This is quite remarkable.
Prepare for the unpredictable
After all, Alice hasn't sent Bob a qubit, just two classical bits. Obviously, neither Alice nor Bob receives that information. This is, in my opinion, a most curious and surprising state of affairs. We'll understand more deeply how it works below. In the introduction I said that this essay is in a new form, a mnemonic medium. That means the medium is designed to make it essentially effortless for you to remember what you read. The way the mnemonic medium works is this: throughout the essay we'll occasionally pause to ask you a few simple questions, testing you on the material just explained.
In the days and weeks ahead we'll re-test you in followup review sessions.
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By expanding the review schedule, we can ensure you consolidate the answers into your long-term memory, while minimizing the study time required. In particular, the expanding review schedule means that each extra minute spent studying provides more and more benefit, a kind of exponential return.
The review sessions take no more than a few minutes per day, and we'll notify you when you need to review.
The benefit is that instead of remembering how quantum teleportation works for a few hours or days, you'll remember for years; it'll become a much more deeply internalized part of your thinking. That may sound a strange aspiration. But if you're genuinely interested in understanding quantum computing, then having teleportation down cold is necessary. To give you a more concrete flavor of how the mnemonic medium works, let's take a look at three questions reviewing part of what you've just learned. Please indulge me by answering these questions — it'll take just a few seconds.
For each question, think about what you believe the answer to be, click to reveal the actual answer, and then mark whether you remembered or not. If you can recall, that's great.
But if not — and most readers don't get the answers to these questions correct! I said above that most readers don't recall the answers to these questions. It's worth thinking about what this means. The questions ask about some of the most absolutely basic things about the quantum teleportation protocol.
If someone is not getting these questions correct, what are they really learning about quantum teleportation? If you're in this boat, I challenge you to name three specific things that you've learned about quantum teleportation so far. Genuine learning requires paying close attention to what you're reading. In fact, it's not difficult to learn any of the three things tested in the questions above, if you're paying attention. I don't mean to be a downer. But I also think it's important to be realistic. Most people myself included learn very little from most of what we read, unless we're paying close attention.
The reading may be entertaining, or produce a brief illusion of understanding. But you can only learn if you pay attention.
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The questions are good way of monitoring whether that's the case. And reviewing them again in the future will help you internalize this understanding for the long term. It's tempting to take refuge in a belief that what you're looking for is a broad, conceptual understanding. Unfortunately, I've never met someone knowledgeable about quantum computing who didn't know the details of teleportation.
You'd think their claim to a broad conceptual understanding of Spanish was hilarious. If you want to understand quantum computing and related subjects, you need to know the details of how the teleportation protocol works. That means knowing things like what state Alice and Bob initially share. What's more, it means not just knowing them immediately after reading. It means internalizing them for the long term. If you're interested in doing that, then I invite you to set up an account by signing in below. If you do so, your review schedule for each question in the essay will be tracked, and you'll receive reminders each day or few days , containing a link which takes you to an online review session.
That review session isn't this full essay — rather, it looks just like the question set you answered above, but contains instead all the questions which are due, so you can quickly run through them. The time commitment is no more than a few minutes per day. You can study on your phone while grabbing coffee, or standing in line, or going for a walk, or in transit.
The return for that small time commitment is an internalized understanding of quantum teleportation, retained for years instead of days. To keep this promise, we'll track your review schedule for each question, and send you occasional reminders to check in, and to run through the questions which are due. You can review on your phone while grabbing coffee, or standing in line, or going for a walk, or on transit.
The return for that small time commitment is an internalized understanding of quantum teleportation; it'll become a part of who you are, retained for years instead of days. Before we verify that the teleportation circuit works, let's briefly discuss one of the most common questions about quantum teleportation: does it enable faster-than-light communication?
It'd be quite marvelous if it enabled faster-than-light communication, since that in turn would give rise to many incredible phenomena, including the ability to send information backward in time. But while it would be marvelous, it is not possible. You can see the trouble if you think closely about the protocol. The speed of that transmission is limited by the speed of light. Instead, what he has is a distribution over four different possible states.
And while I won't prove it here, it turns out to be possible to prove that with only that distribution over states, no information is transferred from Alice to Bob. It's a pity, but that's the way the world seems to work. Notice that the second question is a more qualitative style of question than the earlier questions.
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Your answer may not exactly match the answer given. It's up to you to decide whether you want to mark yourself correct or not. Ask yourself: have I really understood the core point? If so, mark yourself correct. If not, don't! The point of all the questions is to serve you, and it's up to you to decide how best they can do that. To verify that the teleportation protocol works, we'll mostly use tools already introduced in the earlier essay Quantum Computing for the Very Curious.
But there's one missing piece of background knowledge we need to fill in first. In the earlier essay I explained how to do computational basis measurements for multi-qubit systems. But I didn't explain what happens if you measure just some but not all of the qubits. This is relevant to quantum teleportation, since we're going to be measuring just two of three qubits.
The rule for describing such partial measurements is simple, though slightly cumbersome to describe. First, I'll describe it for a two-qubit system, and then explain how it generalizes. Suppose we have a two-qubit system in the state. Suppose we measure just the first qubit in the computational basis. We'd like to know i what the probabilities for the two measurement outcomes are; and ii what the corresponding resulting state of the second qubit will be. Suppose we measure in the computational basis on the first qubit, and obtain the result 0 0 0.
Similarly, if the result from the computational basis measurement is 1 1 1 , then the corresponding conditional state for the second qubit is. I won't write the general rule out in absolutely full generality, but hopefully it's pretty clear what the rule is. For instance, suppose we measure the first two qubits of a many-qubit system in the computational basis.
To figure out the result, we express the state immediately prior to measurement as. As I said above, this rule is a little bit cumbersome, but with some practice it becomes easy to use fluently. We'll get an opportunity very shortly, as we verify the teleportation protocol. To help you get used to the rule, it's worth taking a few minutes to work through the exercise immediately below. Unlike the review questions, the point of the exercise isn't as an aid to memory.