Probability in Quantum Mechanics

Quantum probability connects a preparation and a measurement to the distribution of its recorded outcomes. The same numerical prediction can be understood as objective chance, rational credence, an equilibrium distribution, or a weight of branches. Agreement about the calculation therefore need not settle what the probability represents.

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

Probabilities need an experiment

For a density operator $\rho$ and a POVM $\{E_a\}$, the formal Born rule gives

$$p(a\mid\rho,M)=\operatorname{Tr}(\rho E_a).$$

The measurement label $M$ matters. The probability of an outcome is not generally a property of $\rho$ alone. Positivity and $\sum_aE_a=I$ make this a normalized distribution for each measurement. Different measurement choices need not be marginals of a single distribution of pre-existing values satisfying all quantum functional relations; contextuality explains the additional assumptions such a representation would require.

Consider $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$. A $Z$ measurement has probabilities $1/2,1/2$, while an $X$ measurement gives $+1$ with certainty. Randomness is specified relative to a question, not assigned to the state without qualification.

Frequencies estimate probabilities

Suppose $N$ independently prepared systems are measured with the same binary POVM, with success probability $p$. If the model includes independence, the count $K$ is binomial:

$$\Pr(K=k)=\binom Nk p^k(1-p)^{N-k}, \qquad \operatorname{Var}(K/N)=\frac{p(1-p)}N.$$

Thus a typical frequency approaches $p$ under this statistical model. Independence, stable preparation, and a sampling protocol are additional assumptions. An experiment with drifting preparation may not have one fixed $p$.

For $p=1/2$ and $N=100$, the standard deviation of the frequency is $0.05$. Obtaining 57 successes is compatible with the model; it does not replace the theoretical probability by $0.57$. Conversely, a single unlikely record does not logically refute a probabilistic theory. Inference concerns a specified family of data and an error criterion.

Three separate explanatory demands

A formal account specifies the assignment $\rho,M\mapsto p$. A statistical account connects that assignment to finite records and uncertainty. An interpretive account says why a probability is appropriate in the underlying physical picture.

Collapse theories supply stochastic dynamics. Pilot-wave theory is deterministic but uses a quantum-equilibrium measure. Everettian accounts must explain the role of weights when several recorded alternatives persist. Quantum Bayesian accounts interpret the probabilities as an agent’s commitments constrained by the theory.

These are different explanatory tasks even where predictions agree. Mathematical uncertainty relations alone do not establish one unique interpretation of chance.

A useful diagnostic

Whenever a derivation promises to explain probability, ask what was assumed to be probabilistic at its start. A theorem about additive measures may identify the only permitted measure while leaving its interpretation open. A law of large numbers already presupposes a measure on sequences. A decision theorem constrains preferences under rationality assumptions. None of these limitations makes the result worthless; each identifies precisely what it explains.

Exercises

For independent binary trials with $p=1/4$, calculate the expected count and its standard deviation for $N=400$. Does determinism of an underlying model by itself contradict these statistics?

Solution

The mean is $Np=100$, and the standard deviation is $\sqrt{Np(1-p)}=\sqrt{75}\approx8.66$. Deterministic trajectories together with an appropriate distribution of initial conditions can produce exactly this sampling law. Determinism and a probability measure are logically compatible; explaining and justifying the measure remains necessary.

References

• D. Dürr, S. Goldstein, and N. Zanghì, “Quantum equilibrium and the origin of absolute uncertainty,” Journal of Statistical Physics 67, 843–907 (1992), author manuscript.
• C. A. Fuchs, N. D. Mermin, and R. Schack, “An introduction to QBism with an application to the locality of quantum mechanics,” American Journal of Physics 82, 749–754 (2014), author manuscript.