Quantum experiments

In the early days of quantum physics, physicists directly reasoned on concrete systems: electrons, electromagnetic waves (photons), etc. Until the 1930s, the debates, notably at the Solvay conferences, focused mostly on the study of the duality between matter and wave and on Heisenberg's indeterminacy principle, which states that one cannot simultaneously know the position and momentum of a particle with arbitrarily high precision. In 1935, the focus moved on to compound systems and entanglement, with the famous debate between Albert Einstein and Niels Bohr.

Over the years a formalism was developed. Two competing frameworks gave similar predictions: the matrix framework (Heisenberg) and wave functions (Schrödinger and his equation). It was later shown that they are in fact equivalent, and they were unified with Dirac's notation (bras and kets).

Qubits

David Bohm was, to our knowledge, the first to realize that the mysteries of quantum theory do not need systems whose measured observables have an infinite number of possible values (such as the position or momentum of a particle); measurements with just two possible values (such as 0 and 1, or -1 and 1). They are called dichotomic. Physically, such measurement can be implemented with spin ½ particles or polarized photons, but this is in fact irrelevant. And so qubits were born. Bohm reformulated the problem formulated by Einstein, Podolsky, and Rosen in 1935 (known as EPR) using qubits instead of positions and momenta.

Local laboratories

Another layer of abstraction that appeared over time in the quantum information community is the representation of (possibly open) system states with so-called density matrices; of the (reversible) evolution of a quantum system with unitary operators, which can be seen as abstracting away from the Schrödinger equation; and of measurements as completely positive trace-preserving maps (CPTP maps, also known as quantum channels in quantum information).

We are not going over the math in this section, but just imagine that all that is involved here is linear algebra, with vectors and matrices, linear superposition, multiplication, tensor products, etc.

Thus, a useful unit of abstraction now commonly found in the quantum foundations literature is the notion of a local laboratory with an experimenter that receives a certain number of input qubits, picks a measurement among a list of choices by turning some knob, reads a measurement outcome on some display, and outputs a certain (possibly different) number of output qubits that depend on the measurement outcome.

Furthermore, there is a rule called the Born rule that, given some input qubits (density matrix) and a description of the lab's measurement device (a CPTP map for each possible position of the settings knob), gives the probabilities of obtaining each possible outcome conditioned on picking each specific setting. An update rule furthermore gives the output qubits (also a density matrix) based on the input qubits, the chosen setting, and the obtained outcome.

Again, do not worry about the math at this point: in a local laboratory, one takes the input qubits, feeds them into some device, turns a knob, reads a result, and produces output qubits. Period.