On some states minimizing uncertainty relations: A new look at these relations

This paper identifies a distinct class of non-eigenstates where the lower bounds of both product and sum uncertainty relations vanish due to zero correlation, thereby revealing a dual nature of the uncertainty principle as both a lower bound on standard deviations and an upper bound on the correlation function.

The Gist

The Big Idea: Uncertainty Has Two Faces

For nearly a century, physicists have taught us the Heisenberg Uncertainty Principle with a simple rule: You can't know everything about two different things (like position and momentum) at the same time. If you try to measure one very precisely, the other becomes fuzzy.

The standard formula says:

Uncertainty of A × Uncertainty of B ≥ A tiny number (usually not zero).

This implies that there is always a "floor" or a minimum limit to how much fuzziness you must have. You can never make both things perfectly clear at once.

However, this paper argues that this rule is incomplete. The author, K. Urbanowski, suggests that the Uncertainty Principle actually has two faces:

1. The Standard Face: It sets a minimum limit on how "fuzzy" two things can be.
2. The Hidden Face: It actually sets a maximum limit on how much two things can be "connected" or correlated.

The Plot Twist: When the Limit Vanishes

The most surprising discovery in this paper is that for certain special states of a quantum system, that "minimum limit" (the floor) can actually drop to zero.

The Analogy: The Dance Floor
Imagine a dance floor where two dancers, Alice (Observable A) and Bob (Observable B), are trying to move.

• The Old Rule: The rulebook says, "Alice and Bob must always be at least 1 meter apart. They can never be perfectly still or perfectly synchronized."
• The New Discovery: The author found a specific type of dance floor (a 3-dimensional space or higher) where Alice and Bob can stand perfectly still and perfectly still at the same time, even though they are not "locked" into a single pose (eigenstates).
• The Catch: They aren't locked together; they are just standing in a way where their movements don't interfere with each other at all. In this specific state, the "fuzziness" of one does not force the "fuzziness" of the other.

The "Correlation" Secret

The paper introduces a concept called the Correlation Function. Think of this as a "connection meter" between Alice and Bob.

• If the meter reads 1, they are perfectly linked (like a couple holding hands; if one moves, the other must).
• If the meter reads 0, they are completely independent (like strangers in a crowd; one moving doesn't affect the other).

The author shows that the Uncertainty Principle is actually a trade-off:

The product of their fuzziness is the "ceiling" for their connection.

If Alice and Bob are very fuzzy (high uncertainty), they can be highly connected. But if they are very precise (low uncertainty), they cannot be highly connected.

The "Zero" State:
The paper proves that there are many states where Alice and Bob are both precise (low uncertainty) AND unconnected (zero correlation). In these states, the "floor" for their uncertainty drops to zero. This means you can have a state where both are sharp, provided they don't influence each other.

Why the Old Math Was Misleading

The paper points out that the famous Heisenberg-Robertson formula (the one everyone uses) is a bit like a blunt instrument.

• It looks at the "commutator" (a mathematical way of checking if A and B fight with each other).
• If the math says the "fight" is zero, the old formula says, "Okay, uncertainty can be zero."
• But the author says: "Wait! Just because the 'fight' is zero doesn't mean they are unconnected. They might still be influencing each other in a subtle, hidden way."

The author uses a more precise tool (the Schrödinger relation) to show that even if the "fight" is zero, the "connection" might still be there, keeping the uncertainty high. However, he also found the rare "sweet spot" states where the connection is truly zero, allowing the uncertainty to vanish.

The "Sum" Rules Don't Help

Recently, scientists tried to fix the old rules by creating "Sum Uncertainty Relations" (adding the uncertainties together instead of multiplying them). They hoped this would solve the problem of the "zero limit."

The author tested these new rules and found they don't work for these special states either.

• Analogy: Imagine trying to measure the height of a building by adding the height of the roof and the foundation. If the foundation is missing (zero), the sum rule just tells you the roof is the height of the building. It doesn't give you any new information about the minimum height. The "Sum" rules just end up saying, "Well, the uncertainty is at least zero," which is a useless answer.

The "Three-Dimensional" Requirement

There is a catch to finding these special "zero uncertainty" states. You can't find them in a simple 2-dimensional world (like a flat sheet of paper).

• Analogy: Imagine trying to stand two sticks perpendicular to a flat sheet of paper. You can only do it if the sticks are in the same plane. But if you have a 3D room, you can have Stick A pointing North, Stick B pointing East, and a third invisible direction pointing Up.
• The author shows that you need a 3-dimensional (or larger) space to find these special states where two things are precise but unconnected. In a 2D world, this is mathematically impossible.

The Takeaway

1. Uncertainty isn't just a limit; it's a relationship. It tells us how much two things can be "in sync."
2. The "Zero" states exist. There are many quantum states where two different properties can be known precisely at the same time, as long as they don't influence each other.
3. The old formulas are too simple. They miss the nuance of "correlation." We need to look at the "connection" between variables, not just the "fight" between them.
4. Practical Question: If we can prepare a system in these special states where two things are precise and unconnected, could we use this for better sensors or quantum computers? The paper leaves this as an exciting question for the future.

In short: The Uncertainty Principle isn't just a wall that stops us from knowing things. It's a rulebook that describes how the "fuzziness" of one thing is tied to the "connection" with another. And in the right 3D dance hall, that connection can be broken, letting us see clearly without the usual blur.

Technical Summary

1. Problem Statement

The paper addresses a specific limitation in the standard interpretation of the Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations.

• The Standard View: The uncertainty principle is typically viewed as a lower bound on the product of standard deviations ($\Delta A \cdot \Delta B$) of two non-commuting observables, $A$ and $B$. The bound is usually given by the expectation value of their commutator: $\Delta A \cdot \Delta B \geq \frac{1}{2}|\langle [A, B] \rangle|$.
• The Issue: It is well known that if a system is in an eigenstate of $A$ or $B$, the right-hand side (RHS) of the HR relation vanishes, implying $\Delta A \cdot \Delta B \geq 0$. However, the author identifies a more subtle and non-trivial case: there exist states $|\phi\rangle$ that are not eigenstates of either $A$ or $B$ (meaning $\Delta A > 0$ and $\Delta B > 0$), yet the lower bound of the product of their standard deviations is zero.
• The Gap: The standard HR relation fails to provide a non-zero lower bound in these specific states, potentially leading to incorrect conclusions about the system's properties. Furthermore, the paper investigates whether "sum uncertainty relations" (recently proposed as improvements) can resolve this issue.

2. Methodology

The author employs rigorous mathematical analysis within the framework of Hilbert space quantum mechanics:

• Derivation Review: The paper re-derives the HR and RS relations starting from the Cauchy–Schwarz inequality applied to the deviation vectors $|\psi_A\rangle = \delta_\phi A |\phi\rangle$ and $|\psi_B\rangle = \delta_\phi B |\phi\rangle$, where $\delta_\phi F = F - \langle F \rangle_\phi I$.
• Decomposition of Correlation: The core of the analysis involves decomposing the inner product $\langle \psi_A | \psi_B \rangle$ into real and imaginary parts. The author distinguishes between the commutator term (imaginary part) and the anti-commutator/covariance term (real part).
• Definition of Correlation Function: The paper defines a quantum correlation function (covariance) $C_\phi(A, B) = \langle \phi | \delta_\phi A \delta_\phi B | \phi \rangle$ and a corresponding correlation coefficient $r_\phi(A, B)$ analogous to the Pearson coefficient.
• Geometric Analysis: The author analyzes the geometric conditions under which the vectors $|\psi_A\rangle$ and $|\psi_B\rangle$ are orthogonal ($\langle \psi_A | \psi_B \rangle = 0$) despite being non-zero.
• Counter-Examples: The paper utilizes Gell-Mann matrices ($\lambda_3, \lambda_4$) acting on a three-level system (qutrit) to construct explicit examples of states satisfying the derived conditions.
• Evaluation of Sum Relations: The triangle inequality in Hilbert space is applied to test "sum uncertainty relations" ($\Delta A + \Delta B \geq \Delta(A+B)$) against the identified problematic states.

3. Key Contributions

The paper makes several significant theoretical contributions:

1. Identification of a Large Set of States ($S_{AB}$): The author proves the existence of a large set of states $S_{AB}$ where:
   • The system is not in an eigenstate of $A$ or $B$ ($\Delta A > 0, \Delta B > 0$).
   • The deviation vectors are orthogonal: $\delta_\phi A |\phi\rangle \perp \delta_\phi B |\phi\rangle$.
   • Consequently, the correlation function $C_\phi(A, B) = 0$.
   • Crucially: This orthogonality requires the Hilbert space dimension to be at least 3 ($\dim H \geq 3$).
2. Refinement of the Uncertainty Principle: The paper demonstrates that the general uncertainty relation is actually an inequality involving the correlation function:
   $$\Delta_\phi A \cdot \Delta_\phi B \geq |C_\phi(A, B)|$$
   This reveals that the product of standard deviations is bounded by the magnitude of the correlation, not just the commutator.
3. The "Two Faces" of the Uncertainty Principle: The author proposes a dual interpretation:
   • Face 1: The product of standard deviations is a lower bound on the correlation magnitude.
   • Face 2: The product of standard deviations acts as an upper bound on the modulus of the correlation function $|C_\phi(A, B)|$.
4. Critique of Sum Uncertainty Relations: The paper demonstrates that "sum uncertainty relations" (e.g., $\Delta A + \Delta B \geq \Delta(A+B)$) fail to provide non-trivial lower bounds for the states in set $S_{AB}$. When $\delta_\phi A |\phi\rangle \perp \delta_\phi B |\phi\rangle$, the triangle inequality becomes an equality, rendering the sum relations trivial ($\Delta A + \Delta B \geq \sqrt{(\Delta A)^2 + (\Delta B)^2}$), offering no new information.
5. Distinction between Sets $S_{AB}$ and $S_{[A,B]}$:
   • $S_{[A,B]}$: States where $\langle [A, B] \rangle = 0$. This includes eigenstates and states where the correlation is purely real.
   • $S_{AB}$: A subset of $S_{[A,B]}$ where the full correlation $C_\phi(A, B)$ vanishes (both real and imaginary parts are zero).
   • The paper shows that for states in $S_{[A,B]} \setminus S_{AB}$, the HR relation gives a zero bound, but the RS relation (involving the real part of the covariance) still provides a non-zero bound. Only for $S_{AB}$ does the bound truly vanish.

4. Results

• Explicit Example: Using Gell-Mann matrices $\lambda_3$ and $\lambda_4$ on a qutrit state $|\phi_1\rangle = \frac{1}{N}(a, b, 0)^T$, the author shows that $\Delta \lambda_3 > 0$ and $\Delta \lambda_4 > 0$, yet $C_{\phi_1}(\lambda_3, \lambda_4) = 0$. Thus, $\Delta \lambda_3 \cdot \Delta \lambda_4 \geq 0$.
• Counter-Example to Commutator Vanishing: Another state $|\phi_2\rangle = \frac{1}{\sqrt{3}}(1, 1, 1)^T$ is shown where $\langle [\lambda_3, \lambda_4] \rangle = 0$ (so HR bound is 0), but $C_{\phi_2}(\lambda_3, \lambda_4) \neq 0$. Here, the RS relation provides a non-zero lower bound ($1/3$), proving that $\langle [A, B] \rangle = 0$ does not imply the uncertainty product can be zero.
• Failure of Sum Relations: For the state $|\phi_1\rangle$ (where vectors are orthogonal), the sum uncertainty relation reduces to an identity, providing no constraint on the individual standard deviations.

5. Significance

• Correcting Misinterpretations: The paper clarifies that a vanishing commutator expectation value does not automatically imply that non-commuting observables can be simultaneously measured with arbitrary precision, unless the full correlation function vanishes.
• Physical Interpretation of Uncertainty: By framing the uncertainty principle as a bound on the correlation function, the author links uncertainty directly to quantum entanglement and correlation. If $C_\phi(A, B) = 0$ (in set $S_{AB}$), the observables are uncorrelated and behave as if they were independent, despite being non-commuting operators.
• Dimensionality Constraint: The finding that such "uncorrelated non-eigenstates" only exist in dimensions $\geq 3$ highlights a fundamental geometric property of quantum state spaces, distinguishing qubits from qutrits and higher-dimensional systems.
• Technical Utility: The results suggest that for systems in states $S_{AB}$, the measurement of one non-commuting observable does not disturb the other (no additional fluctuations generated). This has potential implications for quantum control, error correction, and the design of quantum protocols where minimizing disturbance between specific observables is required.

In conclusion, Urbanowski argues that the standard Heisenberg–Robertson relation is insufficient for characterizing states where the commutator expectation vanishes but the system is not in an eigenstate. The Robertson–Schrödinger relation, interpreted through the lens of the correlation function, provides the necessary precision to describe these states, revealing a "second face" of the uncertainty principle where the product of uncertainties limits the strength of quantum correlations.