# Quantum Coin Tossing

Let me ask the following very simple question. Suppose that I toss a pair of identical coins at the same time, then what is the probability of them both coming up heads? There is no catch here, both coins are fair. There are three possible outcomes, both tails, one head and one tail, and both heads. Assuming that it is completely random so that all outcomes are equally likely, then we could argue that each possibility has a one in three chance of occurring, so that the answer to the question is that the probability is 1/3.

Of course, this is wrong! A fair coin has a probability of 1/2 of showing heads and, by independence, standard probability theory says that we should multiply these together for each coin to get the correct answer of ${\frac12\times\frac12=\frac14}$, which can be verified by experiment. Alternatively, we can note that the outcome of one tail and one head, in reality, consists of two equally likely possibilities. Either the first coin can be a head and the second a tail, or vice-versa. So, there are actually four equally likely possible outcomes, only one of which has both coins showing heads, again giving a probability of 1/4. Continue reading “Quantum Coin Tossing”

# Quantum Entanglement States

In an earlier post, I described four simple thought experiments, involving some black boxes and two or more participants. As described there, the results of these experiments were inconsistent with any classical description, assuming that the boxes cannot communicate. However, I also stated that all of these experiments are consistent with quantum probability, and that I would give the mathematical details in a further post. I will do this now. Continue reading “Quantum Entanglement States”

# Quantum Entanglement

Quantum entanglement is one of the most striking differences between the behaviour of the universe described by quantum theory, and that given by classical physics. If two physical systems interact then, even if they later separate, their future evolutions can no longer be considered purely in isolation. Any attempt to describe the systems with classical logic leads inevitably to an apparent link between them, where simply observing one instantaneously impacts the state of the other. This effect remains, regardless of how far apart the systems become.

As it is a very famous quantum phenomenon, a lot has been written about entanglement in both the scientific and popular literature. However, it does still seem to be frequently misunderstood, with many surrounding misconceptions. I will attempt to explain the effects of entanglement in as straightforward a way as possible, with some very basic thought experiments. These can be followed without any understanding of what physical processes may be going on underneath. They only involve pressing a button on a box and observing the colour of a light bulb mounted on it. In fact, this is one of the features of quantum entanglement. It does not matter how you describe the physical world, whether you think of things as particles, waves, or whatever. Entanglement is an observable property independently of how, or even if, we try to describe the physical processes. Continue reading “Quantum Entanglement”

# Algebraic Probability: Quantum Theory

We continue the investigation of representing probability spaces as states on algebras. Whereas, previously, I focused attention on the commutative case and on classical probabilities, in the current post I will look at non-commutative quantum probability.

Quantum theory is concerned with computing probabilities of outcomes of measurements of a physical system, as conducted by an observer. The standard approach is to start with a Hilbert space ${\mathcal H}$, which is used to represent the states of the system. This is a vector space over the complex numbers, together with an inner product ${\langle\cdot,\cdot\rangle}$. By definition, this is linear in one argument and anti-linear in the other,

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\langle\phi,\lambda\psi+\mu\chi\rangle=\lambda\langle\phi,\psi\rangle+\mu\langle\phi,\chi\rangle,\smallskip\\ &\displaystyle\langle\lambda\phi+\mu\psi,\chi\rangle=\bar\lambda\langle\phi,\chi\rangle+\bar\mu\langle\psi,\chi\rangle,\smallskip\\ &\displaystyle\langle\psi,\phi\rangle=\overline{\langle\phi,\psi\rangle}, \end{array}$

for ${\phi,\psi,\chi\in\mathcal H}$ and ${\lambda,\mu\in{\mathbb C}}$. Positive definiteness is required, so that ${\langle\psi,\psi\rangle > 0}$ for ${\psi\not=0}$. I am using the physicists’ convention, where the inner product is linear in the second argument and anti-linear in the first. Furthermore, physicists often use the bra–ket notation ${\langle\phi\vert\psi\rangle}$, which can be split up into the bra’ ${\langle\phi\vert}$ and ket’ ${\vert\psi\rangle}$ considered as elements of the dual space of ${\mathcal H}$ and of ${\mathcal H}$ respectively. For a linear operator ${A\colon\mathcal H\rightarrow\mathcal H}$, the expression ${\langle\phi,A\psi\rangle}$ is often expressed as ${\langle\phi\vert A\vert\psi\rangle}$ in the physicists’ language. By the Hilbert space definition, ${\mathcal H}$ is complete with respect to the norm ${\lVert\psi\rVert=\sqrt{\langle\psi,\psi\rangle}}$. Continue reading “Algebraic Probability: Quantum Theory”