Foundations of quantum mechanics

The foundations of quantum mechanics study the physical and conceptual status of quantum states, measurement, nonlocal correlations, and possible extensions of quantum theory.

The EPR argument uses correlations of an entangled state and a locality assumption to challenge the completeness of the quantum description. Its Einstein–Podolsky–Rosen criterion of reality identifies quantities predictable with certainty without disturbing a distant system as elements of physical reality.

The Einstein–Podolsky–Rosen criterion says that if one can predict a physical quantity with certainty without disturbing the system, then there is an element of physical reality corresponding to that quantity.

A superselection rule partitions a representation into sectors between which the specified observable algebra has no matrix elements. Relative phases between vectors in different sectors cannot be detected by those observables, so a coherent vector spanning sectors and its sector-diagonal mixture define the same observable state. The restriction is relative to the observable algebra; it is not by itself a physical wave-function collapse. Infinite-system phases can yield disjoint representations and emergent superselection sectors even though each finite system permits global connecting operators.

The certainty of a prediction made without disturbing the system is proposed as sufficient evidence that the predicted quantity corresponds to an element of physical reality. It motivates the discussion of completeness and locality in foundations of quantum mechanics, but is distinct from the conditional factorization assumed by local hidden-variable theory.

A deterministic interpretation with a definite particle configuration guided by the wavefunction. The latter obeys the Time-dependent Schrödinger equation, while the particle follows the guidance equation. A Born rule distribution of initial positions is preserved by quantum equilibrium equivariance. The guidance law restricts admissible initial velocities, even when trajectories are rewritten as a second-order force equation.

For the separation of a two-dimensional central-potential eigenstate, the guidance equation gives zero radial velocity and constant-radius circular motion, with circulation $2\pi \hslash k/m$. A real superposition of the degenerate $k$ and $-k$ eigenstates has zero current away from nodes instead. The speed depends on the actual wavefunction, not just its energy.

The amplitude-dependent extra potential in the Madelung equations. Taking the gradient of their phase equation and using the guidance equation yields $mD\mathbf{v}/Dt=-\nabla (V+Q)$, where $D/Dt$ is the material derivative. At wavefunction nodes this local expression may be singular.

For a spinless particle without a magnetic vector potential, Bohmian mechanics sets $\dot{\mathbf{X}}=\hslash \nabla S/m$, where $S$ is the dimensionless quantum phase. Equivalently the velocity is the probability current divided by the probability density. This formula is local to nonzero wavefunction regions.

Both the ensemble position density transported by the guidance equation and the squared wave amplitude satisfy the same probability continuity equation. Subject to existence and uniqueness of that transport, an initial Born rule distribution stays a Born rule distribution. This is a preservation statement, not a derivation that every initial ensemble is already in equilibrium.

An ontological model associates each quantum state preparation with a probability distribution over underlying physical states $\lambda$. A measurement in quantum mechanics's response probabilities depend on its specification and $\lambda$, sum to one, and reproduce the Born rule when averaged over the preparation distribution. The model need not assign deterministic outcomes. Overlap or disjointness of preparation measures defines the psi-epistemic model versus psi-ontic model distinction.

Under preparation independence, an ontological model of a quantum system reproducing exact quantum quantum measurement probabilities cannot assign overlapping preparation measures to distinct pure states. An exclusion quantum measurement on independent copies makes every outcome impossible for one possible preparation; a common ontic region would require every response to vanish there. Thus the conclusion is a psi-ontic model, not a blanket exclusion of hidden variables or all informational interpretations.

If two pure quantum states overlap by $(1/\sqrt{2}{)}^{1/n}$, their $n$-copy states have overlap $1/\sqrt{2}$. Each block spans an effective qubit, with orthonormal basis $e_0={\Psi }_1$ and $e_1=\sqrt{2}{\Psi }_2-{\Psi }_1$. Applying the two-block PBR exclusion measurement for zero and plus uses $2n$ independently prepared copies. A common single-copy ontic overlap of positive mass remains positive under this finite tensor power.

The product preparations $00,0+,+0,++$ are excluded respectively by the orthonormal vectors $(01+10)/\sqrt{2}$, $(00-01+10+11)/2$, $(00+01-10+11)/2$ and $(00-11)/\sqrt{2}$, written in the computational basis. Each vector has zero inner product with its labelled preparation. This projective measurement gives antidistinguishable quantum states and drives the two-copy PBR theorem contradiction.

Independently operated quantum state preparations are assumed to produce independent underlying physical states. Their joint preparation measure factors into the single-system measures. This ontic factorization is an additional assumption, not a consequence of operational independence alone. It turns a common single-system overlap of mass $\epsilon$ into a common $n$-system overlap of mass ${\epsilon }^n$ in the PBR theorem.

An ontological model of a quantum system is psi-epistemic if some pair of distinct pure states has preparation measures with nonzero overlap. The same physical state can then be compatible with either preparation. This precise measure-overlap definition is stronger than merely interpreting a wave function as information or a computational tool.

An ontological model of a quantum system is psi-ontic if distinct pure states have mutually singular preparation distributions. The physical state then determines which pure quantum state was prepared, almost surely in the pairwise sense. Extra underlying variables may still exist. The PBR theorem establishes this conclusion under preparation independence and exact quantum predictions.

Bell's theorem shows that no theory assigning outcomes through local hidden variables can reproduce every quantum correlation.

A Bell inequality is a constraint on observable correlations satisfied by the specified class of local hidden-variable models. In finite-setting scenarios such constraints bound linear combinations of probabilities or expectation values. The CHSH inequality is one example, following from Bell factorization, measurement independence and bounded outcomes. A violation rules out the conjunction of the assumptions defining that model class; it does not alone identify a unique false premise.

For outcomes in $\{0,\dots ,d-1\}$, let $[x{]}_d$ be their nonnegative modular representative. A deterministic local hidden-variable model satisfiesThe unreduced chain telescopes to $-1$, so its reduced nonnegative sum is at least $d-1$ for each assignment. For binary outcomes and two settings, it is a form of the CHSH inequality: $I_{2,2}=2-(E_{11}+E_{21}+E_{22}-E_{12})/2$.

On $|{\Phi }^{+}⟩$, measure ${\sigma }_z\cos \theta +{\sigma }_x\sin \theta$, assigning bit $a$ to eigenvalue $(-1{)}^a$. Choose Alice's angles $2(i-1)\delta$ and Bob's angles $(2i-1)\delta$, with $\delta =\pi /(2N)$. The Bell state correlation is $\cos (\theta -\varphi )$. Each adjacent disagreement term and the last agreement term therefore has probability ${\sin }^2(\pi /(4N))$. For $N\ge 2$, their total violates the chained modular Bell inequality bound one, and it tends to zero as $N\to \infty$. At $N=1$ the value is exactly one.

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A local hidden-variable theory assigns each party's outcome from a shared hidden state and that party's own setting, independently of the spacelike-separated setting chosen by the other party.

If a successful preparation flag is fixed before freely selected Bell inequality settings, conditioning a local hidden-variable theory on that flag only replaces its hidden-variable distribution by the conditional distribution. The later response functions remain local and the distribution is independent of the later settings, so the CHSH inequality still holds. Setting-dependent rejection during the Bell measurements does not satisfy this argument.

Outcome independence requires $P(A,B\mid a,b,\lambda )=P(A\mid a,b,\lambda )P(B\mid a,b,\lambda )$. It says that, once settings and the proposed complete state are fixed, learning the distant outcome does not change the local outcome probability. Combining it with parameter independence gives Bell factorization. Entangled quantum states generally fail outcome independence even though their observable marginal distributions obey quantum no-signalling.

For spacelike-separated measurement regions, a sufficient specification $\lambda$ of their relevant past should screen each outcome from additional information in the other wing. In a two-setting experiment a Bell-local model has $P(A,B\mid a,b,\lambda )=P(A\mid a,\lambda )P(B\mid b,\lambda )$. Together with measurement independence and bounded outcomes, this implies the CHSH inequality. Bell local causality is stronger than quantum no-signalling and is not an assumption that every outcome was predetermined.

A setting-independent hidden variable fixes responses for every measurement choice, and each response depends only on the local setting. Stochastic local hidden-variable theories can be represented as mixtures of these assignments by including local random seeds in the hidden variable. Deterministic counterfactual assignments are the starting point of the chained modular Bell inequality.

Outcome determinism says that a complete hidden state and the measurement settings fix each measurement outcome with probability one.

Parameter independence says that, conditional on the complete hidden state, the probability distribution of one party's outcome is independent of the other party's measurement setting.

Measurement independence says that the distribution of the hidden state is independent of the measurement settings: $\rho (\lambda |a,b)=\rho (\lambda )$. Its failure is the measurement-dependence loophole in a Bell test.

For local outcomes $A_0,A_1,B_0,B_1\in \{-1,1\}$, the Clauser–Horne–Shimony–Holt inequality isAn arbitrary no-signalling correlation can reach the algebraic maximum $4$, while quantum correlations are bounded by $2\sqrt{2}$.

For spin-projective measurements on a two-qubit state, let ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$ be the eigenvalues of $T^{T}T$, where $T$ is its Pauli correlation tensor. Bob's sum and difference of unit axes are orthogonal and have lengths $2\cos \varphi$ and $2\sin \varphi$. Maximizing Alice's two axes aligns them with the corresponding images under $T$. Maximizing $\varphi$ then gives $2\sqrt{\Vert T\mathbf{u}{\Vert }^2+\Vert T\mathbf{v}{\Vert }^2}$ for orthonormal $\mathbf{u},\mathbf{v}$. The maximum of this quadratic-form sum is ${\lambda }_1+{\lambda }_2$, by diagonalizing $T^{T}T$ and selecting its top two eigenvectors. Thus the displayed expression is both an upper bound and attainable. The same maximum applies to the form with the sum of two absolute values, since local outcome signs can select their positive branches.

Every entangled pure state of two qubits violates a CHSH inequality for a suitable choice of local measurement axes. If its two Schmidt coefficients are $c_0,c_1>0$, the optimal construction already gives a value $2\sqrt{1+(2c_0{c}_1{)}^2}>2$.

For the Schmidt-basis Pauli correlation tensor, choose Alice's axes ${\mathbf{e}}_z,{\mathbf{e}}_x$ and Bob's $({\mathbf{e}}_z\pm s{\mathbf{e}}_x)/\sqrt{1+s^2}$. The CHSH inequality expression is $2\sqrt{1+s^2}$. It exceeds two exactly when both Schmidt coefficients are nonzero, reaching $2\sqrt{2}$ for maximal quantum entanglement. A product state gives $s=0$ and saturates rather than violates the inequality.

A no-signalling box is a conditional probability distribution $P(x,y|a,b)$ whose marginal distribution for either party is independent of the other party's input. It can therefore exhibit correlations stronger than quantum correlations without transmitting information superluminally.

A Popescu–Rohrlich box is a bipartite no-signalling box attaining the algebraic maximum of the CHSH inequality.

Take a separated Boolean decomposition $f(x,y)={⨁}_{\ell }{u}_{\ell }(x)v_{\ell }(y)$. Feed $(u_{\ell },v_{\ell })$ to one independent PR box, obtaining outputs with $a_{\ell }\oplus b_{\ell }=u_{\ell }{v}_{\ell }$. Alice forms $A={⨁}_{\ell }{a}_{\ell }$, Bob forms $B={⨁}_{\ell }{b}_{\ell }$, and their parities satisfy $A\oplus B=f$. Sending the single bit $A$ lets Bob compute the answer with certainty. The truth-table decomposition needs at most $2^{\min (m,n)}$ boxes for input lengths $m,n$. Each box's local output is uniform, so Bob's uncommunicated output string carries no information about Alice's input; the final parity bit is essential for a general function depending on that input.

An objective-collapse theory treats wave-function collapse as a physical process localized in spacetime, rather than only as an observer's update of information.

The Ghirardi--Rimini--Weber theory modifies unitary time evolution by random Gaussian position localizations. Each constituent collapses at a small fixed rate, while the rate for an entangled macroscopic body grows with its number of constituents.

For localization length $r_C$, the GRW localization operator acting on particle $i$ isA collapse centred at $\mathbf{X}$ sends $|\psi ⟩$ to the normalized state $L_i(\mathbf{X})|\psi ⟩$ with probability density $\Vert L_i(\mathbf{X})|\psi ⟩{\Vert }^2$.

For equal particle rates in the GRW model, the next centre falls in $B$ with probability $N^{-1}{\sum }_i{\int }_B\Vert L_i(c)\Psi {\Vert }^{2}dc$. With zero Hamiltonian operator, two successive centres in $B$ have joint probability $N^{-2}{\sum }_{ij}{\int }_{B}dc{\int }_{B}dd\Vert L_j(d)L_i(c)\Psi {\Vert }^2$. Dividing by the first-event probability gives the conditional second event, including repeated particle labels. Spatial probabilities depend on actual packet mass, not solely the nominal packet centres.

For $N$ entangled constituents, independent localization rates add to approximately $N\lambda$. One constituent's collapse localizes the correlated macroscopic configuration, rapidly suppressing macroscopically separated branches.

A first localization near one macroscopically separated branch suppresses the other branch through the Gaussian GRW localization operator. This selects a correlated pointer alternative. However, if a region contains only packet centres and excludes substantial packet mass, the next centre need not lie in that region's small enlargement. Probabilities $|\alpha {|}^2$ for the first event and approximately one for the next require regions containing essentially all corresponding branch mass. The general GRW collapse-centre event probabilities retain the geometry exactly.

GRW position localization increases the ensemble mean momentum variance. In three dimensions one collapse adds $3{\hslash }^2/(2r_C^2)$ to $⟨{\mathbf{p}}^2⟩$, producing mean kinetic-energy increase $3{\hslash }^2/(4m{r}_C^2)$.

At a spacetime point $x$, the local quantum state is the reduced density operator obtained on spacelike hypersurfaces approaching the past light cone of $x$. It includes the effects of objective collapses in that past light cone and excludes spacelike-separated collapses.

A quantum state readout device at $x$ hypothetically outputs a classical description of the local quantum state under objective collapse without changing that state. Its output can respond to a remote collapse only after the collapse enters the past light cone of $x$, so the device preserves relativistic causality.

A proper mixed state represents classical uncertainty over states selected by an actual preparation or collapse. Once that selecting event lies in the readout point's past light cone, a quantum state readout device returns the selected component rather than merely the ensemble average.

An improper mixed state is the reduced density matrix of a subsystem entangled with unobserved degrees of freedom. A quantum state readout device returns that reduced state until a relevant collapse lies in its past light cone.