“The quantum state of the universe is not a collection of things distributed through space, but something called a wave function.”
“The wave function gives us a way of calculating the outcomes of measurements: whenever we measure an observable quantity like the position or momentum or spin of a particle, the wave function has a value for every possible outcome, and the probability of obtaining that outcome is given by the wave function squared.”
“Mathematically, wave functions are elements of a mathematical structure called Hilbert space. That means they are vectors — we can add quantum states together (the origin of superpositions in quantum mechanics) and calculate the angle (“dot product”) between them.”
“The word “space” in “Hilbert space” doesn’t mean the good old three-dimensional space we walk through every day, or even the four-dimensional spacetime of relativity. It’s just math-speak for “a collection of things,” in this case “possible quantum states of the universe.””
“quantum mechanics says that the real world is best described by a wave function, an element of Hilbert space, evolving through time. (Of course time could be emergent too … something for another day.)”
“The real world simply is quantum-mechanical from the start; it’s not a quantization of some classical system. The universe is described by an element of Hilbert space. All of our usual classical notions should be derived from that, not the other way around.”
“Even space itself. We think of the space through which we move as one of the most basic and irreducible constituents of the real world, but it might be better thought of as an approximate notion that emerges at large distances and low energies.”
“So here is the task we set for ourselves: start with a quantum state in Hilbert space. Not a random or generic state, admittedly; a particular kind of state. Divide Hilbert space up into pieces — technically, factors that we multiply together to make the whole space. Use quantum information — in particular, the amount of entanglement between different parts of the state, as measured by the mutual information — to define a “distance” between them. Parts that are highly entangled are considered to be nearby, while unentangled parts are far away. This gives us a graph, in which vertices are the different parts of Hilbert space, and the edges are weighted by the emergent distance between them.”
“We can then ask two questions:
When we zoom out, does the graph take on the geometry of a smooth, flat space with a fixed number of dimensions? (Answer: yes, when we put in the right kind of state to start with.) If we perturb the state a little bit, how does the emergent geometry change? (Answer: space curves in response to emergent mass/energy, in a way reminiscent of Einstein’s equation in general relativity.)”
“People like Jacobson and Seth Lloyd have argued, on pretty general grounds, that any theory with gravity will locally be described by finite-dimensional Hilbert spaces.”
“Part of me thinks that a picture along these lines — geometry emerging from quantum information, obeying a version of Einstein’s equation in the classical limit — pretty much has to be true, if you believe (1) regions of space have a finite number of degrees of freedom, and (2) the world is described by a wave function in Hilbert space.”