What Happens to the Original Qubit's State After Moving It?

[Elroy]

Just as an introduction, I'm attempting to bootstrap myself up in the language of quantum computers. I've got a ways to go, but I am making some headway. If y'all don't mind, I'd like to use these forums to ask what I hope are some relatively simple questions.

I'm pretty good to go on the concept of a Bloch sphere, and would appreciate answers framed in those terms. I think we can agree that the internal "state" (before being read) of a qubit can be stated as a value for theta and phi in the Bloch sphere (so long as it's some "pure state").

My immediate question has to do with "moving" qubits. The no-cloning theorem states that they cannot be copied, so I will only talk in terms of "moving" them. Here's my question: Once the state of a qubit is moved from one qubit location to another, what's the state of the original location?

My "guess" is that it would be in a "fully mixed state" (i.e., the point at the very center of the Bloch sphere), but I'm not able to verify that.

I look forward to the replies,
Elroy

 

[Physics Monkey]

Hi Elroy,

Welcome to Physics Forums. I would give two answers.

First, if you physically move the atom, or electron, or NV center, or photon, or whatever is instantiating your qubit, then I would say there is no state at the original point after the move (you've moved the physical system carrying the state somewhere else).

Second, if you move only the quantum information, e.g. by teleporting the state of the local qubit to a distant qubit using a Bell pair, then I would say that indeed the state of the original qubit is now at the center of the Bloch sphere. In fact, we can say more than that. The original qubit is now in a maximally entangled state with one of the extra qubits in the Bell pair used to teleport. When you then ignore the extra qubit from the Bell pair you get the maximally mixed state on the original qubit. Check out the teleportation wiki page http://en.wikipedia.org/wiki/Quantum_teleportation if you want to learn more about this version.

If you move the state by some other method, then the state of the original qubit could be anything.

Hope this helps.

 

[VantagePoint72]

Second, if you move only the quantum information, e.g. by teleporting the state of the local qubit to a distant qubit using a Bell pair, then I would say that indeed the state of the original qubit is now at the center of the Bloch sphere. In fact, we can say more than that. The original qubit is now in a maximally entangled state with one of the extra qubits in the Bell pair used to teleport. When you then ignore the extra qubit from the Bell pair you get the maximally mixed state on the original qubit. Check out the teleportation wiki page http://en.wikipedia.org/wiki/Quantum_teleportation if you want to learn more about this version

How do you figure? Teleportation requires a Bell measurement on the original (let's say Alice's) qubit and the half of the EPR pair that Alice entangles it with. This is necessary for Bob to correct the syndrome in his qubit. There is no remaining entanglement after teleportation because it's been destroyed by measurement. So, the original qubit winds up in one of the computational basis states, unentangled with anything. This is only maximally mixed if, for some reason, Alice throws away the result of her measurement after telling Bob what it was. Otherwise, she knows whether the qubit she teleported is now either $|0\rangle$ or $|1\rangle$ ...because she just measured it.

 

[Physics Monkey]

As you said, Alice must make a Bell measurement on the original qubit and half the Bell pair. The outcome is a Bell state of the original qubit and half the Bell pair. Alice transmits the outcome of the measurement to Bob who applies the right unitary to recover the correct state. However, whatever outcome Alice finds, the original qubit is maximally entangled with half the Bell pair (this being a property of all four Bell states). Did I misunderstand your objection?

 

[naima]

You have to read Pati's paper
"We use the quantum state randomization of a qubit as
one example of the bleaching process and show that the missing information can be
fully recovered up to local unitary transformations in the ancilla qubits."
You will see how information can skip from one place (A) to another place (C).
A/B being then maximally entangled.

 

[Elroy]

Hi Monkey,

I actually did read the wikipedia quantum teleportation page earlier this morning before I posted my question. It was a bit confusing, but I think clarity may be slowly setting in. The first of their four proposed steps (to quantum teleportation) was:

1. An EPR pair is generated, one qubit sent to location A, the other to B.

Upon my initial read, I wasn't clear on what "sent to" meant. I now understand that they are talking about physically transporting the device that contains the isolated qubit.

I am actually an excellent "classical" programmer. However, that may be "getting in the way of" understanding this stuff as much as it is helping. In terms of classical bits, I just can't help but think in terms of pulling them off of my hard drive into memory, moving them around in memory, possibly coding them as an RF pulse and sending them to my router via WiFi, and then all over the world.

I imagine doing the same with qubits, with the understanding that only one "copy" will exist at any particular location at any point in time. Maybe this entire concept is just completely wrong-headed, and the entire field of quantum computing just isn't this reductionistic.

In fact, the more I think about it, maybe the whole idea of moving and storing qubits isn't necessary for an "initial" understanding of what quantum computing would be. Maybe I should just think in terms of qubits that aren't moved, as well as "registers" of qubits that are used for various computations.

And thanks for the welcome. This is my absolute first thread to the forum.

Regards,
Elroy

 

[VantagePoint72]

As you said, Alice must make a Bell measurement on the original qubit and half the Bell pair. The outcome is a Bell state of the original qubit and half the Bell pair. Alice transmits the outcome of the measurement to Bob who applies the right unitary to recover the correct state. However, whatever outcome Alice finds, the original qubit is maximally entangled with half the Bell pair (this being a property of all four Bell states). Did I misunderstand your objection?

Oh, we're misunderstanding each other a little—when one "performs a Bell measurement" on a pair of qubits, as I said, you don't actually, in practice, project onto the Bell basis so that you end up with one of $\{|\Psi^\pm\rangle, |\Phi^\pm\rangle\}$. It is a pretty standard—though as this exchange demonstrates, sometimes not made sufficiently explicit—assumption in quantum information that you are only able to make measurements in the computational basis $\{|0\rangle, |1\rangle\}$. Since the ability to measure in the computational basis plus the ability to do arbitrary unitary operations allows one to effectively measure in any basis, this is often glossed over in proofs. So in order to "perform a Bell measurement" Alice has to rotate her two qubits—the one to be teleported and her half of the EPR pair—from the Bell basis to the computational basis. This uses a Hadamard transformation and a controlled-not, the latter of which disentangles the two qubits. So at the end of her "Bell" measurement her two qubits are in a product of computational basis states. Thus, the original qubit is in a pure state. So, we were talking about slightly different protocols. However, the one I describe is the one that would generally be used in any physical implementation of teleportation.

 

[Strilanc]

Other states get swapped into the original location, possibly while being operated on.

Consider a 2 qubit quantum circuit. How do you move the value of the first qubit to the second qubit? The easiest way is to use a swap gate, which has this transition matrix:

$$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

You can show that any 2-wire operation that properly moves qubit 1's value into qubit 2 can be factored into a bunch of operations on the second qubit followed by a swap gate followed by a bunch of operations on the first qubit. So in a 2 qubit system what gets moved into qubit 1 when you move it to qubit 2? Whatever junk was in qubit 2.

What about in a 3 qubit system? Well then you can choose to have qubit 2's junk or qubit 3's junk moved into qubit 1, or any combination/transformation of their junk! What a deal!

It might be helpful, for the purposes of moving a qubit, to think of a fiber-optic cable as a sequence of qubits connected by constantly firing swap gates. That's definitely oversimplified, but I find it intuitively useful.

 

[Elroy]

Strilanc,

The notion of a swap-gate makes a great deal of sense. I know that we're (mostly) talking in theoretical terms, but I'm just trying to get my head around how a quantum computer (or the portion of a computer containing qubits) might work. For me, I'd like to work out the application math and the actual programming techniques. I'll leave it to the engineers to work out how to preserve the quantum states and preventing the wave-function from collapsing before the qubit is read.

Quantum computing is so new, that these divisions haven't been worked out. It's analogous to electronics engineers and programmers. As a programmer, I couldn't care less that a (classical) bit in a ONE state is represented by 5 volts (from ground), and that a ZERO state is represented by ground voltage (or, in the case of CDs and DVDs, light reflections or not, etc, etc).

In terms of qubits, I'd just like to understand them in terms of |0⟩, |1⟩, and all the other "internal" states represented by |Ψ⟩. However, this does encompass an understanding of multiple qubit "registers" as well as how we might get classical bits into and out of qubits. For instance, once I feel that I have clarity on this "moving" issue (or at least a semblance of clarity), I'd also like to ask if my reasoning is correct to claim the following: So we can move 2^n classical bits into n qubits? That seems correct. However, somewhat obviously, we can only move (or, actually copy, in this case) n qubits into n classical bits (because that's all the information that remains after the wave-function collapses).

Once this thread settles down, I'll start a new thread with that one though. I'll eventually move onto gates and polynomial time, possibly building to such things as Shor's algorithm and how an understanding of the math available with qubits will allow other computations not possible in classical terms.

Regards to All,
Elroy

 

[VantagePoint72]

@Elroy, I think you would benefit from being more systematic in your approach to learning quantum computing. Just making a succession of forum posts about each topic as you decide to move on is going to be incredibly inefficient. Particularly since it sounds like many of them will wind up taking the form, "Please teach me about X." The textbook by Nielsen and Chuang is considered the standard text of the field, is beautifully written, and is very accessible to anyone who knows the basics of the quantum mechanics. Particularly since you seem to have a solid classical computing background, it should be pretty easy going. I suggest getting a copy and just working your way through it. You will learn all about everything your post mentions and more.

 

[Strilanc]

We actually know quite a lot about quantum computing. Enough to fill textbooks. For example, https://www.amazon.com/dp/1107002176/?tag=pfamazon01-20 which you can even http://www.johnboccio.com/research/quantum/notes/QC10th.pdf!

Text books can be a bit dry of course, but one of the authors of the textbook I linked was even nice enough to make a youtube series called quantum computing for the determined. Many of your questions are answered in that series.

An online forum isn't the best place to learn a complex subject like quantum computing. You'll end up with large holes in your knowledge where you don't know to learn what you don't know you don't know.

 

[Elroy]

To LastOne and Strilanc,

Oh gosh, I actually am reading a great deal about it (between making these posts). And Strilanc, thanks a LOT for the link. I've got it and will definitely read it. I'll also look into getting a copy of Nielsen and Chuang's book if I feel that I need more clarity.

I'm certainly not one to be a pest, and I will always work hard to formulate intelligent questions. But, if folks don't mind, I sure would like to be a bit social with my learning about this stuff. Regarding classical computers (particularly from a programming perspective), my knowledge is rock solid. Also, I get around fairly well in linear algebra (having taught upper level statistics courses for several years). My geometry is a touch rusty. However, having had to write algorithms for manipulating 2D and 3D images, I'll muddle through.

Understanding how classical computers can be understood (and completely built up) in terms of bits and a handful of gates, I just can't help but believe that qubit calculations can be explained in the same terms. There is tons of redundant language flying around, and the distinction between an engineer and a programmer is FAR from being drawn in quantum computer circles, but I do believe that these issues can be sorted out.

I wish there was a forum here named "Quantum Programming", but maybe that's asking too much, given that we're not even sure that we have working quantum computers just yet. This is just a fascinating area for me, and I'm just trying to take "baby steps" so as to not stumble and wind up falling off a cliff.

Regards,
Elroy

 

[VantagePoint72]

And Strilanc, thanks a LOT for the link. I've got it and will definitely read it. I'll also look into getting a copy of Nielsen and Chuang's book if I feel that I need more clarity.

Just so we're clear: the link Stilanc gave you is for Nielsen and Chuang.

 

[Strilanc]

I wish there was a forum here named "Quantum Programming", but maybe that's asking too much, given that we're not even sure that we have working quantum computers just yet. This is just a fascinating area for me, and I'm just trying to take "baby steps" so as to not stumble and wind up falling off a cliff.

There are a large number of tools for doing quantum programming in existence.

- There's toy circuit simulators like this one and my own.
- The Quantum Computing Playground let's you apply quantum operations with a script and does an... okay... job visualizing the amplitudes.
- There are quantum programming languages / libraries like QuTiP for Python or Quipper for Haskell or the C-like QCL.

But ultimately, without the understanding provided by the textbook, those are all likely to be just a confusing morass of numbers.

Just so we're clear: the link Stilanc gave you is for Nielsen and Chuang.

Yeah, we basically made posts hitting the exact same points within 30 seconds of each other.

 

[Physics Monkey]

Oh, we're misunderstanding each other a little—when one "performs a Bell measurement" on a pair of qubits, as I said, you don't actually, in practice, project onto the Bell basis so that you end up with one of $\{|\Psi^\pm\rangle, |\Phi^\pm\rangle\}$. It is a pretty standard—though as this exchange demonstrates, sometimes not made sufficiently explicit—assumption in quantum information that you are only able to make measurements in the computational basis $\{|0\rangle, |1\rangle\}$. Since the ability to measure in the computational basis plus the ability to do arbitrary unitary operations allows one to effectively measure in any basis, this is often glossed over in proofs. So in order to "perform a Bell measurement" Alice has to rotate her two qubits—the one to be teleported and her half of the EPR pair—from the Bell basis to the computational basis. This uses a Hadamard transformation and a controlled-not, the latter of which disentangles the two qubits. So at the end of her "Bell" measurement her two qubits are in a product of computational basis states. Thus, the original qubit is in a pure state. So, we were talking about slightly different protocols. However, the one I describe is the one that would generally be used in any physical implementation of teleportation.

I'm sorry for being blunt, but I found your reply rather condescending. I know very well these trivial facts (and you seemed to be addressing me instead of the OP). I would further argue that the assumption that one can only measure in the computational basis (which is anyway an arbitrary choice modulo the tensor product structure) is a bad one when speaking about physical implementations.

Sorry, but the italics just really irritated me especially since I was merely describing the standard situation in an attempt to make contact with the OPs guess.

 

[VantagePoint72]

I'm sorry for being blunt, but I found your reply rather condescending. I know very well these trivial facts (and you seemed to be addressing me instead of the OP). I would further argue that the assumption that one can only measure in the computational basis (which is anyway an arbitrary choice modulo the tensor product structure) is a bad one when speaking about physical implementations.

Sorry, but the italics just really irritated me especially since I was merely describing the standard situation in an attempt to make contact with the OPs guess.

Gosh, sorry for choosing such offensive punctuation! Yours, in turn, is one of the most over sensitive, absurd posts I've ever seen on this forum.

If you are aware of the "trivial facts" I mentioned then you were quite simply wrong (oops, did it again!). The standard protocol of teleportation involves a transformation to the computational basis prior to measurement. This is the implementation every experimental application of teleportation has used. In it, there is no remaining entanglement after the protocol has completed.

You really need to get over yourself. It is fundamentally impossible to know the background from which people are posting, and my post was an attempt to be charitable and find common ground. If someone is charitable enough to misread an error on your part as a miscommunication on their's and you take it as an insult, then you need a reality (and ego) check.

 

[Physics Monkey]

Gosh, sorry for choosing such offensive punctuation! Yours, in turn, is one of the most over sensitive, absurd posts I've ever seen on this forum.

I admit that my response had an element of (pointless) anger - far from absurd I would say - but it hardly matters.

If you are aware of the "trivial facts" I mentioned then you were quite simply wrong (oops, did it again!). The standard protocol of teleportation involves a transformation to the computational basis prior to measurement. This is the implementation every experimental application of teleportation has used. In it, there is no remaining entanglement after the protocol has completed.

Here I simply disagree. I don't think it's up to you to declare what is standard and what isn't. The fact is there are many presentations of teleportation which do not make the extra step of explicitly transforming to the computational basis. In particular, I adhered to the language used by wikipedia for convenience.

To someone who is familiar with certain quantum information manipulations, this is a harmless and trivial distinction, but to a newcomer it might make a difference in their understanding.

You really need to get over yourself. It is fundamentally impossible to know the background from which people are posting, and my post was an attempt to be charitable and find common ground. If someone is charitable enough to misread an error on your part as a miscommunication on their's and you take it as an insult, then you need a reality (and ego) check.

As I said, I wrote in part from anger, and I'm sorry. But I maintain that when faced with uncertainty about the backgrounds of others the charitable road is not to lecture them. Especially when one is correcting "errors" that are amount to nothing but a different convention.

 

[VantagePoint72]

As I said, I wrote in part from anger, and I'm sorry. But I maintain that when faced with uncertainty about the backgrounds of others the charitable road is not to lecture them.

I did not "lecture" you. I thought I saw what was responsible for the discrepancy in our posts, and attempted to explain it. I don't accept partial apologies. Your response was completely ridiculous and disproportionate in every respect and I will not apologize for the apparently grave offense of telling you something it turns out you already knew. You spent half your post claiming injury due to a italicized word for God's sake.

Especially when one is correcting "errors" that are amount to nothing but a different convention.

The "different conventions" you refer to lead to completely contradictory answers to the original question. Given one of them is the one that, again, is actually implemented in experiments, pointing out the difference is entirely on point to this post.

 

[Physics Monkey]

Hi Monkey,

I actually did read the wikipedia quantum teleportation page earlier this morning before I posted my question. It was a bit confusing, but I think clarity may be slowly setting in. The first of their four proposed steps (to quantum teleportation) was:

1. An EPR pair is generated, one qubit sent to location A, the other to B.

Upon my initial read, I wasn't clear on what "sent to" meant. I now understand that they are talking about physically transporting the device that contains the isolated qubit.

I am actually an excellent "classical" programmer. However, that may be "getting in the way of" understanding this stuff as much as it is helping. In terms of classical bits, I just can't help but think in terms of pulling them off of my hard drive into memory, moving them around in memory, possibly coding them as an RF pulse and sending them to my router via WiFi, and then all over the world.

I imagine doing the same with qubits, with the understanding that only one "copy" will exist at any particular location at any point in time. Maybe this entire concept is just completely wrong-headed, and the entire field of quantum computing just isn't this reductionistic.

In fact, the more I think about it, maybe the whole idea of moving and storing qubits isn't necessary for an "initial" understanding of what quantum computing would be. Maybe I should just think in terms of qubits that aren't moved, as well as "registers" of qubits that are used for various computations.

And thanks for the welcome. This is my absolute first thread to the forum.

Regards,
Elroy

Hi Elroy,

I don't have much else to say at the moment except that you might enjoy looking at http://arxiv.org/abs/quant-ph/0101061. Werner phrases his introductory discussion in the amusing language of various impossible machines to emphasize some of the crucial distinctions between classical and quantum information along the lines you are suggesting in your post.

 

[Elroy]

*laughs and shakes my head* :p

I may be new to this forum, but I'm not new to posting in forums. *still laughing a bit*

I'm just getting back to this thread and catching up with it. I don't know what it is about these online forums, but we all seem to have a tendency to dig in our heels more so than in face-to-face encounters. And that includes myself. :) There's one specific guy in a programming forum that I splash around in that just drives me up the wall. The problem is, he's absolutely brilliant in a particular area of programming (understanding the most fundamental details of instantiating COM objects). When someone makes the most trivial of mistakes, he has a habit of copying quoted text with the following under them, "WRONG, WRONG, WRONG!". It just drives me nuts. *laughing*

Well, I haven't done enough reading yet to make intelligent indeterminate (pun intended) responses, but I'll be back. Y'all take care.

Elroy