Okay, so today I just want to finish the chapter by telling you something about recent developments

in the field of psychic creativity.

So this will be me telling you some stories essentially.

There are lots of different things that people are trying to do.

Some of them have to do with quantum information processing, some of them have to do with quantum optics,

some of them have to do with quantum simulation, finding things like new redoubt methods that can be used for various things.

And so one of the trends obviously is that you go to more qubits.

So you have your resonator and then you would place more and more of these qubits coupled to the resonator inside the resonator.

And so what can you do if you have more qubits? Well, first you can demonstrate very simple quantum algorithms.

And in particular this includes quantum error correction, so you can demonstrate the elements of quantum error correction.

So that is something people currently try to do.

And then when you have more than just one or two qubits, there is immediately the question of how do you lay them out on the two dimensional surface of the chip.

And we already saw one example which was the cavity grid.

More generally you could say that you want to come up with more advanced architectures that employ the cavities and the qubits and connect them in various ways.

I just note that one example which we looked at, that was the cavity grid.

Now all of these more advanced applications require you to somewhere in the calculation produce a state that is highly entangled, that is no longer a product state, but shows quantum correlations between the different qubits.

So as a sort of benchmark of whether this might work, there is the idea of just producing highly entangled states for the sake of producing them and checking how good you are at doing this.

And here I want to give you one example, and that is an example which we have worked on.

And it has to do with a particular kind of readout that we learned about even in the case of one qubit, which you remember was this kind of dispersive readout where the qubit state shifts the resonance frequency of the microwave resonator, independence on the state.

And that you can extend to a setup such as here and by a clever trick you can use it to generate entangled states.

So how does it work? Well first of all in the case of a single qubit we learned that the frequency shift just depends on the state of the qubit, so I could say the frequency shift just depends on z.

And the appropriate basis. And now it turns out that all of the qubits simultaneously couple to the same mode, so all of them try to shift the resonance frequency.

And then it turns out that the resonance frequency shifts according to the sum of the corresponding excitation operators, so there would be z1, z2, and z3.

And what's more, we can even control the pre-factors that go in front of this, say number 1, number 2, number 3.

Because if you remember that depended for example on the detuning between the cavity and the qubit, it also depended on the coupling strength between the qubit and the cavity.

And the coupling strength in turn depends on the value of the electric field mode at the position of the qubit, so by simply placing the qubits at different points you could also play around with the coupling strength.

So all of this influences these coefficients. And so this would be then the frequency shift experienced by the mode through which you send your measurement beam.

And so that would also be directly proportionate to the phase shift that the photons of the measurement beam pick up.

And so by looking at the phase shift you could measure this superposition of operators, and that now becomes interesting.

Why is it interesting? Well, whenever you measure an operator that does not completely distinguish all your basis states you may end up with superpositions.

And in particular if we look at this operator and just take the special case where all the lambdas are equal and we essentially only measure the sum of the populations,

then we can immediately deduce that a state such as this where the first qubit is excited has exactly the same frequency shift as a state where the second qubit is excited.

And again that has the same frequency shift as a state where the third qubit is excited.

So all of them are indistinguishable by the measurement. They yield exactly the same measurement result.

And so what you can do now is you can start from a state which is not entangled, which is simply a product state,

and which consists of the product state where all of the qubits have been placed in an equal superposition of up and down.

So that would be the first qubit, the second qubit has been placed in the same state, and the third qubit as well.

Now you could expand this and you would see that obviously this consists of many different combinations.

For example, it contains 000, it also contains terms like 001 and so on, it contains all possible combinations.

Now suppose you measure the phase shift that belongs to one of the qubits being excited, that is this series of states.

If you do measure this particular phase shift, then the system automatically collapses onto the state that belongs to this phase shift.

Only in this case it's not a single base state, but it's all three of them.

And it actually collapses into the coherent superposition that will be contained in this original state.

So with a certain probability that you can read off from the amplitudes here, your measurement will yield the signal that belongs to these states,

and if it does so, then you end up collapsed into a wave function that is in this case the equal superposition of all three of them.

And that is an entangled state in the sense that it can no longer be written as a product state with the different qubits factorized.

In fact, this particular entangled state has a name, it's called a W state.

So that is a way of generating entanglement, even though of course it's probabilistic in the sense that sometimes your measurement may yield another result,

which for example belongs to the case where two of the qubits are excited, and then you would have the superposition 110 plus 101 plus 011.

And so that would be another W state actually.