Quantum Laws in Black, White, and Blackandwhite

3.1.3 Quantum Laws in Black, White, and Blackandwhite

Quantum laws are, as far as we know, the most fundamental physical laws; they are inviolable. Here is our version of quantum physics distilled to five key laws.

Quantum is a system like everything else. 

To each physical system there corresponds a Hilbert space (1) of dimensionality equal to the system's maximum number of reliably distinguishable states (2).  

A quantum state is a configuration of the system.

Each direction (ray) in the Hilbert space corresponds to a possible state of the system (3), with two states being reliably distinguishable if and only if their directions are orthogonal (inner product is zero).

A quantum state changes; it naturally wants to evolve, but it can always be undone.

Evolution of a closed system is a unitary (4) transformation on its Hilbert space.

Scaling - how parts make a whole.

The Hilbert space of a composite system is the tensor product of the Hilbert space of the parts (5).

Quantum measurements are probabilistic.

Each possible measurement (6) on a system corresponds to a resolution of its Hilbert space into orthogonal subspaces {Πj} where ∑jΠj=1 . On state |ψ⟩ the result j occurs with probability P(j)=⟨ψ|Πj|ψ⟩ and the state after the measurement is |ψj⟩=Πj|ψ⟩/√P(j) .

These five principles are the foundation for the whole quantum world.

Clarifications

1. A Hilbert space is a linear vector space with complex coefficients and inner products ⟨ϕ|ψ⟩=∑iϕ∗iψi .

2. For a single qubit, there are two standard orthogonal states (computational basis states) that are conventionally denoted  |0⟩=(10) and |1⟩=(01) .

3. Other qubit states include  |+⟩=1√2(11) , |−⟩=1√2(1−1) ,  |↻⟩=1√2(1i) and |↺⟩=1√2(1−i)

4. Unitary means linear and inner product preserving.

5. A two-qubit system can exist in a product state such as |00⟩ or |0+⟩ but also in an entangled state (|00⟩+|11⟩)/√2 , in which neither qubit has a definite state, even though the pair together does. 

6. Measurement causes the system to behave probabilistically and forget its pre-measurement state, unless that state happens to lie entirely with one of the subspaces Πj .