[Paper Review] Uncertainty Relations for Entangled States

Gustavo Rigolin|arXiv (Cornell University)|Aug 23, 2000

Quantum Mechanics and Applications6 references3 citations

TL;DR

This paper proposes a generalized uncertainty relation for entangled particle pairs by enforcing symmetrization rules for physical observables, ensuring operators commute with permutation operators. The resulting relation, derived from total position and momentum operators, shows that entanglement can lead to $Δ Q_2 \Delta P_2 < \hbar/2$, challenging the standard Heisenberg limit in conditional measurements like Kim and Shih's Popper-type experiment.

ABSTRACT

A generalized uncertainty relation for an entangled pair of particles is obtained if we impose a symmetrization rule for all operators that we should use when doing any calculation using the entangled wave function of the pair. This new relation reduces to Heisenberg's uncertainty relation when the particles have no correlation and suggests that we can have new lower bounds for the product of position and momentum dispersions.

Motivation & Objective

• To resolve the apparent violation of the Heisenberg uncertainty principle in Kim and Shih's Popper-type experiment with entangled photons.
• To address the inadequacy of applying single-particle uncertainty relations to entangled systems due to non-commuting observables with permutation operators.
• To establish a new uncertainty relation based on physical observables that commute with permutation operators, ensuring consistency with quantum mechanical symmetrization rules.
• To demonstrate that entanglement introduces quantum covariance terms that modify the standard uncertainty bound, allowing for lower product of dispersions.
• To provide a theoretical framework for analyzing conditional measurements in entangled systems using total system observables rather than individual particle operators.

Proposed method

• Define physical observables as operators that commute with all permutation operators in the system, ensuring compatibility with the symmetrization postulate for identical particles.
• Introduce the total position $Q(1,2) = Q(1)\otimes\mathcal{I}_2 + \mathcal{I}_1\otimes Q(2)$ and total momentum $P(1,2) = P(1)\otimes\mathcal{I}_2 + \mathcal{I}_1\otimes P(2)$ as physical observables, since they commute with the permutation operator $P_{21}$.
• Derive a generalized uncertainty relation: $(\Delta Q(1,2))^2(\Delta P(1,2))^2 \geq \frac{1}{4}|\langle [Q(1,2), P(1,2)] \rangle|^2 = \frac{\hbar^2}{4}$, using only physical observables.
• Express the uncertainty relation in terms of individual particle variances and covariances: $(\Delta Q_1)^2 + (\Delta Q_2)^2 + 2(\langle Q_1 Q_2 \rangle - \langle Q_1 \rangle\langle Q_2 \rangle)$ and similarly for momentum, to capture entanglement effects.
• Apply the relation to a symmetric, entangled 1D wave function $\Psi(x_1,x_2,0)$ with $k_1 + k_2 = 0$, computing $\langle x_1 x_2 \rangle = \frac{k_0^2 a^4}{2}$ and $\langle p_1 p_2 \rangle = -2\hbar^2 k_0^2$ to show non-vanishing quantum covariance.
• Demonstrate that the resulting uncertainty bound (Eq. 8) differs from Heisenberg’s relation due to entanglement-induced covariance terms, allowing $\Delta Q_2 \Delta P_2 < \hbar/2$.

Experimental results

Research questions

• RQ1 Can the apparent violation of the Heisenberg uncertainty principle in Kim and Shih’s entangled photon experiment be explained without invoking non-locality or measurement blurring?
• RQ2 What is the correct form of the uncertainty relation when applying quantum mechanics to entangled systems of identical particles?
• RQ3 How do symmetrization rules and permutation symmetry constrain the choice of physical observables in many-body quantum systems?
• RQ4 Can entanglement lead to a reduction in the product of position and momentum uncertainties below the standard $\hbar/2$ bound?
• RQ5 What role do quantum covariance functions $\langle Q_1 Q_2 \rangle - \langle Q_1 \rangle\langle Q_2 \rangle$ and $\langle P_1 P_2 \rangle - \langle P_1 \rangle\langle P_2 \rangle$ play in modifying uncertainty relations for entangled states?

Key findings

• The generalized uncertainty relation for entangled particles is derived using total system observables $Q(1,2)$ and $P(1,2)$, which commute with the permutation operator $P_{21}$, ensuring consistency with quantum mechanical symmetrization.
• The derived relation (Eq. 8) explicitly includes quantum covariance terms $\langle Q_1 Q_2 \rangle - \langle Q_1 \rangle\langle Q_2 \rangle$ and $\langle P_1 P_2 \rangle - \langle P_1 \rangle\langle P_2 \rangle$, which vanish only for separable (non-entangled) states.
• For the symmetric entangled wave function (Eq. 11), the position covariance is $\langle x_1 x_2 \rangle = \frac{k_0^2 a^4}{2}$ and the momentum covariance is $\langle p_1 p_2 \rangle = -2\hbar^2 k_0^2$, confirming non-zero entanglement effects.
• The relation allows for $\Delta Q_2 \Delta P_2 < \frac{\hbar}{2}$ in conditional measurements, such as the virtual slit scenario in Kim and Shih’s experiment, where the photon’s position is inferred via coincidence with the other photon.
• The standard Heisenberg uncertainty relation fails for entangled systems because individual particle operators $Q(i)$ and $P(i)$ do not commute with the permutation operator and thus are not physical observables.
• The paper concludes that entanglement introduces new lower bounds for the product of dispersions, and that such states can have minimal position uncertainty without divergent momentum uncertainty, challenging the conventional interpretation of the uncertainty principle.