PRUEBA DE LA SIGNIFICANCIA DE LAS HIPÓTESIS

The main inspiration of this work was whether Gaussian unitary operations can reduce the non-Gaussian cost, with respect to photon subtraction/addition schemes, involved in preparing a desired non-Gaussian target. The extent to which this is true is revealed by comparing the non-Gaussian resources required to prepare a direct truncation of |ψ(λ)i with that required for the minimum core state. Essentially, we determine the utility of Gaussian operations by considering the approximate preparation of |ψ(λ)i with and without them. This can be done by comparing the fidelities of the states produced by each method. These fidelities are defined as

FDT(λ, N) =|hψ(λ)|ψNDT(λ)i| 2_{, (7.27)} with FDT(λ, N) =hψ(λ)|ΠˆN|ψ(λ)i= N X n=0 |ψn(λ)|2, (7.28)

whereψn(λ) = hn|ψ(λ)i, for the direct truncation method and

FC =hλ, r, α, θ|ΠˆN|λ, r, α, θi= N X n=0

|cn|2, (7.29)

for the core state method. The superiority of the core state method can be estab- lished on two levels corresponding to the two questions asked in the introduction. Firstly, the core state method is better than the direct truncation method using the same non-Gaussian resources if

hλ, r, α, θ|ΠˆN|λ, r, α, θi>hψ(λ)|ΠˆN|ψ(λ)i. (7.30)

The second condition, if true, that would demonstrate the superiority of the core method usingless non-Gaussian resources over the direct truncation method is

hλ, r, α, θ|ΠˆM|λ, r, α, θi ≥ hψ(λ)|ΠˆN|ψ(λ)i, (7.31)

forM < N. That is, we would expect that one can better the fidelity with the target by our approach with potentially lessnon-Gaussian resources than simply building a truncated target. For the first class of states with finite dimensional cores, this is

obviously true. This is because we can perfectly prepare the state by preparing the core first and then applying unitary Gaussian operations

FC(N) = N X n=0

|cn|2 = 1. (7.32)

In contrast, the direct truncation method yields

FDT(N) = N X n=0

|ψn(λ)| ≤1, (7.33)

because |ψ(λ)i is, in general, an infinite dimensional state and is only reproduced with unity fidelity asN → ∞ and so

lim

N→∞(FDT(N)) =FC(N) = 1. (7.34)

Thus, we would need infinite non-Gaussian resources to perfectly prepare the target by the direct truncation method. The reason for this is because in the direct trun- cation method the non-Gaussian subtractions/additions also contribute to building the Gaussian envelope of the state in addition to its non-Gaussian core. In contrast, in the core method all of the non-Gaussian resources are concentrated into prepar- ing the non-Gaussian part of the state. Thus, for the example of the photon added coherent state (7.15), the core method is superior since none of the non-Gaussian subtractions/additions contribute to the construction of the displacement operator. In contrast, in the direct truncation method, each subtraction/addition contributes to building both the core and the displacement operator.

For the second class of states with infinite dimensional cores, the situation is more subtle since both converge to unit fidelity as N → ∞. However, prov- ing the optimal nature of the core state method to a direct truncation method for arbitrary pure target states is a non-trivial task and will not be tackled here. Ultimately, this is because the core state method is a complicated optimization process that is dependant on the both the non-Gaussian resource N and the de- sired fidelity. We can, nevertheless, gain a limited insight into the superiority of the core state method over the direct truncation method from the following phase space argument. In particular, we note that the core method can only be con-

W

q , p

q

p

W

q , p

W

q , p

W

q , p

MN

Figure 7.2: The superiority of the core state method as compared to the direct trun- cation method is evident from this phase space diagram. By applying Gaussian op- erations to the target’s Wigner functionWψ(λ)(q, p)we can generate the core state’s Wigner function Wλ,α,r,θ(q, p). This latter quasi-distribution is closer to the origin of phase space and has a more symmetric quadrature noise profile. This transforma- tion achieves two simultaneous feats. One the one hand, the first and second order moments of the core can be tuned to resemble the first and second order moments of WΠN(q, p) and so increase its fidelity with WΠN(q, p). Such tuning cannot be per-

formed in the direct truncation method. On the other hand, the core, as a result of this tuning, is closer to WΠN(q, p) than WΠM(q, p) for M > N since it is located

near the origin while being symmetric.

sidered as advantageous if transforming the first and second moments of the tar- get concentrate the state on a smaller finite dimensional subspace as depicted in Fig.7.2. This follows since the Gaussian operations can only effect the first and second order moments Wλ,α,r,θ(q, p) = Wψ(λ)(Q(q, p, α, r, θ), P(q, p, α, r, θ)), where

Q(q, p, α, r, θ) and P(q, p, α, r, θ) are linear functions of q and p. For example, if we restrict ourselves to only squeezing and displacing the target then the core is given by Wλ,α,r(q, p) = Wψ(λ)(er(q−

√

state method is only better in the first sense (i.e. increasing the fidelity over the direct truncation method with the same resource) if Wλ,α,r,θ(q, p) enjoys a better

overlap with WΠN(q, p) =

PN

n=0Wn(q, p) than the original Wψ(λ)(q, p). This can

only happen if the first and second order moments of Wλ,α,r,θ(q, p) can be tuned

to be more like the first and second order moments of WΠN(q, p). Essentially, pro- vided that the Gaussian parameters of the core can be tuned so that the Wigner functionWλ,α,r,θ(q, p) is located at the origin and has a symmetric quadrature noise

profile, as is the case with WΠN(q, p) , then it is an improvement over the direct truncation method. This is because the direct truncation method does not allow us to modify these moments to increase the resemblance withWΠN(q, p). Of course, if

Wψ(λ)(q, p) is already located at the origin with a symmetric quadrature noise profile

then unitary Gaussian operations cannot improve on its overlap with WΠN(q, p). In addition, this argument can also answer the second question whether the core state method could offer an equal or better fidelity to the target for less non-Gaussian resources. This is also evident from Fig.7.2 since the ability to tune the Gaussian parameters of the core offers the possibility of being able to make the first and second moments of the core more likeWΠN(q, p) thanWΠM(q, p). That is, by tuning the first and second moments of the core to be close to the origin and symmetric means that the vacuum has a larger contribution to the core state than in the target. While this argument does not capture the full complexity of the core state method it does allow an simplified picture that suggests the advantageous nature of the core state method. We can build on this sentiment by providing some examples of states for which the core method is indeed superior to direct truncation. Specifically, in the next section, we consider the Schr¨odinger cat states and demonstrate that they support our case.