Dork Scratchings

I'm still a bit obsessed with the GHZ experiment.  This is the one where you prepare a GHZ state $\lvert 000 \rangle + \lvert 111 \rangle$ and send one qubit to each of 3 protagonists: Alice, Bob, and Charlie.  (I will be ignoring shared normalizer constants throughout this post, as I find they don't add anything to the understanding.) If you do the maths it turns out that when all three choose to measure in the $\lvert + \rangle$, $\lvert - \rangle$ basis (shorthand for $\lvert 0 \rangle + \lvert 1\rangle$ and $\lvert 0 \rangle - \lvert 1\rangle$) then they are guarranteed to get a parity zero result.  On the other hand if only one measures in this basis and the other two measure in the $\lvert +i\rangle$, $\lvert -i\rangle$ basis (shorthand for $\lvert 0 \rangle + i\lvert 1\rangle$ and $\lvert 0 \rangle - i\lvert 1\rangle$) they are guarranteed to get a parity one result.  As I showed in an earlier post this appears to be incompatible with the outcomes being prede...

It has occurred to me that the cartoon I drew two days ago will appear to most people as absurd: Alice measures one half of an entangled pair and instantly splits into two versions of herself!  Bob measures the other half and also splits, but each Alice is paired with one of the Bobs in such a way that, should they meet later to discuss their results, everything remains consistent. Yet if you replace Alice and Bob with two qubits - to get a quantum circuit like the one below - no quantum computing expert would find this outcome absurd. The Einstein Podolsky Rosen experiment in circuit form with non-classical Alice and Bob Alice is qubit q0 and Bob is q3 .  Qubits q1 and q2 are entangled into the Bell state $(\lvert 01 \rangle + \lvert 10 \rangle)/\sqrt 2$ and then q1 is shared with Alice, and q2 with Bob.  In the standard (collapse-based, non-multiversal) description Alice and Bob measure q1 and q2 , collapse the q1q2 wavefunction, and find that one gets 0 while the...

I just read this article on phys.org which claimed to "resolve the paradox" that entanglement appears to imply faster than light communication.  I don't think it actually resolves the paradox, it just side steps it.  Here's how the argument goes (and I paraphrase): Alice and Bob share a pair of entangled qubits in a Bell state $\left( \lvert 01 \rangle + \lvert 10 \rangle \right)/\sqrt{2}$. When Alice measures hers it "collapses" the pair's state immediately even if Bob is a great distance away.  However, this does not imply faster than light communication because, although Alice now knows what Bob will measure, Bob does not know until he makes a measurement himself, or receives a message from Alice which can only travel at the speed of light. This argument misses the point somewhat.  If the state of both qubits changes instantly when Alice measures her qubit (e.g from $\left( \lvert 01 \rangle + \lvert 10 \rangle \right)/\sqrt{2}$ to $\lvert 01 \rangl...

In 1964 John Stewart Bell proposed an experiment to determine whether the results of quantum measurements were truly random, or governed by hidden variables, i.e. state that exists prior to the measurement, but which we don’t have access to. The experiment involved creating a large number of EPR pairs, and firing them at two observers, Alice and Bob, who measure their photon’s polarisation, choosing the $\updownarrow$ direction or the $\nearrow\llap\swarrow$ direction at random. Determining the result of the experiment involves doing a complex statistical calculation to see if something called Bell’s inequality is satisfied or violated. The Bell experiment was first performed by in 1982 by Alain Aspect, and the result, as most commonly interpreted, is that hidden variables can only exist if Quantum Mechanics is non-local, i.e. if it supports faster-than-light causality! Some time after Bell proposed his experiment, Greenberger, Horne, and Zeilinger suggested an alternativ...

What's going on right now, 70,000 light years away on the nearest galaxy to our own, Canis Major?  Or put it another way, if we're at $x = y = z = t = 0$, what's happening at $y = z = t = 0$ and $x$ = 70,000 light years? According to Einstein's special theory of relativity it depends who you ask.  The most natural coordinates to use depend on your inertial frame.  Sitting in frame $S$, I may use the coordinates $x,y,z,t$ but someone in frame $S'$ would need to use $x',y',z',t'$ in order to observe the universe operating according to the same physical laws.  To make it more concrete, imagine $S$ and $S'$ share their $y$ and $z$ axes, and $S'$ is moving at velocity $v$ along the $x$ axis of $S$.  In this case the transform for converting between the two reference frames is $$ \begin{align} x' &= \frac{x - vt}{\sqrt{1-\frac{v^2}{c^2}}} \\ y' &= y \\ z' &= z \\ t' &= \frac{t - \frac{vx}{c^2}}{\sqrt{1-\frac{v^2}{c^...

MWQT$^3$ I've just discovered Quantum Tic Tac Toe .  This is a brilliant game designed to give players an intuition for Quantum Mechanics without requiring them to learn any complicated mathematics. Superposition Instead of playing in just one square each player gets to play in two squares at once.   Here player X has played in the 1st & 2nd squares on their first move, the 3rd & 6th squares on their 2nd move, and the 8th & 9th squares on their 3rd move.  Similarly player O has played on two squares each move, and the marks are subscripted to indicate when in the game play they were laid down. Entanglement Ultimately each pair of marks will be replaced with a single mark and the board will look like ordinary Tic Tac Toe - which is how we are able to determine a winner.  But the final location of, say, $O_2$ may need to depend on the final location of, say, $X_1$ if we are to avoid ending up with squares with multiple marks in them.  In this case...

Lockdown reading My lockdown reading list consists of just one book. This might last a long time I thought, so it's my opportunity to make a 2nd stab at understanding Quantum Field Theory. Last time, I bought "Quantum Field Theory for the gifted Amateur" and I learned a lot from it. Mainly that I am not gifted! I got three chapters through it and then gave up on the book, and on quantum field theory. This time round I did my research better and found a much more gentle book: Student Friendly Quantum Field Theory, by Robert D. Klauber . It covers the same material, but takes pains not to lose the reader, by spelling out every ambiguity and subtlety. I'm half way through and feeling quite chuffed with myself. Here I am studying hard, on a sunny day in Lockdown Britain: No, the weights are not mine In this book, and every other in the QFT literature there is a concept of some particles being virtual. What's this about?  Why are some virtual and o...

It may not be entirely obvious from the photograph (which I took at night, while hashing ) but this is a wrought iron gate. It opens on to a front garden on Maid's Causeway, Cambridge.  Why?  I don't know.  Has an important physicist lived there?  There's no Blue Plaque , so maybe it's just an enthusiast, like me! What does it mean?  I don't know for certain, but I suspect it is a reference to something similar to the GHSZ variant of the Bell Inequality Test .  The results of this test demonstrate that there are no hidden variables in quantum mechanics.  I say "similar" because, in GHSZ instead of 0 and 1 the spins $\downarrow$ and $\uparrow$ are used, and there's a minus instead of a plus. If you know anything more, please tell me!

Top left box evolves into a superposition of the other two When introduced to quantum physics the first example we encounter is usually that of a single particle.  We are shown that experiment demonstrates a particle left alone evolves into a superposition of states.  These states may be position states or they may be momentum states, or if the particle has spin it may be a superposition of spin states.  It doesn't matter, the point is that fundamental particles can be in a superposition of states.  That's because they are small, we are told, so you wouldn't expect them to behave like big things do, we are told. The next thing we are shown is how the Schroedinger equation governs the evolution of this superposition.  The particle is not usually in every state equally, it is more in one position state (or momentum state, or spin state) than it is in another.  The distribution over these so-called basis states evolves with time and the ...

Clockwise from top left: Occam, Deutsch, Everett, and Dirac Occam's razor The anthropic principle Forward reasoning Lack of any consistent alternative 1. Occam's razor Hugh Everett's 1956 thesis The Theory of the Universal Wavefunction opens with a mathematical summary of the then widely accepted Copenhagen Interpretation. "... there are two fundamentally different ways in which the state function can change:    Process 1:  The discontinuous change brought about by the observation of a quantity with eigenstates $\phi_1, \phi_2,...,$ in which the state $\psi$ will be changed to the state $\phi_j$ with probability $\lvert(\psi,\phi_j)\rvert^2$. Process 2: The continuous, deterministic change of state of the (isolated) system with time according to a wave equation $\frac{\partial \psi}{\partial t} = U\psi$, where $U$ is a linear operator." The 1st process is commonly known as the "collapse of the wavefunction" and ...