Quantum mechanics vs. probability

A few similarities and differences between quantum systems and probabilistic systems.

• A probability distribution on a set $X$ is a function $\rho: X \rightarrow \mathbb{R}$ such that $\int_X \rho(x) \; dx = 1$. A quantum state in a set $X$ is a norm 1 element of the free vector space $\mathbb{C}\{X\}$. An operator on a probability distribution is a stochastic matrix, i.e. a matrix of non-negative real entries whose columns sum to 1. An operator on a quantum system is a unitary matrix.
• Given two probability distributions on $X$ and $Y$, one can form the product $\rho_X \times \rho_Y: X \times Y \rightarrow \mathbb{R}$, which has $\int_{X \times Y} \rho_X \rho_Y = 1$ by Fubini's theorem. Given two quantum states $v \in V$ and $w \in W$, one can form $v \otimes w \in V \otimes W$, which is also a unit vector.
• Given a probability distribution on a product $X \times Y$, one can project the distribution to either $X$ or $Y$ by integration. However, given a tensor product of vector spaces $V \otimes W$, there is no way to project to $V$ or $W$.
• There is a notion of conditional probability, corresponding to the notion of partial measurement of a quantum state. Note that we have to renormalize the vector to be a unit vector in this case.
• The property of independence of distributions corresponds to the property of a quantum state being separable, i.e. a rank one tensor. The notion of having non-zero correlation corresponds to entanglement.
• Stochastic matrices often have no stochastic square root, i.e. one can not take half steps. The square roots of unitary matrices are always unitary.
• Stochasitc matrices can never turn uncertainty into certainty, i.e. there is no stochastic matrix $A$ such that $A \begin{pmatrix} 1/2 \\ 1/2 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$. However, unitary matrices act transitively on the unit vectors, e.g. the Hadamard operator $H = \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{pmatrix}$ has $H \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$.
• We can measure the the probability of some event $E \subset X$ by taking $P(E) = \int_E \rho(x) \; dx$. For quantum systems, measurements on a complex inner product space $V$ correspond to Hermitian projections (i.e. self-adjoint idempotents).
  • More precisely, a measurement should be viewed as a "complete" collection of such self-adjoint idempotents that sum to the identity. This is mostly semantic and refers to the fact that we usually do not simply ask if a coin flip is heads, we ask if it is heads or tails. Of course, any non-complete collection of "orthogonal" projections can be completed by adding the projection to the orthogonal complement of the sums of the images.