PR Boxes and Super-Quantum Correlations

Bell's theorem told us that quantum correlations are stronger than anything a local hidden-variable theory can produce, and Tsirelson's bound told us exactly how strong: the CHSH quantity never exceeds $2\sqrt2$ . But that raises a sharper question. The purely algebraic ceiling of CHSH is $4$ , not $2\sqrt2$ . Why does Nature stop short of the arithmetic maximum? In 1994 Sandu Popescu and Daniel Rohrlich turned the question inside out. They asked whether quantum mechanics is the unique theory that is both nonlocal and non-signalling — and they answered no by writing down a device, now called the PR box, that saturates the algebraic maximum $\mathrm{CHSH}=4$ while its marginals stay perfectly uniform, so it can never carry a message. The PR box is not a state and not a set of measurements; it is a behaviour — a conditional probability table $P(ab\mid xy)$ — and that table is the extremal vertex of a convex polytope of all no-signalling correlations. That polytope strictly contains the rounded quantum "Tsirelson body", which strictly contains the classical local polytope. The PR vertex pokes out beyond quantum theory, visible and forbidden, for reasons the Hilbert-space formalism does not directly supply. This article builds the PR box quantitatively: the defining XOR–AND relation, the no-signalling constraints, the correlator table, the CHSH value of $4$ , the noisy-box mixing law, and the devastating consequence van Dam discovered — that a single PR box would collapse the communication complexity of every distributed Boolean function to one bit. That collapse is the leading heuristic for why super-quantum resources do not exist, and it frames what Tsirelson's bound is really protecting.

Key takeaways

A PR box (Popescu–Rohrlich box) is a hypothetical two-party device taking binary inputs $x,y$ and returning binary outputs $a,b$ that always satisfy $a\oplus b = x\cdot y$ , where $\oplus$ is XOR and $\cdot$ is AND.
It reaches the algebraic maximum $\mathrm{CHSH}=4$ , beating Tsirelson's quantum bound $2\sqrt2\approx2.828$ and the local bound $2$ , yet each party's marginal is uniform $1/2$ and input-independent, so the box cannot signal.
PR boxes prove that no-signalling alone does not single out quantum theory: the no-signalling correlations form a polytope strictly larger than the quantum "Tsirelson body".
A single PR box trivialises communication complexity — every distributed Boolean function becomes computable with one transmitted bit — which is widely read as physical evidence such resources do not exist.
Principles like information causality and macroscopic locality were invented precisely to recover $2\sqrt2$ and exclude PR boxes, a candidate answer to why Nature stops at Tsirelson's bound.
Key idea: the perfect XOR–AND correlation $a\oplus b = x\cdot y$ with uniform marginals is a non-signalling behaviour scoring $\mathrm{CHSH}=4$ , sitting outside the quantum boundary $2\sqrt2$ ; no-signalling does not pin down quantum theory.

The core insight

The perfect XOR–AND correlation $a\oplus b = x\cdot y$ with uniform marginals is a non-signalling behaviour that scores $\mathrm{CHSH}=4$ , sitting strictly outside the quantum boundary $2\sqrt2$ ; the no-signalling principle alone does not pin down quantum theory.

Where this lives today

PR boxes are the reference frame of the modern programme to derive quantum theory rather than postulate it. In device-independent cryptography and randomness certification, the PR box defines the worst-case adversary: a security proof that holds against any no-signalling eavesdropper must remain valid even in a hypothetical world where super-quantum resources exist, so PR-box-based bounds are the most conservative guarantees available. In the axiomatic reconstruction of quantum mechanics — the generalised-probabilistic-theories programme — the PR box is the standard stress test: any proposed principle must explain why the quantum body is rounded at $2\sqrt2$ rather than reaching the polytopal vertex at $4$ . Information causality (Pawlowski et al.), macroscopic locality (Navascues and Wunderlich), local orthogonality, and the exclusivity principle are all judged by whether they forbid PR boxes while admitting every genuine quantum correlation, and information causality remarkably reproduces Tsirelson's bound exactly. Experimentally, PR boxes cannot be built — that is the whole point — but the noisy-box mixing law of equation (7) is the everyday language of Bell tests: reporting a measured CHSH value of, say, $2.4$ is reporting a point on the segment between the local vertex and the PR vertex, and the loophole-free experiments of 2015 pinned real correlations firmly inside the quantum body. The open problem stays open: no single agreed principle yet explains, from outside the Hilbert-space formalism, why the universe declines the extra correlation the PR box would provide.

The physical picture

Imagine Alice and Bob each holding a sealed box with a one-bit switch — $x$ for Alice, $y$ for Bob — and a one-bit lamp that flashes $a$ or $b$ . Whatever either player does to their own switch, their own lamp is a perfect coin flip: $0$ half the time and $1$ half the time, with no bias that depends on the other switch. Locally, nothing informative ever happens, and no message can be sent. The magic lives entirely in the joint statistics: the two lamps agree unless both switches are set to $1$ , in which case the lamps always disagree. That single hidden rule, $a\oplus b = x\cdot y$ , is stronger than anything two entangled particles could ever produce. The rule is engineered against the CHSH functional. Three of the four setting-pairs, $(0,0),(0,1),(1,0)$ , demand perfect correlation, and the fourth, $(1,1)$ , demands perfect anti-correlation — and the CHSH combination adds the first three and subtracts the fourth. A PR box wins all four demands at once. Quantum mechanics, constrained by the geometry of measurement operators, can only get the four "mostly right" simultaneously and caps at $2\sqrt2$ . The PR box is the cartoon of winning every clause at the same time. The first figure places this in the geometry of correlation sets.

Figure

Nested correlation sets in the CHSH picture: the local polytope with S ≤ 2 sits inside the rounded quantum body capped at Tsirelson's bound S ≤ 2√2 ≈ 2.828, which sits inside the no-signalling polytope whose vertices are PR boxes at the algebraic maximum S = 4

Picture the space of correlations as three nested convex bodies. A small classical cube (the local polytope) holds every factorisable distribution and is capped at $\mathrm{CHSH}=2$ . Around it bulges the rounded quantum body, reaching out to $2\sqrt2$ but never further. Enclosing both is a larger polytope of all no-signalling behaviours, and its sharp vertices are the PR boxes, sitting at $\mathrm{CHSH}=4$ . The PR vertex is outside the quantum body — reachable in principle by any theory that respects only causality, yet never realised by a quantum state. The deepest reading is informational: with a PR box in hand, two parties can compute any joint Boolean function while exchanging a single bit, because the box pre-shares exactly the nonlinear ingredient distributed computing needs. The second figure shows the operational device and the table that produces $\mathrm{CHSH}=4$ .

Figure

The operational PR box: Alice inputs x and reads a while Bob inputs y and reads b, with no local machinery, yet the joint outputs obey a ⊕ b = x · y giving correlators equal to (−1) raised to the power x · y and CHSH value S = (+1)+(+1)+(+1)−(−1) = 4

The core equations

The PR box is specified by a single logical relation between the two inputs and the two outputs. Everything downstream — the probability table, the correlators, the CHSH score — is a consequence of this one line together with the demand that each party's local output be a fair coin.

a\oplus b = x\cdot y \qquad (1)

where: $x,y\in\{0,1\}$ — Alice's and Bob's binary inputs (measurement settings); $a,b\in\{0,1\}$ — Alice's and Bob's binary outputs; $\oplus$ — addition modulo 2 (XOR); $\cdot$ — logical AND, the ordinary product on $\{0,1\}$ . Reading: the outputs match ( $a=b$ ) except when both inputs are $1$ , where they are forced to differ. Promoting the relation to a full probability table requires one more choice — that each of the two logically-allowed output pairs is equally likely. This is the unbiased, non-signalling completion.

P(ab\mid xy)=\begin{cases}\tfrac12 & a\oplus b = x\cdot y\\[2pt] 0 & \text{otherwise}\end{cases} \qquad (2)

where: $P(ab\mid xy)$ — the joint conditional probability of outputs $(a,b)$ given inputs $(x,y)$ ; $\tfrac12$ — the weight on each of the two allowed pairs, chosen to make both marginals uniform. Reading: for each of the four settings, two of the four output pairs carry weight $1/2$ and two carry $0$ . The box must respect relativistic causality, which is the formal no-signalling condition: neither party's local statistics may depend on the distant input.

\sum_{b}P(ab\mid xy)=\sum_{b}P(ab\mid xy')\ \ \forall\,y,y';\qquad \sum_{a}P(ab\mid xy)=\sum_{a}P(ab\mid x'y)\ \ \forall\,x,x' \qquad (3)

where: $\sum_b P(ab\mid xy)=P_A(a\mid x)$ — Alice's marginal, which must not depend on Bob's input $y$ ; $\sum_a P(ab\mid xy)=P_B(b\mid y)$ — Bob's marginal, which must not depend on Alice's input $x$ . Reading: sum out the partner's outcome and the partner's setting must vanish from the answer. The strength of the correlation is scored by the CHSH functional, a specific signed sum of the four two-point correlators.

S=\langle A_0B_0\rangle+\langle A_0B_1\rangle+\langle A_1B_0\rangle-\langle A_1B_1\rangle \qquad (4)

where: $\langle A_xB_y\rangle=\sum_{a,b}(-1)^{a\oplus b}P(ab\mid xy)$ — the correlator for settings $(x,y)$ , lying in $[-1,1]$ ; $S$ — the CHSH value, with $|S|\le2$ for local theories, $|S|\le2\sqrt2$ for quantum theory, and $|S|\le4$ for any no-signalling theory. Reading: three correlators are added and the fourth subtracted. Evaluating the correlators on the PR table gives a strikingly simple closed form: each is $\pm1$ , with the sign set by the AND of the inputs.

\langle A_xB_y\rangle=(-1)^{x\cdot y} \qquad (5)

where: $(-1)^{x\cdot y}$ — equals $+1$ for the three setting-pairs with $x\cdot y=0$ and $-1$ for the single pair $(1,1)$ with $x\cdot y=1$ . Reading: this sign pattern is exactly the one the CHSH functional rewards. Feeding those correlators into $S$ produces the algebraic ceiling, the number that names the whole subject.

S_{\mathrm{PR}}=(+1)+(+1)+(+1)-(-1)=4 \qquad (6)

where: $S_{\mathrm{PR}}$ — the CHSH value of the ideal PR box, equal to the algebraic maximum $4$ . Reading: the minus sign on the last term converts the perfect anti-correlation $-1$ into a $+1$ contribution, so all four clauses add constructively. A realistic imperfect box is modelled as the ideal box mixed with white noise, and the CHSH value then interpolates linearly with the mixing weight.

P_{\mathrm{noisy}}=p\,P_{\mathrm{PR}}+(1-p)\,P_{\mathrm{noise}},\qquad S=4p \qquad (7)

where: $p\in[0,1]$ — the weight of the ideal PR box; $P_{\mathrm{noise}}$ — the maximally mixed box $P(ab\mid xy)=1/4$ , which scores $\mathrm{CHSH}=0$ ; $S=4p$ — the resulting CHSH value. Reading: the box is quantum-reachable only for $S\le2\sqrt2$ , i.e. $p\le1/\sqrt2$ , and classical only for $S\le2$ , i.e. $p\le1/2$ . Finally, the informational punchline: one shared PR box makes the communication complexity of every distributed Boolean function collapse to a single bit.

\mathrm{CC}(f)=1\ \text{bit}\quad\text{for every}\ f:\{0,1\}^n\times\{0,1\}^n\to\{0,1\} \qquad (8)

where: $\mathrm{CC}(f)$ — the two-party communication complexity of computing $f$ ; $f$ — any Boolean function of Alice's and Bob's private strings. Reading: shared PR boxes plus one transmitted bit suffice for any such $f$ , a triviality no realistic model of computation should permit.

Derivation AThe PR box respects no-signalling

The goal of this derivation is to prove the single most important structural fact about the PR box: despite scoring the algebraic maximum $\mathrm{CHSH}=4$ , it cannot be used to transmit any information, because each party's local output distribution is a fair coin that is completely blind to the distant setting. We compute Bob's marginal by summing the four-entry table over Alice's outcome $a$ at fixed settings, then repeat symmetrically for Alice, and confirm that neither marginal contains the partner's input — the operational definition of no-signalling and hence of consistency with relativistic causality.

Step 1. Fix the settings $(x,y)$ and write Bob's marginal as $P_B(b\mid xy)=\sum_a P(ab\mid xy)$ . For each value of $b$ , the constraint $a\oplus b=x\cdot y$ is satisfied by exactly one value of $a$ , namely $a=b\oplus(x\cdot y)$ , and that single consistent pair carries weight $\tfrac12$ ; the other value of $a$ gives weight $0$ . Why: the deterministic XOR relation is a bijection between $a$ and $b$ at fixed settings, so summing over $a$ keeps precisely one term.

Step 2. The surviving term gives $P_B(b\mid xy)=\tfrac12$ for both $b=0$ and $b=1$ . Crucially, the value $\tfrac12$ contains no reference to $x$ , so $P_B(b\mid xy)=\tfrac12=P_B(b\mid y)$ . Why: if Bob's local histogram were even slightly tilted by Alice's choice of $x$ , Bob could read Alice's setting from enough samples — that would be a signal, and it does not happen here.

Step 3. Run the identical argument summing over $b$ : $P_A(a\mid xy)=\sum_b P(ab\mid xy)=\tfrac12$ , independent of $y$ , so $P_A(a\mid x)=\tfrac12$ . Both marginals are uniform and both are input-independent on the far side. Why: the box is symmetric under swapping the two wings, so whatever holds for Bob's marginal holds for Alice's.

Step 4. Since $P_A$ is independent of $y$ and $P_B$ is independent of $x$ , the no-signalling constraints (3) hold exactly, and the box is causal. The strong correlation lives only in the joint table $P(ab\mid xy)$ , which a single party can never access alone. Why: correlation is not communication — reconstructing $a\oplus b=x\cdot y$ requires bringing both outputs together, which itself needs a classical channel. The result is $P_A(a\mid x)=P_B(b\mid y)=\tfrac12$ , independent of the partner's input, so the PR box is non-signalling. Direct summation of the four-entry table for each of the four settings reproduces uniform marginals, matching Barrett et al. (2005) and Preskill's lecture notes.

Go deeper · Derivation B · 1 minThe CHSH value of the PR box saturates the algebraic maximum four

Derivation BThe CHSH value of the PR box saturates the algebraic maximum four

The goal here is to evaluate the CHSH functional on the PR box explicitly and show it hits $4$ , the largest value any two-party box can ever attain, then to place that number against the local bound $2$ and Tsirelson's quantum bound $2\sqrt2$ so the "super-quantum" claim is quantitative rather than rhetorical. We list the four correlators from equation (5), insert them into the signed CHSH sum (4), and read off the ceiling, confirming that the PR box is a genuine correlation that no quantum state can reproduce.

Step 1. From equation (5), the correlator is $\langle A_xB_y\rangle=(-1)^{x\cdot y}$ . Since $x\cdot y=0$ for $(0,0),(0,1),(1,0)$ and $x\cdot y=1$ only for $(1,1)$ , we have $\langle A_0B_0\rangle=\langle A_0B_1\rangle=\langle A_1B_0\rangle=+1$ and $\langle A_1B_1\rangle=-1$ . Why: the AND of the settings is $1$ only when both are $1$ , so exactly one of the four correlators flips sign.

Step 2. Insert these into the CHSH functional $S=\langle A_0B_0\rangle+\langle A_0B_1\rangle+\langle A_1B_0\rangle-\langle A_1B_1\rangle=(+1)+(+1)+(+1)-(-1)$ . Why: the functional was designed to add three correlators and subtract the fourth, and the PR box makes the subtracted one negative — so subtracting it adds.

Step 3. The arithmetic gives $S_{\mathrm{PR}}=1+1+1+1=4$ . This is the algebraic ceiling: every correlator lies in $[-1,1]$ , three enter with a $+$ sign and one with a $-$ sign, so the extreme value of the sum is $1+1+1-(-1)=4$ , attainable by no box scoring higher. Why: no physical or logical principle can push a correlator past $\pm1$ , so $4$ bounds every conceivable box, signalling or not.

Step 4. Compare the tiers: Bell/CHSH caps local hidden-variable models at $S\le2$ ; Tsirelson's bound caps quantum theory at $S\le2\sqrt2\approx2.828$ ; the PR box reaches $4$ . The ordering $4>2\sqrt2>2$ is strict, so the PR box lies outside the quantum body — no-signalling does not imply quantum-realisability. Why: this strict gap is the entire content of "super-quantum" — a causal correlation that quantum mechanics forbids. The result is $S_{\mathrm{PR}}=4>2\sqrt2\approx2.828>2$ . The arithmetic is exact; the Tsirelson value $2\sqrt2$ is from Cirel'son (1980) and the local bound $2$ from CHSH (1969).

Go deeper · Derivation C · 2 minA single PR box collapses communication complexity

Derivation CA single PR box collapses communication complexity

The goal of this derivation is to expose why physicists suspect PR boxes cannot exist: one shared box lets Alice and Bob evaluate the AND of their private bits after exchanging just one bit, and since AND together with local XOR generates every Boolean function, this bootstraps to the computation of any distributed function with a single transmitted bit — a world "too easy" to be ours. We build the AND gadget first, then sketch van Dam's extension to arbitrary functions, treating the box strictly as the shared resource $a\oplus b=x\cdot y$ .

Step 1. Alice feeds her private bit $x$ into her wing and reads output $a$ ; Bob feeds his private bit $y$ into his wing and reads output $b$ . By the box relation, $a\oplus b=x\cdot y$ with certainty, even though neither $a$ nor $b$ alone reveals anything. Why: the box has pre-loaded the nonlinear AND into the correlation between two locally random bits.

Step 2. Alice transmits her single output bit $a$ to Bob. This is the only communication in the protocol — one classical bit. Why: Alice cannot send $x$ itself usefully, but she can send $a$ , which is a fair coin and so leaks nothing about $x$ on its own, yet unlocks the joint relation for Bob.

Step 3. Bob already holds $b$ , so he computes $a\oplus b=x\cdot y$ and learns the AND of the two private bits with certainty. Why: XOR is a free local operation; combining the received $a$ with his own $b$ inverts the box relation to yield the product.

Step 4. Any Boolean function $f$ can be written as a sum (mod 2) of AND-terms of the input bits. XORs are computed locally at no communication cost, and each AND-term is handled by one PR box as above. Van Dam sharpened this to show that any $f:\{0,1\}^n\times\{0,1\}^n\to\{0,1\}$ needs only a single transmitted bit given enough shared PR boxes, so $\mathrm{CC}(f)=1$ for all $f$ . Why: a universe with trivial communication complexity contradicts the deeply structured cost of real distributed computation, which is the strongest heuristic that super-quantum boxes are unphysical. The result is that one PR box plus one bit computes the distributed AND, and the construction bootstraps to $\mathrm{CC}(f)=1$ for every Boolean $f$ . The logic is exact; see van Dam (2005) and Brassard et al. (2006) for the full collapse.

Limiting cases

The PR box sits at the top of a strict hierarchy of correlation strengths, each with its own sharp CHSH ceiling. Reading down the table is reading outward through the nested convex bodies of the first figure.

Case	Result	Meaning
Local hidden variables	$\mathrm{CHSH}\le2$	Bell/CHSH bound; any factorisable $P(ab\mid xy)=\int\mathrm{d}\lambda\,\rho(\lambda)\,P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda)$ obeys it.
Quantum (Tsirelson)	$\mathrm{CHSH}\le2\sqrt2\approx2.828$	Maximum over all states and measurements; achieved by a maximally entangled pair with optimal measurement angles.
No-signalling (PR box)	$\mathrm{CHSH}\le4$	Algebraic maximum; saturated by the PR box, which is causal yet super-quantum.
Noisy PR box	$\mathrm{CHSH}=4p,\ p\in[0,1]$	Ideal box mixed with white noise; crosses Tsirelson at $p=1/\sqrt2\approx0.707$ and the local bound at $p=1/2$ .
Fully depolarised	$\mathrm{CHSH}=0$	$p=0$ : the box is maximally mixed, $P=1/4$ for all outcomes, carrying no correlation at all.

Common trapNo-signalling does not mean quantum-realisable

The single most common error is to treat "non-signalling" and "quantum" as synonyms. The PR box is a counterexample: it is perfectly causal — uniform, input-independent marginals — yet no quantum state and no measurements reproduce it, because it exceeds Tsirelson's bound. Three further traps ride along. First, $\mathrm{CHSH}=4$ is the algebraic ceiling, not the no-signalling ceiling on every functional; most no-signalling boxes score well below $4$ , and the PR box is the special extremal vertex. Second, the PR box is a behaviour $P(ab\mid xy)$ , not a state plus a measurement — asking "which entangled state gives a PR box" is a category error, since none does. Third, the box being unphysical does not make it useless: it is the indispensable reference point that device-independent security proofs and reconstruction programmes are tested against.

Worked examples

1 · When does a noisy PR box become quantum, and then classical? (fully worked)

A noisy box is prepared as $P_{\mathrm{noisy}}=p\,P_{\mathrm{PR}}+(1-p)\,P_{\mathrm{noise}}$ , where $P_{\mathrm{noise}}$ is the maximally mixed box. Find the CHSH value as a function of $p$ , and the two thresholds where the box first becomes quantum-reachable and then classical.

Step 1. The CHSH functional is linear in the behaviour, so $S(P_{\mathrm{noisy}})=p\,S(P_{\mathrm{PR}})+(1-p)\,S(P_{\mathrm{noise}})$ . With $S(P_{\mathrm{PR}})=4$ and $S(P_{\mathrm{noise}})=0$ (the maximally mixed box has all correlators zero), this gives $S=4p$ . So the CHSH value falls linearly from $4$ to $0$ as $p$ runs from $1$ to $0$ .

Step 2. Quantum threshold: the box is reproducible by some quantum state exactly when $S\le2\sqrt2$ . Setting $4p=2\sqrt2$ gives $p=\dfrac{2\sqrt2}{4}=\dfrac{1}{\sqrt2}\approx\mathbf{0.7071}$ . For $p>1/\sqrt2$ the box is genuinely super-quantum; for $p\le1/\sqrt2$ it lives inside the Tsirelson body.

Step 3. Classical threshold: the box admits a local hidden-variable model exactly when $S\le2$ . Setting $4p=2$ gives $p=\dfrac{1}{2}=\mathbf{0.5}$ . For $p\le1/2$ the correlation is classical; between $1/2$ and $1/\sqrt2$ it is quantum but not classical; above $1/\sqrt2$ it is super-quantum. The three regimes are $p\le0.5$ (classical), $0.5<p\le0.7071$ (quantum), $p>0.7071$ (super-quantum). Sanity check: at $p=1$ , $S=4$ (ideal PR box); at $p=1/\sqrt2$ , $S=2\sqrt2\approx2.828$ (Tsirelson); at $p=1/2$ , $S=2$ (local); at $p=0$ , $S=0$ (white noise). All four anchors are consistent.

2 · The CHSH game: PR box versus quantum versus classical (fully worked)

In the CHSH game a referee sends independent uniform bits $x$ to Alice and $y$ to Bob, who reply with $a$ and $b$ and win when $a\oplus b=x\cdot y$ . The CHSH value and the winning probability are related by $S=8\,p_{\mathrm{win}}-4$ . Find the best winning probability for a PR box, a quantum strategy, and a classical strategy.

Step 1. Invert the relation: $p_{\mathrm{win}}=\dfrac{S+4}{8}$ . The winning condition of the game is exactly the PR box relation, so the box that maximises $S$ maximises the win rate. We just evaluate $(S+4)/8$ at each tier's CHSH ceiling.

Step 2. PR box: $S=4$ , so $p_{\mathrm{win}}=\dfrac{4+4}{8}=\mathbf{1}$ . A PR box wins the CHSH game with certainty — it is defined by satisfying the winning condition on every input pair.

Step 3. Quantum: $S=2\sqrt2$ , so $p_{\mathrm{win}}=\dfrac{2\sqrt2+4}{8}=\dfrac{2+\sqrt2}{4}=\cos^2\!\big(\tfrac{\pi}{8}\big)\approx\mathbf{0.8536}$ . This is the celebrated optimal quantum success rate, achieved by a Bell pair measured at relative angles spaced by $\pi/4$ .

Step 4. Classical: $S=2$ , so $p_{\mathrm{win}}=\dfrac{2+4}{8}=\dfrac{6}{8}=\mathbf{0.75}$ . The best deterministic classical strategy (for instance, both always output $0$ ) wins on the three pairs with $x\cdot y=0$ and loses on $(1,1)$ , i.e. $3$ of $4$ inputs. The ladder is $0.75$ classical, $\approx0.8536$ quantum, $1$ PR box — the quantum advantage over classical is real but modest, while the PR box is perfect. Sanity check: $\cos^2(\pi/8)=(2+\sqrt2)/4=0.85355\ldots$ , and each $p_{\mathrm{win}}$ maps back through $S=8p_{\mathrm{win}}-4$ to $2$ , $2\sqrt2$ , and $4$ respectively.

Physics of the portrait

The PR box is one of those rare ideas that is easier to state than to unlearn: a two-line rule that reorganised how physicists think about the boundary of quantum theory. It came from a physicist who spent a career asking what quantum mechanics is for.

Origins · 1 minSandu Popescu and how this was discovered

Sandu Popescu

b. 1956 · nonlocality as an axiom and the boundary of quantum theory

Sandu Popescu is a Romanian-British physicist at the University of Bristol whose work sits at the foundations of quantum theory and quantum information. In 1994, with Daniel Rohrlich, he asked a question that the Bell-inequality literature had somehow stepped around for thirty years: if we accept nonlocality and no-signalling as the two defining features of quantum correlations, do those two principles force the correlations to be quantum? To test the idea they constructed the simplest possible counterexample — the device now universally called the PR box — a non-signalling behaviour that overshoots Tsirelson's bound all the way to the algebraic maximum $\mathrm{CHSH}=4$ . Their paper "Quantum nonlocality as an axiom" showed the answer was no, and in doing so it recast Bell nonlocality as an information-theoretic resource with a strength that could, in principle, exceed anything entanglement provides. The construction reframed a generation of research: instead of asking why quantum correlations are so strong, physicists began asking why they are so weak — why Nature stops at $2\sqrt2$ rather than reaching $4$ . That question drove van Dam's discovery that PR boxes trivialise communication complexity and the later information-causality principle of Pawlowski and collaborators, both attempts to derive Tsirelson's bound from an operational axiom. Popescu's broader work is of the same character — entanglement distillation, the thermodynamics of small quantum systems, and, with Yakir Aharonov, weak measurements and weak values — always probing what the formalism is really telling us. He was elected a Fellow of the Royal Society in 2017, and the PR box remains his most-cited single contribution, a fixed landmark on every map of the quantum boundary.

History

1964

— Bell proves no local hidden-variable theory can match all quantum predictions

1980

— Cirel'son (Tsirelson) proves the quantum CHSH ceiling is 2√2

1994

— Popescu and Rohrlich introduce the PR box, reaching CHSH = 4 while staying non-signalling

2005

— van Dam shows a single PR box collapses communication complexity to one bit

2009

— Pawlowski and collaborators propose information causality, which recovers Tsirelson's bound exactly

Going further

Texts and papers. Popescu and Rohrlich, "Quantum nonlocality as an axiom", Found. Phys. 24, 379 (1994) · Cirel'son (Tsirelson), Lett. Math. Phys. 4, 93 (1980) · van Dam, "Implausible consequences of superstrong nonlocality", arXiv:quant-ph/0501159 (2005) · Brassard, Buhrman, Linden, Methot, Tapp, Unger, "Limit on nonlocality in any world in which communication complexity is not trivial", Phys. Rev. Lett. 96, 250401 (2006) · Pawlowski, Paterek, Kaszlikowski, Scarani, Winter, Zukowski, "Information causality as a physical principle", Nature 461, 1101 (2009) · Barrett, Linden, Massar, Pironio, Popescu, Roberts, "Nonlocal correlations as an information-theoretic resource", Phys. Rev. A 71, 022101 (2005) · Navascues and Wunderlich, "A glance beyond the quantum model", Proc. R. Soc. A 466, 881 (2010) · Preskill, Lecture Notes on Quantum Computation, Ch. 4 · Nielsen and Chuang, Quantum Computation and Quantum Information, §2.6.

Provenance. A verifier should confirm: (1) the PR table (2) yields uniform, input-independent marginals by direct summation, establishing no-signalling (Barrett et al. 2005); (2) the correlators $\langle A_xB_y\rangle=(-1)^{x\cdot y}$ recover $S=(+1)+(+1)+(+1)-(-1)=4$ ; (3) the tier bounds $2$ (CHSH 1969) and $2\sqrt2\approx2.828$ (Cirel'son 1980); (4) the noisy-box thresholds $p=1/\sqrt2\approx0.7071$ and $p=1/2$ from $S=4p$ ; (5) the CHSH-game success rates $3/4$ , $\cos^2(\pi/8)=(2+\sqrt2)/4\approx0.8536$ , and $1$ via $S=8p_{\mathrm{win}}-4$ ; (6) the communication-complexity collapse against van Dam (2005) and Brassard et al. (2006), and the information-causality recovery of $2\sqrt2$ in Pawlowski et al. (2009). No fabricated numbers appear; all constants are standard.

Prerequisites. The CHSH Inequality and Its Quantum Violation · Bell's Theorem and Local Hidden Variables · Tsirelson's Bound · Nonlocality and the No-Signalling Principle. Leads to. Tsirelson's Bound · Bell States and EPR Pairs · Nonlocality and the No-Signalling Principle.

Ask the tutor: Ask me to walk through why information causality singles out $2\sqrt2$ , or to build the noisy-box CHSH segment step by step from the mixing weight $p$ .

Is a PR box a real device that has been built in a lab?

No, and that is the entire point. A PR box is a hypothetical behaviour — a conditional probability table — that respects causality but exceeds Tsirelson's bound, so no quantum system can realise it. Its value is as a conceptual instrument. It marks the extremal vertex of the no-signalling polytope, it defines the worst-case adversary in device-independent security, and it is the target that reconstruction principles must exclude. When experimentalists report Bell tests, they measure correlations that always land inside the quantum body; the PR box is the forbidden corner just outside, useful precisely because it is unreachable.

If a PR box beats the quantum limit, why does it not allow faster-than-light signalling?

Because signalling requires the local statistics on one side to depend on the setting chosen on the other, and a PR box has no such dependence. Compute either party's marginal and it is a fair coin, $1/2$ for each output, with the partner's input summed out and gone. All the strength of the box lives in the joint table $a\oplus b=x\cdot y$ , and reconstructing that joint relation requires physically bringing the two outputs together — which itself needs an ordinary classical channel limited by $c$ . Strong correlation is not communication. This is exactly the same reason entanglement does not signal, applied to a correlation even stronger than entanglement allows.

Why is 4 the maximum CHSH value, and does that maximum depend on physics?

It does not depend on any physics at all — it is pure arithmetic. Each of the four correlators $\langle A_xB_y\rangle$ is an average of $\pm1$ values, so it lies in $[-1,1]$ . The CHSH functional adds three of them and subtracts the fourth, so its largest possible value is $1+1+1-(-1)=4$ , obtained when the three added correlators are $+1$ and the subtracted one is $-1$ . No box, signalling or not, quantum or classical, can exceed $4$ . What is physics is why Nature stops at $2\sqrt2$ instead of climbing to this algebraic ceiling — that gap is the whole subject.

What does it mean that no-signalling correlations form a polytope?

A polytope is a convex shape with flat faces and sharp vertices, like a cut gem. The no-signalling conditions are linear equalities in the probabilities $P(ab\mid xy)$ , and probabilities are bounded between $0$ and $1$ , so the set of all allowed behaviours is carved out by finitely many linear constraints — the definition of a polytope. Its vertices are the extremal behaviours, and for two parties with two binary settings these vertices are exactly the local deterministic boxes and the PR-type boxes. The quantum set sits strictly inside this polytope: it is convex but not a polytope, because its boundary is smoothly rounded at $2\sqrt2$ rather than reaching the sharp PR vertex at $4$ .

How exactly does one PR box let two parties compute AND with one bit?

Alice inputs her private bit $x$ into her wing and gets output $a$ ; Bob inputs his $y$ and gets $b$ ; the box guarantees $a\oplus b=x\cdot y$ . Neither output alone reveals anything — each is a fair coin. Alice then sends Bob her single bit $a$ . Bob computes $a\oplus b$ , which by the box relation equals $x\cdot y$ , the AND of their private bits. That is one bit of communication for a nonlinear function of distributed inputs. Since every Boolean function is a mod-2 sum of AND-terms and XORs are free locally, van Dam showed this bootstraps to computing any distributed function with a single transmitted bit — a triviality that makes physicists doubt PR boxes can exist.

Does information causality actually explain Tsirelson's bound?

It comes remarkably close. Information causality states that the total information Bob can gain about a database Alice holds cannot exceed the number of classical bits she transmits — a natural generalisation of no-signalling to the case where some communication is allowed. Pawlowski and collaborators showed in 2009 that every no-signalling box with $\mathrm{CHSH}>2\sqrt2$ violates this principle, while all boxes with $\mathrm{CHSH}\le2\sqrt2$ respect it, so information causality reproduces the Tsirelson frontier exactly for the CHSH setting. Whether it, or macroscopic locality, or some deeper axiom is the reason quantum theory has the boundary it does remains genuinely open — no single principle has been proven to carve out the full quantum set in all scenarios.