Understanding Device Identification of Non-Orthogonal Quantum States

[RobikShrestha]

I am a little confused about exercise 1.2 in the book "Quantum Computation And Quantum Information" By Michael Nielson.

The question is:

Explain how a device which, upon input of one of two non-orthogonal quantum states |a> or |b> correctly identified the state, could be used to build a device which cloned the states |a> and |b>, in violation of no-cloning theorem. Conversely, explain how a device for cloning could be used to distinguish non-orthogonal quantum states.

It asks us to explain how a device which upon input of one of two non-orthogonal quantum states correctly "identified" the state could be used to build a cloning device. What does "identify" mean? Does it mean, we know the exact state? Or does it mean we know it is state #1 vs state #2 but not the exact state?

Second part asks us if we had cloning device how would we distinguish non-orthogonal quantum states. For that, can't we clone the state a large number of times and then measure them all to find the state, with error decreasing as no. of clones increases?

 

[naima]

Suppose that Bob and Alice know how to prepare two states a> and b>. they decide that Bob will send Alice a random sequence like a b b a a a b a b a b b ...
I think that the author says that Alice will be able to write the name of the received particle sequence: a b b a a a ...

 

[RobikShrestha]

Ok then that means Alice knows exactly what states a and b are. If we know the exact state, then we can prepare the state in principle right?

So the answer to how to build cloning device would be:
1. Detect the state
2. Prepare the detected state
?
Or is the answer trickier than that?

And what about the second part? Cloning it large number of times and then measuring to detect the state, is that right?

 

[naima]

the author does say that the state to be cloned is a or b.
I have not the answer. I am reading Box 2.3 which proves that non othogonal states cannot be reliably distinguished.

 

[RobikShrestha]

Ok. I haven't got that far into the book, but I think the answer has to be more "thorough" than just saying prepare the state which you have detected.

 

[naima]

the author does not tell if Alice can repeat the measurement which distinguishes the state to be cloned and another state. are the states modified?

 

[RobikShrestha]

Yes the author does not specify if it can be repeated. The author also does not specify what "detecting" means? I mean it could be that the device outputs 1 for state a and 0 for state b, or it could output entire description to fully define the state. That's why I was confused.

One way I was thinking about the solution is device outputs:
|a>|0> when input is |a>
and |b>|1> when input is |b>
Now, those two outputs are orthogonal and thus can be cloned right? After cloning, may be we could do some post processing to retrieve original state.

 

[atyy]

Suppose that Bob and Alice know how to prepare two states a> and b>. they decide that Bob will send Alice a random sequence like a b b a a a b a b a b b ...
I think that the author says that Alice will be able to write the name of the received particle sequence: a b b a a a ...

Ok then that means Alice knows exactly what states a and b are. If we know the exact state, then we can prepare the state in principle right?

So the answer to how to build cloning device would be:
1. Detect the state
2. Prepare the detected state
?
Or is the answer trickier than that?

And what about the second part? Cloning it large number of times and then measuring to detect the state, is that right?

That's also my understanding from looking at http://arxiv.org/abs/quant-ph/9601025v1. I think in the last part one may need to make measurements of non-commuting observables http://arxiv.org/abs/quant-ph/0511044.

 

[RobikShrestha]

No idea what "non-commuting observables" means. But if we had a cloning device, we could clone it multiple times right? Then measure them.

 

[atyy]

No idea what "non-commuting observables" means. But if we had a cloning device, we could clone it multiple times right? Then measure them.

Yes, if there was a cloning device we could clone the state, and by making measurements on the state determine the state. By non-commuting, I meant that to identify a particular wave function (for example), one might have to make measurements as well as momentum to identify the wave function completely.

 

[naima]

I wrote the specifications of the distinguisher device and i saw that it would be also a cloner device.
It is a quantum gate (like a CNOT gate) with two input and two ouput channels.
it receives a control state C and a target state T. the output control state is equal to the input control state and the target output channel is the interesting result.
the result is bilinear in the outputs.
if C = T the result is 0>
if T is orthogonal to C the result is 1>
So let us use it to see how to distinguish two states.
I always send a 0> to the target input (it is a particle on a ground state).
If the control state to be compared is 0> the device returns 0> (equality) if it is 1> it returns 1> (orthogonality)
If i send a 0> + b v> as it is linear it returns a 0> + b v>.
so the distinguisher is a cloner!

 

[RobikShrestha]

@atty

By non-commuting, I meant that to identify a particular wave function (for example), one might have to make measurements as well as momentum to identify the wave function completely.

Make measurements as well as momentum? Do you mean measure position and momentum, violating Heisenberg's principle?

 

[atyy]

@atty

Make measurements as well as momentum? Do you mean measure position and momentum, violating Heisenberg's principle?

No. :) I mean, since you have an ensemble of many copies of the state, you can measure position on some of the copies, and measure momentum on a different subensemble of the copies.

 

[RobikShrestha]

@naima

Thank you for your answer.
A couple questions though:

the output control state is equal to the input control state and the target output channel is the interesting result.

Here, we are doing an "observation" to determine target output. Once we observe, the state might be lost right? So, what mechanism will ensure that output control state will be same as input control state?

To verify, we also need to prove that if non-orthogonal states were not distinguishable then, the device you proposed would fail to clone states.

 

[RobikShrestha]

No. :) I mean, since you have an ensemble of many copies of the state, you can measure position on some of the copies, and measure momentum on a different subensemble of the copies.

Ah ok. Re-thinking about the solution about cloning it multiple times and measuring them, it really is for determining exact state. But may be there is some other way in which we don't have measure the entire state to just "distinguish" the two states. I mean, the question only asks to distinguish between states.

Conversely, explain how a device for cloning could be used to distinguish non-orthogonal quantum states.

 

[naima]

This device cannot exist, we are in a dream world so we can suppose that the control state remains unchanged.
the cloner does not clone any target state in the control state.
It only works with the 0> target. 1> will give an orthonormal state to the control state. Take two input states equal to a 0> + b 1> and compute the result (use linearity). You get 0> it is the symbolic answer for equality.

 

[RobikShrestha]

@naima @atyy

we can suppose that the control state remains unchanged.

I was hoping that we would only make assumption the author asks us to make i.e. about a non-orthogonal states being distinguishable. I mean, if input control state is observed, then can we really guarantee it remains unchanged? I mean, if we had some mechanism to do that, wouldn't we have cloning device already?

I propose the following solution:

1. We assign a "marker" qubit to distinguish states.

When input is |a> output |a>|0>
and when input is |b> output |b>|1>
This process converts non-orthogonal states to orthogonal states which is possible only because non-orthogonal states can be distinguished (according to author's instruction).

2. Since we can build cloning device for orthogonal states, we can clone them

3. Finally, we retrieve back |a> and |b> by ignoring the "marker" qubits from the cloned states.

Also note that if non-orthogonal states were not distinguishable (as is the reality), we would not be able to do the first step, so this cloning device would not work in that scenario.

 

[RobikShrestha]

Re-thinking about it now, seems like, my solution also needs to preserve states |a> and |b>. So, step #1, "detection mechanism" is really a supposition.

 

[naima]

Look at the no cloning theorem
http://en.wikipedia.org/wiki/No-cloning_theorem
they write that there woul be 2 ways to make such a device.
The first would be to observe the state but it would change it.
The remaining possibility would be to control the hamiltonian so that:

They write $|e>_B$ What i wrote |0> This is the target particle that receives the result.

Of course they say later that it is not possible. But they notice that the only remaining chance is "unchanged control state".

 

[RobikShrestha]

The proof in the wiki link seems to be proof by contradiction. Like, first it assumes existence of a cloning device and then, it figures out that such a device would only work when a=b or <a|b>=0 (orthogonal).
Our task is to describe how a cloning device would work when non-orthogonal states can be distinguished.
It would definitely be better if we could mathematically describe the mechanism or at least the reason by which the control state remains unchanged.