It’s time I blogged some physics instead of filler like mouse transportation. There’s a lot of physics stuff going on around here but right now it’s kind of hush-hush and I can’t tell you about it. Which reminds me, I found an older version (perhaps a reader will disavow me of the notion that it is the oldest) of the line used in Top Gun, “I could tell you, but then I’d have to kill you“: Alexandre Dumas, in The Man in the Iron Mask aka The Vicomte de Bragelonne, writes:
“It is a state secret,” replied d’Artagnan, bluntly; “and as you know that according to the King’s orders it is under the penalty of death that any one should penetrate it, I will, if you like, allow you to read it and have you shot immediately afterwards.”
“The man in the iron mask” was a mysterious 17th century prisoner of the reign of Louis IV in France. Will I spoil the book if I tell you that in it, the state secret is that the man in the iron mask is the exact twin of the King of France? I hope not. It’s germane; in this post we will discuss what one would have to do to make the twin (or clone) of a quantum object, a [state] secret that evaded science until quite recently.
I will explain why this is of interest, and how this comes about in the language of quantum mechanics. For us, the quantum object will be an electron, and it’s state will be its spin.
From a classical point of view, there is nothing strange going on here. The “classical state” of an electron would be its position, velocity, and angular momentum. If you wish to clone the electron, you simply measure these things, and then adjust another electron to be the same (other than position, I suppose).
If the original electron is at Alice’s Restaurant, and we wish to make a clone at Bob’s Diner, then Alice measures her electron, sends the information to Bob, and Bob creates a clone. The cloning is by way of using classical information. The classical information can be sent any old way, for example, by heliograph.
In quantum mechanics, the situation is different, particularly with the intrinsic angular momentum (spin). The cloning of quantum spin turns out to be impossible. The proof against this is called the no cloning theorem. By “cloning” we mean taking a particle with an arbitrary, unknown, quantum state, and copying that state onto another particle without changing the state of the original particle. On the other hand, it is fairly easy to create a large number of particles all with the same state. This last is regularly done, most famously in the strange gas state, the Bose-Einstein Condensate.
If these theorems were to apply to people instead of quantum objects like the electron, and the quantum state of a person was their political persuasions, then it would be impossible to copy one person’s political persuasions onto another person (without knowing this persuasion beforehand), but it would be possible to arrange for a lot of people to have the same, known, persuasion. The limitation arises because in the process of asking a person political questions, you can change their political persuasion. I.e. all quantum questions are a sort of push polling as in the question “do you support Candidate A for office even though he is a member of Group W?”
Quantum mechanics dates to the 1920s, but the “no cloning theorem” remained a quantum state secret until 1982 when Wootters, Zurek, and Dieks removed the mask. Quantum teleportation was not discovered until 1993, by Bennett, Brassard, Crépeau, Jozsa, Peres, and Wootters. Pay attention and study hard, young physics graduate students; these subjects are barely older than you are, and there are many things still undiscovered out there.
The basic reason that, in quantum mechanics, spin cannot be impossible is that a spin state contains more information than can be extracted from the state by a single measurement. It requires multiple measurements to get all the information in a spin state but every single measurement destroys all information (and leaves the particle in the measured state).
To make an analogy with humans as quantum particles, we could think of measurement as asking a question. Before the question is asked, the person is in a state where he could answer either yes or no, either way. But after you have asked the question, the person remembers the answer, and if you ask again, you will again get that same answer. Asking the question destroys the original state of the person and replaces it with a “yes man” or “no man”. In academia, this is known as “never having to admit you were wrong.” Anyway, to figure out a person’s political persuasion before they are asked leading questions, you must obtain a large sample of identical people and ask them the question, and then calculate percentages.
But the analogy does not quite do justice to the bizarre behavior of quantum spin. Rather than a single question, to extract complete information on a spin-1/2 state, we must ask three different questions. And after asking any one of these questions, in addition to losing the prior information on that question, we also lose all information on the other two questions.
Physically, we can take the three questions to be “what is your spin along the x axis”, “what is your spin along the y axis”, and “what is your spin along the z axis”. The answers to any of these questions is either “with” or “against”. That is, if we ask the particle, “what is your spin along the x axis”, then the only two possible answers are “my spin is with the x-axis”, or “my spin is against the x axis”. And after we ask this question once, we will get the same answer each time we ask it again (so long as we don’t ask any other questions along the way).
But if after asking “what is your spin along the x axis”, and getting an answer, we then ask “what is your spin along the y axis”, then we will get an answer of “with” or “against” with 50% probability each. This is what is meant by “the information is destroyed”, after you ask about spin along any axis, the answers to the question of spin along the other axes will be 50% “with,” and 50% “against.” You might as well flip a coin.
What’s a human analogy to this? If this applied to humans it would be as if the human brain only had room for keeping track of one opinion. When you ask him a different question, he forgets his old opinion and chooses a new one (according to the probability laws). You can ask him the same question twice and get consistent answers, so long as you haven’t asked either of the two other question between.
To define a spin-1/2 state, we need the probabilities of getting “with” and “against” for each of the three axes. An example set of probabilities is:
Spin along the x axis? With: 50% Against: 50%,
Spin along the y axis? With: 80% Against: 20%,
Spin along the z axis? With: 10% Against: 90%.
Mathematically, the description of a spin-1/2 state is a 3-dimensional real unit vector, . For the above probabilities, the 3-d real unit vector works out to be . The technical reason for this is that the probability that a spinor gives you an answer of “with” is equal to , where is the angle between the spin direction and the direction (axis) along which you ask the question. And for a vector , the cosine of the angle between this vector and the x, y, and z axes are just .
This is highly unlikely to be the way you will have spin explained to you in class, but you can look up the details in various places. The 3-d vector maps out what is called the Bloch sphere. The probabilities for “with” answers are then , and . Mixed states, which require density matrices, fill the interior of the sphere.
As a student of physics, it is somewhat unfortunate that this is not the way you were / will be introduced to spin-1/2. The reason it is not used is that the above description only works for spin-1/2, it does not generalize well to higher spins. So instead they will give you a method of dealing with spin that is more complicated, but more general, that of spinors. In doing this, they end up creating physicists who are clones on this topic.
Instead of the Bloch sphere, spin-1/2 states will be encoded as 2-d complex vectors. The vector becomes the “spinor”:
providing . (Can the reader write a similar solution that works for ? If not, see the density matrix inspired calculation that I typed into the wikipedia article on the Pauli spinors a few years ago.)
In a certain intuitive way, spinors amount to square roots of vectors: As a rotation operator that operates on a matrix, spinors must be applied twice, both on the left and on the right. And the square roots of complex numbers have an arbitrary sign; we could take or . Spinors being a sort of square root also have this ambiguity but it is worse. One can multiply a spinor by an arbitrary complex phase and the result is an equivalent spinor. That is, multiplying a spinor by a complex phase does not change what physical state it represents.
From the probability rule, it is clear that measuring a spinor on any one axis only gives information about its spin along that axis. Measuring a spin-1/2 particles spin along the x axis gives information only about . And the information is probabilistic only; to get a good estimate of we have to repeat the measurement on a lot of identical states. And then (in general) we have to do the same thing for y and z.
And this is the long description of why it is impossible to clone spin-1/2 states using measurement. The actual proof of why one cannot clone a spin-1/2 state using other methods is quite short, and is given in the wikipedia article.
However, it is possible to use entanglement, along with measurement, to “teleport” a quantum state by transmitting information classically. And this is what we will discuss in the next (planned) blog post. Entanglement requires that we move from the modeling of single particle quantum states (the electron we wish to clone), to the modeling of quantum states with two particles.
Bit from Trit and spin-1/2
The idea of bit from trit is that the spin-1/2 fermions actually are made up of preons with three orthogonal orientations. This makes it a little more natural that it requires three measurements to get all the information on a spin-1/2 state.