If a tree falls in a forest, and nobody is there to hear it, does it make a sound? The answer was obvious to my 12-year-old self — of course it made a sound. More specifically, something ranging from a thud to a thump. There doesn’t need to be an animal present for the tree to jiggle air molecules. Classical physics for the win! Around the same time I was exposed to this thought experiment, I read Michael Crichton’s Timeline. The premise is simple, but not necessarily feasible: archeologists use ‘quantum technology’ (many-worlds interpretation and quantum teleportation) to travel to the Dordogne region of France in the mid 1300s. Blood, guts, action, drama, and plot twists ensue. I haven’t returned to this book since I was thirteen, so I’m guaranteed to have the plot wrong, but for better or worse, I credit this book with planting the seeds of a misconception about what ‘quantum teleportation’ actually entails. This is the first of a multi-part post which will introduce readers to the one-and-only way we know of how teleportation works.
Before getting there, given that this is my first post, I should introduce myself: my name is Shaun Maguire and I’m a 2nd year graduate student in the quantum information group at Caltech. I agree with some of the earlier sentiments that it’ll be interesting to see where this blog goes — especially considering that they gave the keys to hooligans like myself. I’m new to this field and certain to get details wrong, but I’ll try to make up for this by providing boundless enthusiasm and a fresh perspective.
Anyways, back to quantum teleportation. To me, the word teleportation is synonymous with faster than light transport. So that begs the question, where is the ‘faster than light’ in quantum teleportation? The meta/quantum answer is that it’s both there and it’s not. It’s there in that we can effect instantaneous change over arbitrarily large spatial distances. It’s not there in that we need to transmit classical information between these two locations, abiding by the rules of special relativity, before we can take advantage of the teleported information. Let me explain the process in more detail, trying to do so using both words and notation in parallel.
The quantum coin: The story of quantum teleportation starts with two foundational elements of quantum information: qubits and entanglement. In what follows, I will attempt to explain these two concepts at a level targeting bright aspiring quantum physicists (experts know these topics way better than myself and people in between have resources like Nielsen and Chuang and John Preskill’s notes.)
For now, qubits (short for quantum bits) are to quantum information what bits are to classical information (the language in which computers operate). There are many different implementations for a classical bit (the familiar “0” and “1”). It can be represented by any system which has two states. It is convenient (for later use) to introduce notation which represents such a state abstractly as and Examples include: current in a circuit above/below a threshold (classical computer), right/left circular polarization of photons (optical computer) and sequences of base pairs in DNA (molecular computer). Caltech has been a pioneer in all three of these forms of computing, with seminal contributions from the research groups of: Carver Mead, Amnon Yariv and Erik Winfree (this list is far from comprehensive regarding the contributions coming from Caltech). I would like to think that IQIM is making contributions of a similar scale to quantum computing, but obviously only time will tell.
A classical bit is always in a definite state — for example it is either: on/off, left/right or A/T/C/G (not quite a bit, but close enough) — where for our purposes this physical underpinning is abstracted away and represented by 0s and 1s. A qubit is a different beast entirely. It exists in a superposition , where and are complex numbers such that , and is the probability that a measurement of this qubit would yield (while is the probability of getting the state after the measurement). This is easy to say, but what does it mean? You can think of a superposition as a ‘mixture’ of the two states. It’s neither one nor the other, it’s a ‘mixture’ of the two. However, when a scientist measures the qubit, it collapses to a definite state. It becomes either a zero or a one. In technical language, the outcomes of measurements are eigenfunctions, even though the state before the measurement can be a linear combination of different eigenfunctions. The subtlety of this business is evidenced by the great number of books that have been written on the topics of Schrödinger’s cat and different interpretations of quantum mechanics. Much of this literature stems from confusion about what’s going on in the time between preparing a quantum state and measuring it.
It’s my personal opinion that for the time being, you’d be best served by sweeping these subtleties under the rug and taking it on faith that qubits are both probabilistic and don’t have a defined state until they are measured. I promise that your intuition improves as you work with quantum systems (but the difficulty in explaining these subtleties to the non-initiated remains!) The first condition, that qubits are probabilistic, means that even if there was a machine which prepared millions of identical qubits, and stored them in little boxes (wrapped like Holiday presents?), then when you measured different qubits, you wouldn’t necessarily obtain the same result! The probability of measuring the state corresponding to would be and the probability of getting would be . You might be thinking that this is perfectly acceptable, maybe the machine flipped a coin and chose a state accordingly, before preparing the boxes? The probabilistic aspect of qubits has a classical analog in the flipped coin. But it’s the second property of qubits, when you try to understand what happens in the time between preparing a quantum state and measuring it, where your intuition really begins to deviate from classical physics. The important thing is that the qubit hadn’t ‘made up its mind’ before the measurement. We can perform experiments which conclusively show that the qubit wasn’t in a definite state before the measurement — it wasn’t yet a 0 or a 1.
To make the differences between bits and qubits more concrete, I’ll introduce a silly conception of a two-state system which is near and dear to my heart — I run frequently and I therefore eat lots of dessert, so my mind thinks in terms of cookies and brownies. Using this model, bits are EITHER cookies OR brownies. Qubits are superpositions of cookies and brownies. For example, a qubit could be in an unequal superposition : . Let’s imagine there are classical and quantum factories which produce bits and qubits respectively. To say that the classical factory is probabilistic might mean that there’s a long assembly line which can produce both cookies and brownies. Let’s say that when you show up at the factory to buy a dozen cookies you have the option of ordering the house specialty — ‘the probabilistic dozen.’ What this means is that every time someone orders another ‘probabilistic dozen,’ an attendant flips an unfair coin (1/4 heads, 3/4 tails) twelve times and packs the box accordingly. Without looking in the box, you take it home to serve to your family after dinner (or to selfishly devour the whole box after a long run.) This particular box may have three cookies, it may have four, or any other number between zero and twelve. You don’t know before you open the box — however, the number of cookies in the box is very well defined — the attendant who packed the box knows exactly how many cookies vs brownies are in there, even though you don’t know.
The quantum factory is very different. In this case, there are two ways that the factory could prepare something akin to a ‘probabilistic dozen.’ First, the attendant could do exactly the same thing as before, flipping a coin and then putting either cookies or brownies in the box accordingly. A cookie would have the state , and a brownie the state . When you open the box, you’ll have a certain number of cookies and a certain number of brownies and the attendant will know exactly how many you will obtain*. Moreover, the cookies and brownies were prepared differently. The second, fundamentally different way to serve up a ‘probabilistic dozen’ would be to package twelve identical quantum brown-kies in the superposition: . Again, the attendant knows exactly what they packed for you — twelve identical quantum brown-kies — but they don’t know how many cookies and brownies you’ll obtain when you open your box. When you open the box for the first time, each quantum brown-kie is forced to collapse into a definite cookie or brownie state. It will then take this form forever after, unless some additional quantum magic is performed. Also, it’s worth noting that the quantum brown-kies don’t have to ‘make up their mind’ every time you open the box. Only the first time — the first measurement.
In summary, a qubit is a superposition of two states: . It’s not exclusively in either state, it’s still in limbo, with a proclivity towards each of its two possible states governed by and . It chooses between the two when it’s measured for the first time. This would be a very natural segue into quantum computation, but I’m going to stick with quantum teleportation for now. That means I also need to explain entanglement.
My next post will attempt to do just that: explain entanglement. Why should you read it? Because at the end of this series, you’ll be able to understand how we built a quantum teleportation machine at Caltech.