[LW meta: This is a very brief overview of what quantum mechanics says and how it relates to ordinary reality. Many readers here will probably already be familiar with this material, especially if they've read Eliezer's quantum physics sequence. Also note that there are diagrams in the linkpost that I didn't include because of problems with the editor]
2.5.1 The Formalism of Quantum Mechanics
2.5.1.1 Quantum mechanics describes the states of physical systems, their time evolution, and the outcome of measurements, using the following postulates:
2.5.1.1.1 Physical systems are represented as (norm-1) vectors in a Hilbert space . Hilbert spaces are complex vector spaces which are complete with respect to an inner product. They can be either finite or infinite dimensional.
2.5.1.1.2 Time evolution of states is generated by the differential equation . here is a self-adjoint operator(concretely, a Hermitian matrix for finite-dimensional systems). may vary in time. Equivalently, time evolution between two times is given by a unitary operator , so .
2.5.1.1.2.1 Said in more familiar language, time evolution corresponds to rotations of the complex vector space.
2.5.1.1.3 Measurements of a system are described by orthogonal direct-sum decompositions of the Hilbert space ; different measurement outcomes correspond to elements of the direct sum. The probability of obtaining a given outcome is equal to the amplitude squared of the projection of the state onto the corresponding subspace. Given a measurement outcome, the state of the system is then projected onto the corresponding subspace and normalized.
2.5.1.1.3.1 Said in more familiar language, measurements correspond to different bases of Hilbert space, together with a grouping of basis vectors according to whether they represent the same measurement outcome. When you take a measurement, the state "collapses" to one of the possible basis-vector-subspaces with probability corresponding to the total squared amplitude on that basis-vector-set when the state is written in terms of the given basis.
2.5.1.1.3.2 There is an alternative formulation in which measurements correspond to self-adjoint linear operators. The spectral theorem implies these are the same, as for any such operator(given certain technical conditions) there is an orthonormal basis of eigenvectors of the operator. We take subspaces corresponding to different eigenvectors as being elements of the direct sum, and their corresponding eigenvalue as the measurement value.
2.5.1.1.3.3 The above applies to discrete observables(those with a finite or countable set of possible values). What about continuous measurements like position? Arguably, any measurement device has a finite resolution so no true continuous observables are ever measured, only some finite coarse-graining thereof. There exists a mathematical formalism for continuous measurements, but it is more complex than the formalism for discrete measurements so I won't be covering it here.
2.5.1.1.4 The Hilbert space describing the joint state of 2 systems with state-spaces , is given by the tensor product .
2.5.1.2 We can illustrate the formalism in the simplest case, that of a quantum system with 2 discrete degrees of freedom. This represents the state space of a qubit(projected onto 2 real dimensions) with basis vectors , ( is bra-ket notation denoting vectors in Hilbert space). Thus, represents one possible measurement we could take of the system. Another, non-commuting measurement we could take is represented by the blue basis vectors . A possible unitary time evolution would be a rotation taking the red basis set to the blue basis set, i.e. , .
2.5.2 Relation to Mundane Reality
2.5.2.1 OK, but what does the above formalism have to do with the mundane reality we see around us, with tables and chairs and air and so on?
2.5.2.2 Firstly, an important example of a quantum system to keep in mind is the quantum particle. Here, the Hilbert space is the set of complex-valued functions on some subset of a space, say or .
2.5.2.2.1 (Real particles typically also have a discrete degree of freedom called spin, but let's ignore that for now.)
2.5.2.3 These complex-valued functions are in fact what we perceive as particles. Why do we perceive them as points with independent velocity? Typically the functions are quite tightly localized to a given part of space so it's impossible to perceive their wave-like nature(which in any case would cause them to collapse to a subspace).
2.5.2.3.1 As for velocity, the Hamiltonian used to evolve quantum particles in time has a component that causes complex waves(you could think of these as small spirals turning as you move in a certain direction) to turn and move in that direction; this causes small bumps(Gaussian wavepackets) of these sorts of waves to turn and move in a given direction(and also spread out a bit). This is the underlying source of the velocity of objects.
2.5.2.4 There's an additional subtlety when thinking about the state of many quantum particles, which is that(by 2.5.1.1.4) the Hilbert space associated with many of them consists of complex functions on their joint configuration space. What do I mean by this? If you have three non-identical particles in 3D space, their joint configuration space is 9-D -- the space of possible ways of arranging all 3 particles. Their joint Hilbert space consists of complex functions on this space.
2.5.2.4.1 One's naive expectation for the Hilbert space associated with many particles might be specifying complex functions for each particle independently. The set of complex functions on joint configuration space is much larger!
2.5.2.4.1.1 For example, one possible state of Hilbert space might place a high amplitude on particle A being in position 1 and particle B being in position 3, and also a high amplitude on particle A being in position 2 and particle B being in position 4, but not have a high amplitude on particle A being in position 1 and particle B being in position 4. This correlation between the states of distant particles is known as entanglement.
2.5.2.5 So in essence, the true state of the universe consists of something like a complex-valued probability distribution over all possible arrangements of matter. This raises the question: why don't we perceive all this stuff? Why does the universe look "normal"?
2.5.2.6 Basically the reason is that the state of the system associated to our universe is relatively tightly-peaked around a "normal-looking" state. i.e: particles are relatively localized(at least to the level of accuracy of macroscopic objects) and there isn't much long-range entanglement. The only very long-range entanglement in our world is likely due to human experiments.
2.5.2.6.1 On a microscopic level, particles can be more spread out. For instance, electrons in metals have fairly spatially-extended wavefunctions. Even particles in air have relatively-extended wavefunctions -- it's just hard to notice because typical interactions with air don't have a particle-size resolution.
2.5.2.6.2 Thus you can think of the states of some particles in such objects as being like thermal states, spread out relatively uniformly over the configuration space available to them(while being consistent with conservation laws etc.).
2.5.2.7 Okay, so that's a (very brief!) description of how mundane reality is consistent with the weird ontology of quantum mechanics. This however leaves a few questions: (a) Given that QM is mostly consistent with normal reality, what divergent predictions does it make? Why do we think it's true? (b) How is this state of relative normalcy maintained? Why don't we drift into weirder corners of Hilbert space? For instance, wavepackets spread out as they travel, so why don't they eventually become macroscopically large and noticeable? (c) What's going on when we "measure" a system and the wavefunction "collapses"? First let's consider question (a).[1]
2.5.3 Consequences: Quantized Measurements, Bell Inequalities, Contextuality
2.5.3.1 So what are the actual consequences of QM that differ from a classical world-model?
2.5.3.2 Quantized Measurements. This is what brought people's attention to the possibility of QM in the first place. The possible values of many observables, as explained in the last section, are quantized, that is they can only take on a range of discrete values, because their corresponding operators have a discrete spectrum(and furthermore, not quantized in such a way that they become approximately continuous upon taking more sensitive measurements).
2.5.3.2.1 Example: Stern-Gerlach experiment. Spin is a discrete degree of freedom associated with all quantum particles. It is called spin because it has analogous properties to the spin(rotational angular momentum) of an extended object. Unlike that spin however, quantum spin is a discrete quantity -- hence measuring it(in whatever chosen basis) gives one of a finite list of outcomes.
2.5.3.3 Bell Inequalities. Quantized measurements give us some reason to believe the ontology of discrete collapse to a set of eigenvectors. What about the more extravagant parts of QM, the vast enlargement of state space and hence possibility of "entanglement"? Here too we can run experiments. In particular, we can deliberately entangle particles separated at a large distance(by kilometers, say) and perform experiments measuring both "at the same time"(more precisely, at a space-like separation), producing correlations which would be impossible if the particles had independent states.
2.5.3.3.1 Ah, but how do we know that there isn't some random hidden state produced at the moment we entangle the particles, capable of producing non-independent correlations without the need for the elaborate quantum state space? As an example, we might imagine that we randomly flip a coin and then put letters either labeled "A" or "B" in two envelopes, then separate them. The state of the two envelopes would then be "entangled", in that they are not independent in the overall probability distribution over worlds, but nothing "quantum" needs to happen.
2.5.3.3.2 The answer comes from our ability to perform non-commuting measurements on a system. In particular, we imagine entangling the particles, then measuring in one of two bases at location 1, and another set of two bases at location 2, producing a total of 4 possible joint measurements. For some choices of measurements, the overall correlation among the different measurement results produced in this manner cannot be produced by any classical hidden variables theory. The inequality that any classical hidden variables theory must obey, and which quantum systems violate, is called the Bell Inequality.
2.5.3.3.3 These experiments have in fact been carried out in this way and produced the desired inequality violation. This is probably the strongest single piece of evidence that we live in an intrinsically quantum world.
2.5.3.4 Quantum computing. It has been proposed to use the vastly larger state-space of quantum mechanics to run computations that would be impossible in the classical world. Note that QM does not allow you to try running exponentially many versions of an algorithm in parallel and picking the best one -- instead it's better to think of quantum algorithms as exploiting interference between a vast number of computations at once.
2.5.3.4.1 The most famous quantum algorithm is Shor's algorithm which allows factoring large numbers, giving an exponential speedup over the best-known classical algorithms, breaking many crypto schemes. Another well-known quantum algorithm is Grover Search, which gives a quadratic speedup on a more generic class of problems.
2.5.3.4.2 No quantum computer capable of running Shor's or Grover's algorithm on a non-trivial instance size has been constructed(the largest number factored so far is 21). If such an algorithm could be run, it would be a powerful demonstration that we live in a quantum world. Efforts are ongoing to construct a quantum machine that does a computational task believed to be infeasible for classical machines(this is known as "quantum supremacy"/"quantum advantage").
2.5.3.5 Contextuality. This is a somewhat more subtle property, a generalization of the Bell inequalities. Given a set of measurements and a state, contextuality quantifies the extent to which we can think of the state as having a definite(perhaps probabilistic) value on each measurement, regardless of whether it is measured or not. If this is not the case, then we cannot think of the system as existing totally independently of the measurements chosen to probe it -- in other words, it partially depends on its context, hence the term.
2.5.3.6 Miscellaneous. QM is used to predict and explain many properties of materials on the microscale. E.g. the periodic table is explained by the quantum-mechanical theory of the atom. Covalent bonds can be modeled as eigenstates of the 2-electron state space under the potential function of a pair of nuclei. Quantum mechanics explains the stability of matter. It is this ubiquitous usefulness which is perhaps the strongest argument in favor of quantum mechanics; almost everything in chemistry and physics relies on it, and it gives a remarkably powerful and self-consistent picture of the world.
2.5.3.7 So quantum mechanics is well-attested by experiment. However, although the properties above are interesting, they don't seem to have any obvious relevance for the questions this text is interested in, namely the existence of life and sentience(although perhaps a more determined investigator could draw out some relations). For this, let's instead look towards the processes by which apparent classicality is maintained.
Another foundational problem which I will not address is the origin of the Born rule: why do measurement events/branches have probabilities given by their total amplitude squared? My perspective is that we can view this rule as part of the laws, needing no more special explanation than the time evolution rules. ↩︎
[LW meta: This is a very brief overview of what quantum mechanics says and how it relates to ordinary reality. Many readers here will probably already be familiar with this material, especially if they've read Eliezer's quantum physics sequence. Also note that there are diagrams in the linkpost that I didn't include because of problems with the editor]
2.5.1 The Formalism of Quantum Mechanics
2.5.1.1 Quantum mechanics describes the states of physical systems, their time evolution, and the outcome of measurements, using the following postulates:
2.5.1.1.1 Physical systems are represented as (norm-1) vectors
2.5.1.1.2 Time evolution of states is generated by the differential equation
2.5.1.1.3 Measurements of a system are described by orthogonal direct-sum decompositions of the Hilbert space
2.5.1.1.3.1 Said in more familiar language, measurements correspond to different bases of Hilbert space, together with a grouping of basis vectors according to whether they represent the same measurement outcome. When you take a measurement, the state "collapses" to one of the possible basis-vector-subspaces with probability corresponding to the total squared amplitude on that basis-vector-set when the state is written in terms of the given basis.
2.5.1.1.3.2 There is an alternative formulation in which measurements correspond to self-adjoint linear operators. The spectral theorem implies these are the same, as for any such operator(given certain technical conditions) there is an orthonormal basis of eigenvectors of the operator. We take subspaces corresponding to different eigenvectors as being elements of the direct sum, and their corresponding eigenvalue as the measurement value.
2.5.1.1.3.3 The above applies to discrete observables(those with a finite or countable set of possible values). What about continuous measurements like position? Arguably, any measurement device has a finite resolution so no true continuous observables are ever measured, only some finite coarse-graining thereof. There exists a mathematical formalism for continuous measurements, but it is more complex than the formalism for discrete measurements so I won't be covering it here.
2.5.1.1.4 The Hilbert space describing the joint state of 2 systems with state-spaces
2.5.1.2 We can illustrate the formalism in the simplest case, that of a quantum system with 2 discrete degrees of freedom. This represents the state space of a qubit(projected onto 2 real dimensions) with basis vectors
2.5.2 Relation to Mundane Reality
2.5.2.1 OK, but what does the above formalism have to do with the mundane reality we see around us, with tables and chairs and air and so on?
2.5.2.2 Firstly, an important example of a quantum system to keep in mind is the quantum particle. Here, the Hilbert space is the set of complex-valued functions on some subset of a space, say
2.5.2.2.1 (Real particles typically also have a discrete degree of freedom called spin, but let's ignore that for now.)
2.5.2.3 These complex-valued functions are in fact what we perceive as particles. Why do we perceive them as points with independent velocity? Typically the functions are quite tightly localized to a given part of space so it's impossible to perceive their wave-like nature(which in any case would cause them to collapse to a subspace).
2.5.2.4 There's an additional subtlety when thinking about the state of many quantum particles, which is that(by 2.5.1.1.4) the Hilbert space associated with many of them consists of complex functions on their joint configuration space. What do I mean by this? If you have three non-identical particles in 3D space, their joint configuration space is 9-D -- the space of possible ways of arranging all 3 particles. Their joint Hilbert space consists of complex functions on this space.
2.5.2.4.1 One's naive expectation for the Hilbert space associated with many particles might be specifying complex functions for each particle independently. The set of complex functions on joint configuration space is much larger!
2.5.2.5 So in essence, the true state of the universe consists of something like a complex-valued probability distribution over all possible arrangements of matter. This raises the question: why don't we perceive all this stuff? Why does the universe look "normal"?
2.5.2.6 Basically the reason is that the state of the system associated to our universe is relatively tightly-peaked around a "normal-looking" state. i.e: particles are relatively localized(at least to the level of accuracy of macroscopic objects) and there isn't much long-range entanglement. The only very long-range entanglement in our world is likely due to human experiments.
2.5.2.6.1 On a microscopic level, particles can be more spread out. For instance, electrons in metals have fairly spatially-extended wavefunctions. Even particles in air have relatively-extended wavefunctions -- it's just hard to notice because typical interactions with air don't have a particle-size resolution.
2.5.2.6.2 Thus you can think of the states of some particles in such objects as being like thermal states, spread out relatively uniformly over the configuration space available to them(while being consistent with conservation laws etc.).
2.5.2.7 Okay, so that's a (very brief!) description of how mundane reality is consistent with the weird ontology of quantum mechanics. This however leaves a few questions: (a) Given that QM is mostly consistent with normal reality, what divergent predictions does it make? Why do we think it's true? (b) How is this state of relative normalcy maintained? Why don't we drift into weirder corners of Hilbert space? For instance, wavepackets spread out as they travel, so why don't they eventually become macroscopically large and noticeable? (c) What's going on when we "measure" a system and the wavefunction "collapses"? First let's consider question (a). [1]
2.5.3 Consequences: Quantized Measurements, Bell Inequalities, Contextuality
2.5.3.1 So what are the actual consequences of QM that differ from a classical world-model?
2.5.3.2 Quantized Measurements. This is what brought people's attention to the possibility of QM in the first place. The possible values of many observables, as explained in the last section, are quantized, that is they can only take on a range of discrete values, because their corresponding operators have a discrete spectrum(and furthermore, not quantized in such a way that they become approximately continuous upon taking more sensitive measurements).
2.5.3.3 Bell Inequalities. Quantized measurements give us some reason to believe the ontology of discrete collapse to a set of eigenvectors. What about the more extravagant parts of QM, the vast enlargement of state space and hence possibility of "entanglement"? Here too we can run experiments. In particular, we can deliberately entangle particles separated at a large distance(by kilometers, say) and perform experiments measuring both "at the same time"(more precisely, at a space-like separation), producing correlations which would be impossible if the particles had independent states.
2.5.3.3.1 Ah, but how do we know that there isn't some random hidden state produced at the moment we entangle the particles, capable of producing non-independent correlations without the need for the elaborate quantum state space? As an example, we might imagine that we randomly flip a coin and then put letters either labeled "A" or "B" in two envelopes, then separate them. The state of the two envelopes would then be "entangled", in that they are not independent in the overall probability distribution over worlds, but nothing "quantum" needs to happen.
2.5.3.3.2 The answer comes from our ability to perform non-commuting measurements on a system. In particular, we imagine entangling the particles, then measuring in one of two bases at location 1, and another set of two bases at location 2, producing a total of 4 possible joint measurements. For some choices of measurements, the overall correlation among the different measurement results produced in this manner cannot be produced by any classical hidden variables theory. The inequality that any classical hidden variables theory must obey, and which quantum systems violate, is called the Bell Inequality.
2.5.3.3.3 These experiments have in fact been carried out in this way and produced the desired inequality violation. This is probably the strongest single piece of evidence that we live in an intrinsically quantum world.
2.5.3.4 Quantum computing. It has been proposed to use the vastly larger state-space of quantum mechanics to run computations that would be impossible in the classical world. Note that QM does not allow you to try running exponentially many versions of an algorithm in parallel and picking the best one -- instead it's better to think of quantum algorithms as exploiting interference between a vast number of computations at once.
2.5.3.4.1 The most famous quantum algorithm is Shor's algorithm which allows factoring large numbers, giving an exponential speedup over the best-known classical algorithms, breaking many crypto schemes. Another well-known quantum algorithm is Grover Search, which gives a quadratic speedup on a more generic class of problems.
2.5.3.4.2 No quantum computer capable of running Shor's or Grover's algorithm on a non-trivial instance size has been constructed(the largest number factored so far is 21). If such an algorithm could be run, it would be a powerful demonstration that we live in a quantum world. Efforts are ongoing to construct a quantum machine that does a computational task believed to be infeasible for classical machines(this is known as "quantum supremacy"/"quantum advantage").
2.5.3.5 Contextuality. This is a somewhat more subtle property, a generalization of the Bell inequalities. Given a set of measurements and a state, contextuality quantifies the extent to which we can think of the state as having a definite(perhaps probabilistic) value on each measurement, regardless of whether it is measured or not. If this is not the case, then we cannot think of the system as existing totally independently of the measurements chosen to probe it -- in other words, it partially depends on its context, hence the term.
2.5.3.6 Miscellaneous. QM is used to predict and explain many properties of materials on the microscale. E.g. the periodic table is explained by the quantum-mechanical theory of the atom. Covalent bonds can be modeled as eigenstates of the 2-electron state space under the potential function of a pair of nuclei. Quantum mechanics explains the stability of matter. It is this ubiquitous usefulness which is perhaps the strongest argument in favor of quantum mechanics; almost everything in chemistry and physics relies on it, and it gives a remarkably powerful and self-consistent picture of the world.
2.5.3.7 So quantum mechanics is well-attested by experiment. However, although the properties above are interesting, they don't seem to have any obvious relevance for the questions this text is interested in, namely the existence of life and sentience(although perhaps a more determined investigator could draw out some relations). For this, let's instead look towards the processes by which apparent classicality is maintained.
Another foundational problem which I will not address is the origin of the Born rule: why do measurement events/branches have probabilities given by their total amplitude squared? My perspective is that we can view this rule as part of the laws, needing no more special explanation than the time evolution rules. ↩︎