Heisenberg's uncertainty principle is usually taken to express a limitation of operational possibilities imposed by quantum mechanics.Here we demonstrate that the full content of this principle also includes its positive role as a condition ensuring that mutually exclusive experimental options can be reconciled if an appropriate trade-off is accepted.The uncertainty principle is shown to appear in three manifestations, in the form of uncertainty relations: for the widths of the position and momentum distributions in any quantum state; for the inaccuracies of any joint measurement of these quantities; and for the inaccuracy of a measurement of one of the quantities and the ensuing disturbance in the distribution of the other quantity.Whilst conceptually distinct, these three kinds of uncertainty relations are shown to be closely related formally.Finally, we survey models and experimental implementations of joint measurements of position and momentum and comment briefly on the status of experimental tests of the uncertainty principle.
In recent years, novel quantifications of measurement error in quantum mechanics have for the first time enabled precise formulations of Heisenberg's famous but often challenged measurement uncertainty relation. These relations take the form of a trade-off for the necessary errors in joint approximate measurements of position and momentum and other incompatible pairs of observables. Here we review some of these error measures, examine their properties and suitability, and compare their relative strengths as criteria for "good" approximations.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
The uncertainty principle is commonly regarded as a fundamental limitation ofquantum mechanics, often attributed directly to Heisenberg’s 1927 analysis.In this work, we revisit the original formulation of the principle and carefullydistinguish Heisenberg’s measurement–disturbance argument from the latermathematical uncertainty relations derived by Kennard, Robertson, andSchrödinger. We show that the standard variance-based formulation relies onthe prior introduction of fluctuation operators and ensemble averages, therebypresupposing statistical indeterminacy rather than deriving it from firstprinciples. From a physical perspective, Heisenberg’s argument is shown toreflect constraints imposed by invasive measurement techniques rather than auniversal prohibition on simultaneous knowledge of conjugate variables.Motivated by this distinction, we propose a conjecture that, in the absence ofmeasurement-induced energy transfer, the simultaneous determination ofposition and momentum may, in principle, be achievable. Our analysis suggeststhat quantum uncertainty, as traditionally formulated, encodes limitations ofcurrent observational methods rather than an intrinsic property of nature. \section{Heisenberg’s original observation and formulation (1927)} In his seminal 1927 paper \emph{Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik} \cite{Heisenberg1927}, Werner Heisenberg sought to clarify the physical meaning of quantum mechanics rather than to establish a rigorous mathematical theorem. His analysis was based on thought experiments concerning the act of measurement, most notably the
γ-ray microscope. Heisenberg argued that any attempt to localize a particle within a spatial region
Δx requires radiation of wavelength comparable to
Δx, which inevitably transfers an uncontrollable momentum
Δp to the particle. As a consequence of this measurement-induced disturbance, he proposed that the product of the inaccuracies in position and momentum measurements satisfies a relation of the order\begin{equation}\Delta x\, \Delta p \sim h .\end{equation}Here,
Δx and
Δp denote operational measurement imprecisions arising from the interaction between the measuring apparatus and the system, rather than statistical variances of quantum observables. Heisenberg did not introduce operators, Hilbert space, or probability distributions, nor did he provide a formal inequality with a precise lower bound. The argument was explicitly heuristic and qualitative, emphasizing limitations imposed by the measurement process itself. Consequently, the original uncertainty principle was formulated as a statement about measurement disturbance and experimental resolution, not as a universal mathematical constraint on quantum states. \section{Standard mathematical proof of the position--momentum uncertainty relation} \subsection{Setup and definitions} Let
H be a Hilbert space and let
∣ψ⟩∈H be anormalized state,
⟨ψ∣ψ⟩=1. Let
A and
B beself-adjoint operators (observables) on a common dense domain containing
∣ψ⟩, such that all expressions below are well-defined. Define expectation values\cite{Kennard1927,Robertson1929,Schrodinger1930,Ballentine1998,Sakurai1994}\begin{equation}\langle A \rangle := \langle \psi | A | \psi \rangle,\qquad\langle B \rangle := \langle \psi | B | \psi \rangle .\end{equation}Define the fluctuation (centered) operators\begin{equation}\Delta A := A - \langle A \rangle, \qquad \Delta B := B - \langle B \rangle .\end{equation}The variances are\begin{equation}\sigma_A^2 := \langle (\Delta A)^2 \rangle = \langle \psi | (\Delta A)^2 | \psi \rangle,\qquad\sigma_B^2 := \langle (\Delta B)^2 \rangle = \langle \psi | (\Delta B)^2 | \psi \rangle.\end{equation} \subsection{Key inequality: Cauchy--Schwarz in Hilbert space} Define vectors\begin{equation}|\phi\rangle := \Delta A |\psi\rangle, \qquad |\chi\rangle := \Delta B |\psi\rangle .\end{equation}By the Cauchy--Schwarz inequality,\begin{equation}\langle \phi|\phi\rangle \, \langle \chi|\chi\rangle \ge |\langle \phi|\chi\rangle|^2 .\end{equation}Compute each factor:\begin{equation}\langle \phi|\phi\rangle= \langle \psi| (\Delta A)^\dagger (\Delta A) |\psi\rangle= \langle \psi| (\Delta A)^2 |\psi\rangle= \sigma_A^2,\end{equation}and similarly
⟨χ∣χ⟩=σB2. Therefore\begin{equation}\sigma_A^2 \sigma_B^2 \ge \left| \langle \psi| \Delta A \Delta B |\psi\rangle \right|^2= |\langle \Delta A \Delta B \rangle|^2 .\label{eq:cs_step}\end{equation} \subsection{Robertson--Schr\"odinger decomposition} Decompose
ΔAΔB into commutator and anticommutator parts:\begin{equation}\Delta A \Delta B=\frac{1}{2}\bigl[\Delta A,\Delta B\bigr]+\frac{1}{2}\bigl\{\Delta A,\Delta B\bigr\},\end{equation}where
[X,Y]=XY−YX and
{X,Y}=XY+YX. Taking expectation values gives\begin{equation}\langle \Delta A \Delta B \rangle=\frac{1}{2}\langle [\Delta A,\Delta B]\rangle+\frac{1}{2}\langle \{\Delta A,\Delta B\}\rangle .\label{eq:decomp_exp}\end{equation} Now note the following facts for self-adjoint
A,B:\begin{itemize}\item The operator
[ΔA,ΔB] is anti-self-adjoint, hence
⟨[ΔA,ΔB]⟩ is purely imaginary.\item The operator
{ΔA,ΔB} is self-adjoint, hence
⟨{ΔA,ΔB}⟩ is real.\end{itemize}Therefore the two terms in \eqref{eq:decomp_exp} are orthogonal in the complex plane(real vs.\ imaginary), and we may write\begin{equation}|\langle \Delta A \Delta B \rangle|^2=\left(\frac{1}{2}|\langle [\Delta A,\Delta B]\rangle|\right)^2+\left(\frac{1}{2}\langle \{\Delta A,\Delta B\}\rangle\right)^2 .\label{eq:RS_identity}\end{equation} Substituting \eqref{eq:RS_identity} into \eqref{eq:cs_step} yields theSchr\"odinger uncertainty relation:\begin{equation}\boxed{\sigma_A^2 \sigma_B^2 \ge\left(\frac{1}{2}|\langle [\Delta A,\Delta B]\rangle|\right)^2+\left(\frac{1}{2}\langle \{\Delta A,\Delta B\}\rangle\right)^2}.\label{eq:schrodinger}\end{equation} Since constants commute with everything,\begin{equation}[\Delta A,\Delta B] = [A-\langle A\rangle,\; B-\langle B\rangle] = [A,B].\end{equation}Hence \eqref{eq:schrodinger} can be written as\begin{equation}\boxed{\sigma_A^2 \sigma_B^2 \ge\left(\frac{1}{2}|\langle [A,B]\rangle|\right)^2+\left(\frac{1}{2}\langle \{\Delta A,\Delta B\}\rangle\right)^2}.\end{equation} \subsection{Robertson inequality (weaker but widely quoted)} Dropping the nonnegative anticommutator term gives Robertson's inequality:\begin{equation}\boxed{\sigma_A \sigma_B \ge \frac{1}{2}|\langle [A,B]\rangle|}.\label{eq:robertson}\end{equation} \subsection{Specialization to position and momentum} Take
A=x and
B=p (one spatial dimension), with canonical commutation relation\begin{equation}[x,p] = i\hbar.\end{equation}Therefore,\begin{equation}|\langle [x,p]\rangle|= |\langle i\hbar \rangle|= \hbar,\end{equation}and \eqref{eq:robertson} yields the standard position--momentum uncertainty relation\begin{equation}\boxed{\sigma_x \sigma_p \ge \frac{\hbar}{2}.}\end{equation} \subsection{Condition for equality (minimum-uncertainty states)} Equality in Cauchy--Schwarz holds iff the vectors are linearly dependent:\begin{equation}\Delta A|\psi\rangle = \lambda \, \Delta B|\psi\rangle\quad \text{for some complex } \lambda.\end{equation}For
A=x and
B=p, this condition leads (in the
x-representation) to afirst-order differential equation whose normalized solutions are Gaussianwave packets. Hence Gaussian states saturate
σxσp=ℏ/2. \subsection{Definition of fluctuation operators and their role in the uncertainty relation} In the mathematical formulation of quantum uncertainty developed after Heisenberg’s original 1927 heuristic proposal, the quantities entering the uncertainty relation were redefined in statistical terms. In particular, in his 1929 paper, Robertson introduced the operators. \noindent\textbf{Critical observation.}The introduction of the fluctuation operators \begin{center}\fcolorbox{red}{white}{
ΔA=A−⟨A⟩,ΔB=B−⟨B⟩}\end{center} \textbf{constitutes a nontrivial mathematicalassumption rather than a derived physical necessity. In the Robertson--Schrödinger formalism, the uncertainty of observables is defined \emph{a priori} in terms ofstatistical deviations from expectation values, thereby presupposing the verynotion of indeterminacy that the uncertainty principle is intended to justify.In other words, the operators
ΔA and
ΔB encode uncertainty byconstruction, since their variances measure ensemble spread relative to a mean.The subsequent derivation of a lower bound on
σAσB thereforeestablishes a constraint on these predefined statistical quantities, rather than demonstrating that physical observables must be uncertain from first principles.Consequently, the uncertainty relation obtained in this manner is logicallyconditional upon the initial choice of variance-based definitions and does not,by itself, prove the existence of intrinsic measurement indeterminacy or disturbance in individual quantum events.} \textit{They actully assumed and defined the spread and uncertainity initial in both the quantities and proved the uncertainity principle.} where
A and
B are self-adjoint operators representing observables and
⟨A⟩=⟨ψ∣A∣ψ⟩ denotes the expectationvalue in a normalized quantum state
∣ψ⟩ \cite{Robertson1929}.These operators describe deviations of observables from their mean valuesand are introduced in direct analogy with classical statistical fluctuations.The corresponding variances,
σA2=⟨(ΔA)2⟩ and
σB2=⟨(ΔB)2⟩,measure the statistical spread of measurement outcomes over an ensembleof identically prepared systems