The Tsirelson Bound
Quantum correlations violate Bell inequalities but only up to a precise ceiling, revealing that nature is nonlocal yet not maximally so.
The quantum ceiling
The CHSH quantity S is bounded by 2 for local hidden-variable theories. Quantum mechanics can exceed this, but not without limit: Tsirelson's bound states that |S| <= 2 sqrt(2) for any quantum state and any measurements. This ceiling is achieved by a maximally entangled pair with optimal settings, and no quantum system can do better.
Why 2 sqrt(2)
The bound follows from the operator structure of quantum measurements. The CHSH operator is B = A0(B0 + B1) + A1(B0 - B1) with each observable squaring to the identity. A short computation gives B^2 = 4 I + [A0,A1][B0,B1], and since the commutator norms are bounded, ||B|| <= 2 sqrt(2). The value is dictated entirely by how noncommuting observables can be, not by any additional assumption.
Super-quantum correlations
One can imagine hypothetical correlations that respect no-signaling yet reach the algebraic maximum S = 4. The Popescu-Rohrlich box is such an object. It is consistent with relativity but not realizable in quantum mechanics. Tsirelson's bound is thus what separates quantum theory from more strongly nonlocal but still causal alternatives, and understanding why nature stops at 2 sqrt(2) is an active foundational question.
- Local realism: |S| <= 2
- Quantum: |S| <= 2 sqrt(2) approx 2.828 (Tsirelson)
- No-signaling only: |S| <= 4 (PR box)
Practical role
Because the maximum is known and unique, observing S close to 2 sqrt(2) certifies that a device is behaving as an ideal maximally entangled system. This self-testing property lets one verify quantum hardware and entanglement without trusting the internal workings of the apparatus, a foundation for device-independent certification of states and measurements.