. Inverting the entropy. The following elementary inversion applies to (I) and (II). If d>0 and U,V are fixed, and
 S(t)=g(dt)-Ut^2(L-log d)+Vt^2+O(t^3 L),
then
 g^{-1}(S(t))=dt-Ut^2+Vt^2/(L-log d)+O(t^3).
To verify this, insert x=dt-Ut^2+Vt^2/(L-log d) into g. Use g'(dt)=L-log d+O(t) and g''(x)=-1/[x(1+x)]. Taylor's remainder is O(t^3), and the resulting entropy error is O(t^3 L). Both the proposed x and the true inverse are comparable to dt: this follows first by comparing S(t) with g(dt/2) and g(2dt). On that interval g' is comparable to L, so the mean value theorem converts the entropy error to O(t^3).

Thus (I) gives
 g^{-1}(S(rho_{d,z}(t)))=dt-|z|^2t^2-d^2t^2/(L-log d)+O(t^3),
and (II) gives
 g^{-1}(S(rho_C(t)))=st-Tt^2+[K(1-log(K/s^2))-s^2]t^2/(L-log s)+O(t^3).
Subtract the weighted input expressions. The order-t terms cancel by the definition of s; the order-t^2 coherence terms cancel by the definition of T. Replacing each 1/(L-log d) by 1/L introduces only O(1/L^2). The remaining coefficient is
 eta a^2+qb^2-s^2+K(1-log(K/s^2))
 =eta q(a-b)^2+K(1-log(K/s^2)),
which proves the asserted asymptotic expansion.

5. Strict sign. The exact identity s^2-2K=(eta a-qb)^2>=0 implies 0<K/s^2<=1/2. Therefore 1-log(K/s^2)>=1+log 2>0, and the coefficient is at least K(1+log 2)>0. The little-o remainder then proves strict positivity of the EPnI gap for every sufficiently small positive t.

REMAINING GAP

The original EPnI for arbitrary independent finite-energy inputs remains unresolved. This lemma covers only one-mode states supported on |0>,|1> with fixed positive population slopes and coherences proportional to t. It does not cover other coherence scalings, arbitrary low-temperature families, nonperturbative inputs, or multimode correlations.
