CHSH Inequality (S-Parameter) Equation, Derivation & Rearrangements

Core idea

Overview

The CHSH inequality is a mathematical formulation used in quantum mechanics to test Bell's theorem, providing a criterion to distinguish between local hidden variable theories and quantum entanglement. It calculates the S-parameter based on correlation values of particle properties measured at different detector orientations.

When to use: Use this formula when conducting Bell tests to verify the presence of non-local correlations in a bipartite system. It assumes that two observers, typically named Alice and Bob, each choose between two measurement settings for their respective particles.

Why it matters: It serves as the experimental proof that the universe does not obey local realism, as quantum systems can violate the classical bound of 2. This principle is fundamental for secure quantum communication and the development of quantum computers.

Symbols

Variables

S = S-Parameter, E_1 = E(a,b), E_2 = E(a,b'), E_3 = E(a',b), E_4 = E(a',b')

S

S-Parameter

V a r iab l e

E_{1}

E(a,b)

V a r iab l e

E_{2}

E(a,b')

V a r iab l e

E_{3}

E(a',b)

V a r iab l e

E_{4}

E(a',b')

V a r iab l e

Walkthrough

Derivation

Formula: CHSH Inequality S-Parameter

The S-parameter tests Bell's inequality. Quantum mechanics violates the classical CHSH bound of 2.

Four correlation functions E(a,b) measured with settings a, a', b, b'.
Classically |S| ≤ 2; quantum mechanics allows |S| ≤ 2√2 ≈ 2.828.

1

Define the CHSH Parameter:

Each E(x,y) is the correlation between measurement outcomes at settings x and y.

S = E (a, b) - E (a, b^{'}) + E (a^{'}, b) + E (a^{'}, b^{'})

2

Quantum Maximum (Tsirelson's Bound):

If the measured |S| > 2, classical hidden variables cannot explain the correlations — the system is entangled.

∣ S ∣ \leq 22 \approx 2.828

Result

∣ S ∣ \leq 22 \approx 2.828

Source: University Quantum Computing — Bell Inequalities

Free formulas

Rearrangements

Solve for $S$

Make S the subject (CHSH S-Parameter)

S

This problem defines the S-Parameter, S, which is used in the CHSH inequality to test Bell's theorem. Since S is already isolated, the task is to identify its definition.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph displays a sinusoidal wave pattern as the independent variable varies, reflecting the periodic nature of the correlation functions. The S-parameter oscillates between specific bounds, with the shape determined by the trigonometric relationship between the measurement settings.

Graph type: sinusoidal

Why it behaves this way

Intuition

Imagine two distant observers, Alice and Bob, each choosing one of two measurement settings for their particle; the S-parameter statistically combines the average agreement or disagreement of their outcomes across all

S

The CHSH S-parameter, a statistical quantity used to test Bell's theorem and quantify non-local correlations.

This value indicates how strongly the measurement outcomes of two particles are correlated in a way that cannot be explained by classical local realism. A value greater than 2 implies quantum entanglement.

E (x, y)

The expectation value (average) of the product of measurement outcomes when Alice uses setting 'x' and Bob uses setting 'y'.

This term quantifies the statistical agreement or disagreement between Alice's and Bob's results for a specific pair of measurement choices.

a, a^{'}

Alice's two distinct possible measurement settings (e.g., orientations of a polarizer or spin analyzer).

These represent the choices Alice makes about how to measure her particle, which influence the observed correlations.

b, b^{'}

Bob's two distinct possible measurement settings.

These represent the choices Bob makes about how to measure his particle, which influence the observed correlations.

Signs and relationships

- E(a, b'): The specific negative sign for this term, along with the positive signs for the others, is a crucial part of the algebraic construction of the CHSH operator.

Free study cues

Insight

Canonical usage

The CHSH S-parameter is a dimensionless quantity used to quantify correlations in Bell tests, with its value indicating whether local realism is violated.

Common confusion

Students might mistakenly search for physical units for the S-parameter or its constituent expectation values, but they are fundamentally dimensionless quantities representing correlations.

Dimension note

The S-parameter is a measure of correlation, derived from expectation values of products of measurement results. Since measurement results are assigned dimensionless values (e.g., +1 or -1 for spin or polarization), the

Unit systems

$S$ dimensionless · The CHSH S-parameter is a sum and difference of expectation values of products of measurement outcomes, making it inherently dimensionless.

$E (x, y)$ dimensionless · Each E term represents an expectation value of the product of two measurement outcomes. Measurement outcomes are typically assigned dimensionless values (e.g., +1 or -1), so their product and subsequent expectation value

Ballpark figures

Quantity:
Quantity:

One free problem

Practice Problem

A quantum optics experiment measures four correlation values: E(a, b) = 0.707, E(a, b') = -0.707, E(a', b) = 0.707, and E(a', b') = 0.707. Calculate the resulting S-parameter to determine if it reaches Tsirelson's bound.

E(a,b)0.707

E(a,b')-0.707

E(a',b)0.707

E(a',b')0.707

Solve for: s

Hint: Plug the values into the formula S = e1 - e2 + e3 + e4 and watch the double negative.

The full worked solution stays in the interactive walkthrough.

Study smarter

Tips

The classical limit for the S-parameter is always between -2 and 2.
The maximum quantum violation, known as Tsirelson's bound, is approximately 2.828 (2√2).
Pay close attention to the negative sign on the second term (e2) in the sum.
Correlation values E range from -1 to 1.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The S-parameter tests Bell's inequality. Quantum mechanics violates the classical CHSH bound of 2.

Use this formula when conducting Bell tests to verify the presence of non-local correlations in a bipartite system. It assumes that two observers, typically named Alice and Bob, each choose between two measurement settings for their respective particles.

It serves as the experimental proof that the universe does not obey local realism, as quantum systems can violate the classical bound of 2. This principle is fundamental for secure quantum communication and the development of quantum computers.

The classical limit for the S-parameter is always between -2 and 2. The maximum quantum violation, known as Tsirelson's bound, is approximately 2.828 (2√2). Pay close attention to the negative sign on the second term (e2) in the sum. Correlation values E range from -1 to 1.

References

Sources

Wikipedia: CHSH inequality
Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information (10th Anniversary Edition). Cambridge University Press.
Clauser, J. F., Horne, M. A., Shimony, A., & Holt, R. A. (1969). Proposed Experiment to Test Local Hidden-Variable Theories.
Griffiths, David J. Introduction to Quantum Mechanics. 3rd ed. Cambridge University Press, 2018.
Nielsen and Chuang Quantum Computation and Quantum Information
Clauser et al. Proposed Experiment to Test Local Hidden-Variable Theories
Bell On the Einstein Podolsky Rosen Paradox
CHSH inequality Wikipedia

CHSH Inequality (S-Parameter)