Figure 3.6: GEF( ˆρAE): Gaussian entanglement of formation between modeAand the environ-
ment, J←( ˆρ
AE): classical correlations between modesA andB as measured by modeB, S( ˆρA):
entropy of modeA, all for Vx =Vp = 32. These are plotted against attenuation for the state in
Eq. (3.17).
discord increase. As loss acts on mode B, the entanglement between mode A and the environment increases. This results in a decrease in classical correlations between modes
A and B, which is accompanied by discord increase. Olivares and Paris [157] studied the relationship between Gaussian discord and Gaussian entanglement of formation in a three-mode pure state, where they found the relationship
D←( ˆρAB) +S( ˆρE) =S( ˆρB) +EF( ˆρAE). (3.20)
This demonstrates the relationship between the dynamics of discord and entanglement with the environment. As expected, in this work this equation is satisfied, with the increase in entanglement accompanied by discord increase. Only when loss becomes high, andS( ˆρB) drops rapidly, does the discord start to decrease. This shows how important the
behaviour of entanglement with the environment is to discord dynamics, however the local entropies also play a part. Therefore the behaviour of discord depends on an interesting mixture of entanglement with the environment and local entropies.
In the qubit case, system-environment interactions have also been shown to have some relationship to discord increase. Discord increase can be achieved between two partiesA
andB by applying an entangling operation to modeAand some additional environmental mode [202]. In this work they also show that the discord increase emerges as a by-product of changes to the other correlation measures. A recent experiment has also been carried out that studies the flow of correlations between a two-qubit state and the environment [7], where they show that the decay of entanglement in a system is accompanied by the growth of multipartite entanglement and discord.
3.5
Alternative analysis of discord increase
The method of purification used in Section 3.3 also opens up a new avenue to interpret the phenomenon of discord increase. In what follows I consider the case whereVx0 =Vp0=V0to ease discussion, however the results would be similar if this restriction was lifted. During the purification, we saw that the discord increase scheme can be rewritten as simply splitting a thermal state on a beamsplitter, where both the size of the thermal state and the transmission of the beamsplitter depend on the level of attenuation. The situation is represented in Fig. 3.7 and this means that discord increase with loss can instead be
Figure 3.7: ModeA0 is a thermal state with varianceVx0 =Vp0 that depends on the loss. It is split on a beamsplitter with transmissionT2that also depends on the loss. The final state can be
used to study discord increase as long as both the initial variance and the transmission are varied correctly.
interpreted in terms of splitting a thermal state on an asymmetric beamsplitter.
To do this analysis one has to be aware of how the initial variance and transmission depend on loss. From Eq. (3.15), the variance V0 is a decreasing function of loss. It decreases from V with no loss to V2+1 for complete attenuation. From Eq. (3.14), the transmission increases from T22 = 12 for no loss to T22 = 1 for complete attenuation. To study the behaviour of discord, we can consider both of these effects individually.
First we look at how varying the transmission of the beamsplitter affects the discord behaviour. We only need to consider the case where T2
2 ≥ 12, since those are the values
that are needed to compare to the discord increase scheme. In fact,T22 ≤ 12 corresponds to the case when loss is applied to modeA, or it gives the behaviour ofD→( ˆρAB) when loss
is applied to mode B. Therefore by considering all values of transmission we can study the right and left-discord simultaneously.
Fig. 3.8 shows the discord as a function ofT22 for a range of thermal states. Increasing
T22 from 0.5 to 1 is the same as applying loss in the original discord increase scheme. As can be seen in Fig. 3.8 (a), increasing the variance of the thermal state increases the total amount of discord, and it also increases the level of the discord increase. For larger thermal noise, the maximum of discord also occurs at a higher value of transmission, which explains why the maximum of discord was achieved at a higher loss level in the original discord increase scheme. In this case, increasingT2 means modeB has less thermal noise,
which gives a higher level of discord since the basis states making up the mixture are less distinguishable. In Fig. 3.8 (b) the discord as a function ofT2
2 is extended to the full range
of transmission values. Reducing transmission from T22 = 0.5 to T22 = 0 represents the behaviour of the right-discord in the discord increase scheme as loss is increased. Clearly the discord always decreases during this change, which, combined with the fact that the starting variance also reduces with loss, means that the right-discord can never increase as a result of loss on mode B.
It is important to note that any thermal state with V0 > 1 will give an increase in discord as T22 is increased from 0.5 to 1. However discord increase through loss only hap- pens if the variance is high enough. This is because the variance of the initial thermal state reduces to model increasing loss. This reduces the total discord, so there is a bal- ance between increasing the discord with increasing T2, and decreasing the discord with
decreasing V0. Fig. 3.9 shows a contour plot demonstrating this behaviour. The black lines are the path, from bottom to top, followed by a state as loss is increased in the discord increase scheme. The value of discord varies along the black line in the same way as it varies in the discord increase scheme with increasing loss. In the left-most black line
3.5. Alternative analysis of discord increase
Figure 3.8: Discord D←(ρ0AB) as a function of T2
2 for the scheme in Fig. 3.7. Red: V0 = 8,
Orange: V0 = 16, Green: V0=32, Blue: V0 = 128, Purple: V0 = 1024. (a) shows the behaviour between 0.5≤T2
2 ≤1. (b) extends the graph down toT22= 0.
corresponding to V = 4, it can be seen that the discord immediately begins to fall. For the third line from the left V = 12 it is clear that discord initially rises. This increase becomes even larger for lines further to the right, indicating the larger increase in discord for larger thermal states.
Figure 3.9: Contour plot ofD←( ˆρ0AB) againstV0 andT2
2. Black lines: from bottom to top, the
path of a state withV=4, 8, 12, 16, 20, 26, 32, 38, 44 from left to right. The discord change along that path is the discord change observed in the discord increase scheme. Note that V = V0 at
T2
2 = 0.5. V is the variance from the original discord increase scheme and remains constant along
each black line.
This graph gives a visualisation of the balance between two different effects related to discord increase. From T22 = 0.5, discord increases as you move up and to the right of the graph. Therefore moving up along the black line contributes to discord increase, but moving to the left results in a decrease in discord. At low V, the movement to the left cancels out the small increase in discord that would be observed from moving up the graph, however for larger V, discord increases faster moving up the graph than the reduction caused by moving to the left. Therefore discord increase is observed for largerV
Figure 3.10: A mixture of two coherent states with equal amplitude is sent through a balanced beamsplitter. Loss is then applied to modeB by a variable attenuating beamsplitter.
under local loss can be understood by considering the simpler case of splitting a thermal state.