Qubits ("quantum bits") are the quantum counterpart of the classical "bit" of a digital computer. The "bit" is an abbreviation of "binary digit."

Where classical bits can have the value "0" or "1," qubits are in a coherent superposition or linear combination of |0> or |1> quantum states, as Paul Dirac described in his 1930 text Principles of Quantum Mechanics.

Measurement of a qubit projects the qubit randomly into either single-particle state |0> or |1>, with probability 1/2 following Dirac's "projection postulate."

The 1/2 probability of each state is the square of the "probability amplitude" 1/√2.

When two qubits a and b are entangled, their wave function Ψ is a linear combination (or superposition of two-particle "product" states |01> and |10>,

The two-particle wave function Ψ_ab describes the behavior of two entangled particles (electrons, photons, or atoms) with spin angular momentum up |↑> or down |↓> that have traveled far apart.

When either a or b is measured, the two-particle wave function Ψ_ab collapses, both particles are individually projected into random states up (|↑> or down |↓>), but jointly they always appear perfectly correlated in one of the two product states up-down |↑↓> or down-up |↓↑>.

Either of these product states conserves the total spin angular momentum of the initial prepared entangled state, although particular spin directions are not defined.

We describe this conserved total angular momentum as a hidden constant of the motion.

The perfect correlations between widely separated objects are understandably and frequently misinterpreted as the measurement of one particle, say A, instantaneously and non-locally causing the other particle, say B, to become correlated.

But quantum mechanics finds no such "spooky action at a distance, as Albert Einstein called it. There is however nonlocal behavior.

Einstein discovered examples of nonlocality starting in 1905. In 1916 he showed that quantum processes can be totally random, the result of chance. But all his life he hoped for a "reality" in which properties of physical objects would have values independent of our observing them.

This nonlocality and randomness are qubit properties at the core of quantum information science, including quantum cryptography, quantum teleportation, and quantum computing.

Since two entangled qubits are in a superposition of |0> or |1>, or |↓> or |↑>, they are not in a definite quantum state before a measurement and the state resulting from a measurement did not "exist" before the measurement. This bothered Einstein all his life.

Reacting to the 1935 Einstein-Podolsky-Rosen Paradox paper, Erwin Schrödinger told Einstein that the entangled particles cannot be separated and in a definite product state before measurement. He said they are in a superposition of such states, as defined by Dirac.

Encouraged by Einstein, David Bohm in 1952 proposed that local hidden variables could account for the perfect correlations and the nonlocality at the heart of the EPR paradox.

In 1964 John Bell proposed a test of what he called an "inequality" that would distinguish between local and non-local hidden variables. Multiple experiments starting in 1972 have confirmed that any such variables would need to be non-local.

We cannot understand how a purely mathematical and immaterial abstract wave function, the solution of the linear Schrödinger equation, can cause material concrete particles to move to positions and acquire properties that perfectly match the statistical predictions of the wave function.

In 1964 Richard Feynman said that "nobody understands quantum mechanics." We see the central difficulty as how the math describes (or controls) matter in motion. Feynman also said "The question now is, how does it really work? What machinery is actually producing this thing? Nobody knows any machinery."(The Character of Physical Law, p.144)

Feynman also said that the "one and only mystery" of quantum mechanics can always be explained in terms of the two-slit experiment, in which a single particle appears to travel through both slits, better a wave function that is a superposition of partial waves going through the slits that somehow "controls" the location of a particle that comes through either slit.

Following Feynman's suggestion, we can explain many puzzling experiments with two-particle wave functions (or entangled qubits). They include John Wheeler's delayed choice, the quantum eraser, the Mach-Zender interferometer, the Fresnel-Arago effect, and of course Schrödinger's Cat.