[Paper Review] General and Stronger Uncertainty Relation

Vishnu M. Bannur|arXiv (Cornell University)|Mar 2, 2015

Numerical methods in inverse problems3 citations

TL;DR

This paper generalizes Maccone and Pati's stronger uncertainty relations by deriving an infinite family of variance-based uncertainty inequalities using a free parameter α. By minimizing over α, the authors identify a single optimal inequality that reduces to standard Heisenberg-Robertson and Schrödinger relations in limiting cases and consistently yields nontrivial lower bounds, even when standard relations fail.

ABSTRACT

Recently, Maccone and Pati [Phys. Rev. Lett. {\bf 113}, 260401 (2014)] derived few inequalities among variances of incompatible operators which they called stronger uncertainty relations, stronger than Heisenberg-Robertson or Schrodinger uncertainty relations. Here we generalize their study to get infinite number of such inequalities and propose that only one of them may be the correct uncertainty relation between incompatible operators. We get back well known uncertainty relations of Heisenberg-Robertson and Schrodinger under certain limits. We also reexamine the conclusions of Maccone and Pati and comment on their work.

Motivation & Objective

• To address the limitation of standard uncertainty relations, which can yield trivial (zero) lower bounds for incompatible observables in certain states.
• To generalize Maccone and Pati's stronger uncertainty relations into an infinite set of such inequalities using a free parameter α.
• To identify the single optimal inequality among the infinite family by minimizing over α, ensuring the tightest possible bound.
• To demonstrate that the resulting relation reduces to the Heisenberg-Robertson and Schrödinger uncertainty relations in appropriate limits.
• To verify the nontriviality and strength of the derived relation through explicit examples, such as spin-1 systems.

Proposed method

• Introduce a complex linear combination |ϕ⟩ = |ψ_A⟩ + iα|ψ_B⟩ with a real parameter α, where |ψ_A⟩ and |ψ_B⟩ are deviations of operators A and B from their expectation values.
• Apply the Cauchy-Schwarz inequality to |ϕ⟩ and an arbitrary normalized state |ψ^⊥⟩ orthogonal to |ψ⟩, leading to a complex inequality involving α.
• Separate the inequality into real and imaginary parts to derive a general uncertainty relation depending on α, with the form [ΔA² - |⟨ψ^⊥|A|ψ⟩|²] + α²[ΔB² - |⟨ψ^⊥|B|ψ⟩|²] + iα[⋯] ≥ 0.
• Minimize the resulting expression over α to find the optimal value that yields the tightest lower bound, leading to a specific α that defines the true stronger uncertainty relation.
• Derive the final inequality in the form (ΔA² - |⟨ψ^⊥|A|ψ⟩|²)(ΔB² - |⟨ψ^⊥|B|ψ⟩|²) ≥ ¼|⟨[A,B]⟩ - [⋯]|², which reduces to known relations in limits.
• Verify the result using explicit examples, such as spin-1 systems with |ψ⟩ = |0⟩ and |ψ^⊥⟩ = cosθ|+⟩ + sinθ|−⟩, showing nontrivial bounds for the derived relation.

Experimental results

Research questions

• RQ1 Can Maccone and Pati's stronger uncertainty relations be generalized beyond their specific inequalities to an infinite family of such relations?
• RQ2 Is there a unique optimal uncertainty relation among the infinite family, and if so, how can it be identified?
• RQ3 Does the optimal relation reduce to the standard Heisenberg-Robertson and Schrödinger uncertainty relations in appropriate limits?
• RQ4 Can the derived relation yield nontrivial lower bounds even when standard uncertainty relations give a trivial (zero) bound?
• RQ5 How does the choice of the orthogonal state |ψ^⊥⟩ and the parameter α affect the strength and validity of the uncertainty bound?

Key findings

• The paper derives an infinite family of uncertainty inequalities by generalizing Maccone and Pati's approach using a free parameter α, showing their relations are special cases.
• The true stronger uncertainty relation is identified by minimizing over α, yielding a single optimal inequality that provides the tightest nontrivial lower bound.
• The optimal relation reduces to the Heisenberg-Robertson and Schrödinger uncertainty relations in the limits where the state |ψ⟩ is an eigenstate of A or B, respectively.
• For the spin-1 system with |ψ⟩ = |0⟩ and |ψ^⊥⟩ = cosθ|+⟩ + sinθ|−⟩, the derived relation gives a nontrivial lower bound of 2ℏ²cos²θ, while the standard HRS relation gives a trivial bound of zero.
• The inequality remains nontrivial and consistent across different choices of |ψ^⊥⟩ and α, with equality achieved in specific cases such as α = ±1 for certain spin states.
• The method confirms that the derived relation is stronger than Maccone and Pati's original relations, as it yields a tighter bound and is universally nontrivial when the state is not an eigenstate of either observable.