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| <html lang="en"> | |
| <head> | |
| <meta charset="UTF-8" /> | |
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| <title>Entanglement Reference Sheet — E(Δ) = −V·cos Δ</title> | |
| <style> | |
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| header{padding:22px 16px;text-align:center} | |
| header h1{margin:0;font-size:1.4rem} | |
| header p{margin:6px 0 0;color:var(--muted)} | |
| main{max-width:1100px;margin:0 auto;padding:16px;display:grid;gap:16px} | |
| @media(min-width:960px){main{grid-template-columns:360px 1fr}} | |
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| h2{margin:0 0 10px;font-size:1.05rem} | |
| label{display:block;margin:10px 0 4px;color:var(--muted)} | |
| input[type=range]{width:100%} | |
| input[type=number]{width:100%;padding:8px;border-radius:8px;border:1px solid #1f2937;background:#0a1220;color:var(--ink)} | |
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| .btn.primary{background:var(--accent);border-color:#0891b2;color:#001018;font-weight:700} | |
| .kpi{display:grid;grid-template-columns:repeat(3,1fr);gap:10px} | |
| .kpi .box{background:#0a1220;border:1px solid #1f2937;border-radius:12px;padding:10px;text-align:center} | |
| .val{font-size:1.05rem;font-weight:700;color:var(--accent)} | |
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| canvas{width:100%;height:240px;background:#07101d;border:1px solid #1f2937;border-radius:12px} | |
| .ok{color:var(--ok)} .warn{color:var(--warn)} .err{color:var(--err)} | |
| table{width:100%;border-collapse:collapse} | |
| th,td{border:1px solid #1f2937;padding:8px;text-align:center} | |
| th{background:#0a1220;color:#cbd5e1} | |
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| code{background:#091324;padding:2px 6px;border-radius:6px;border:1px solid #1f2937} | |
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| .badge.warn{background:rgba(245,158,11,0.15);color:var(--warn)} | |
| .badge.err{background:rgba(239,68,68,0.15);color:var(--err)} | |
| </style> | |
| </head> | |
| <body> | |
| <header> | |
| <h1>Entanglement Reference Sheet — <code>E(Δ) = − V·cos Δ</code></h1> | |
| <p>No-signaling + rotational symmetry + information causality → quantum (Tsirelson) correlations</p> | |
| </header> | |
| <main> | |
| <!-- Controls --> | |
| <section class="card"> | |
| <h2>🎛 Controls</h2> | |
| <label>Visibility V (0…1): <span id="Vlab">1.00</span></label> | |
| <input type="range" id="V" min="0" max="1" step="0.01" value="1"> | |
| <div class="row"> | |
| <div style="flex:1"> | |
| <label>A0 (°): <span id="A0lab">0.0</span></label> | |
| <input type="range" id="A0" min="0" max="180" step="0.1" value="0"> | |
| </div> | |
| <div style="flex:1"> | |
| <label>A1 (°): <span id="A1lab">45.0</span></label> | |
| <input type="range" id="A1" min="0" max="180" step="0.1" value="45"> | |
| </div> | |
| </div> | |
| <div class="row"> | |
| <div style="flex:1"> | |
| <label>B0 (°): <span id="B0lab">22.5</span></label> | |
| <input type="range" id="B0" min="0" max="180" step="0.1" value="22.5"> | |
| </div> | |
| <div style="flex:1"> | |
| <label>B1 (°): <span id="B1lab">67.5</span></label> | |
| <input type="range" id="B1" min="0" max="180" step="0.1" value="67.5"> | |
| </div> | |
| </div> | |
| <div class="row" style="margin-top:10px"> | |
| <button class="btn" id="setOpt">Set CHSH-optimal angles</button> | |
| <button class="btn primary" id="recalc">Recalculate</button> | |
| <span class="pill">Δ = <span id="DeltaLab">A0–B0 = 22.5°</span></span> | |
| <span id="violBadge" class="badge warn">status</span> | |
| </div> | |
| <div class="kpi" style="margin-top:10px"> | |
| <div class="box"> | |
| <div class="small">Correlation E(A0,B0)</div> | |
| <div id="E00" class="val">–</div> | |
| </div> | |
| <div class="box"> | |
| <div class="small">S (CHSH)</div> | |
| <div id="Sval" class="val">–</div> | |
| </div> | |
| <div class="box"> | |
| <div class="small">CHSH game win prob</div> | |
| <div id="pwin" class="val">–</div> | |
| </div> | |
| </div> | |
| <div class="row" style="margin-top:10px"> | |
| <span class="pill">Regime: <b id="regime">–</b></span> | |
| <span class="pill">Local ≤ 2 • Quantum ≤ 2√2 • PR = 4</span> | |
| </div> | |
| </section> | |
| <!-- Results & Plots --> | |
| <section class="card"> | |
| <h2>📊 Joint Distribution & Fisher Information</h2> | |
| <div class="row"> | |
| <div style="flex:1"> | |
| <label>Single-pair angle Δ (°) for E & P(a,b): <span id="Dlab">22.5</span></label> | |
| <input type="range" id="D" min="0" max="180" step="0.1" value="22.5"> | |
| </div> | |
| <div style="width:180px"> | |
| <label>Δ direct input (°)</label> | |
| <input type="number" id="Dnum" min="0" max="180" step="0.1" value="22.5"> | |
| </div> | |
| </div> | |
| <div class="kpi" style="margin-top:10px"> | |
| <div class="box"> | |
| <div class="small">E(Δ)</div> | |
| <div id="Edelta" class="val">–</div> | |
| </div> | |
| <div class="box"> | |
| <div class="small">Fisher 𝓕(Δ)</div> | |
| <div id="Fdelta" class="val">–</div> | |
| </div> | |
| <div class="box"> | |
| <div class="small">Max 𝓕 over Δ</div> | |
| <div id="Fmax" class="val">–</div> | |
| </div> | |
| </div> | |
| <canvas id="bars"></canvas> | |
| <div class="small">Bars show joint probabilities P(++), P(+−), P(−+), P(−−) with fair marginals (no-signaling). Computed via <code>P(a,b)=¼(1+a b·E)</code>. Marginals: P(a=+1)=<span id="PaPlus">0.5000</span>, P(b=+1)=<span id="PbPlus">0.5000</span>.</div> | |
| </section> | |
| <!-- Monogamy --> | |
| <section class="card"> | |
| <h2>🔗 Monogamy Check (Toner–Verstraete)</h2> | |
| <div class="row"> | |
| <div style="flex:1"> | |
| <label>Enter candidate S_AC</label> | |
| <input type="number" id="SAC" value="2.0" min="0" max="4" step="0.01"> | |
| </div> | |
| <div style="flex:1;display:flex;align-items:flex-end"> | |
| <button class="btn" id="monoCheck">Check S_AB² + S_AC² ≤ 8</button> | |
| </div> | |
| </div> | |
| <div class="row" style="margin-top:10px"> | |
| <span class="pill">S_AB = <b id="SABpill">–</b></span> | |
| <span class="pill">S_AC = <b id="SACpill">–</b></span> | |
| <span class="pill">Sum = <b id="sumMono">–</b></span> | |
| <span class="pill">Status: <b id="monoStatus">–</b></span> | |
| </div> | |
| </section> | |
| <!-- Comparison --> | |
| <section class="card"> | |
| <h2>📚 Comparison Table</h2> | |
| <table> | |
| <thead> | |
| <tr><th>Model</th><th>Max CHSH S</th><th>Correlation form</th><th>Signaling?</th></tr> | |
| </thead> | |
| <tbody> | |
| <tr><td>Local Hidden Variables</td><td>≤ 2</td><td>Piecewise linear</td><td>No</td></tr> | |
| <tr><td>Quantum (this work)</td><td>≤ 2√2 ≈ 2.828</td><td><code>E(Δ)=−V cos Δ</code></td><td>No</td></tr> | |
| <tr><td>PR-Box (supra-quantum)</td><td>4</td><td>±1 (discontinuous)</td><td>No</td></tr> | |
| </tbody> | |
| </table> | |
| <p class="small" style="margin-top:8px">Information causality rules out PR-box correlations; rotational symmetry fixes the cosine form; maximum entropy selects the unique joint distribution.</p> | |
| </section> | |
| </main> | |
| <footer> | |
| Interactive reference for entanglement correlations • © 2026 Quantum Hub • | |
| </footer> | |
| <script> | |
| const V = document.getElementById('V'); | |
| const Vlab = document.getElementById('Vlab'); | |
| const A0 = document.getElementById('A0'), A1 = document.getElementById('A1'), | |
| B0 = document.getElementById('B0'), B1 = document.getElementById('B1'); | |
| const A0lab = document.getElementById('A0lab'), A1lab = document.getElementById('A1lab'), | |
| B0lab = document.getElementById('B0lab'), B1lab = document.getElementById('B1lab'); | |
| const E00 = document.getElementById('E00'), Sval = document.getElementById('Sval'), pwin = document.getElementById('pwin'); | |
| const regime = document.getElementById('regime'); | |
| const DeltaLab = document.getElementById('DeltaLab'); | |
| const violBadge = document.getElementById('violBadge'); | |
| const D = document.getElementById('D'), Dnum = document.getElementById('Dnum'); | |
| const Dlab = document.getElementById('Dlab'); | |
| const Edelta = document.getElementById('Edelta'), Fdelta = document.getElementById('Fdelta'), Fmax = document.getElementById('Fmax'); | |
| const PaPlus = document.getElementById('PaPlus'), PbPlus = document.getElementById('PbPlus'); | |
| const bars = document.getElementById('bars'); | |
| const setOpt = document.getElementById('setOpt'), recalc = document.getElementById('recalc'); | |
| const SACin = document.getElementById('SAC'), SABpill = document.getElementById('SABpill'), | |
| SACpill = document.getElementById('SACpill'), sumMono = document.getElementById('sumMono'), monoStatus = document.getElementById('monoStatus'); | |
| function deg2rad(x){ return x*Math.PI/180; } | |
| function cosd(x){ return Math.cos(deg2rad(x)); } | |
| function sind(x){ return Math.sin(deg2rad(x)); } | |
| function wrap180(d){ return ((d % 180) + 180) % 180; } | |
| // correlation law | |
| function E_of(V, ddeg){ return - V * cosd(wrap180(ddeg)); } | |
| // CHSH | |
| function S_from(V, a0,a1,b0,b1){ | |
| const E00 = E_of(V, a0-b0); | |
| const E01 = E_of(V, a0-b1); | |
| const E10 = E_of(V, a1-b0); | |
| const E11 = E_of(V, a1-b1); | |
| const S = Math.abs(E00 + E01 + E10 - E11); | |
| return {S,E00,E01,E10,E11}; | |
| } | |
| // CHSH game win probability | |
| function pwin_from_S(S){ return 0.5 + S/8; } | |
| // joint distribution with fair marginals: P(a,b) = 1/4 (1 + a b E); here E = -V cos Δ | |
| function joint_from_E(E){ | |
| const Ppp = 0.25*(1 + E); | |
| const Ppm = 0.25*(1 - E); | |
| const Pmp = 0.25*(1 - E); | |
| const Pmm = 0.25*(1 + E); | |
| return [Ppp,Ppm,Pmp,Pmm]; | |
| } | |
| // draw bars | |
| function drawBars(probs){ | |
| const ctx = bars.getContext('2d'); | |
| const w = bars.width = bars.clientWidth*2, h = bars.height = bars.clientHeight*2; | |
| ctx.clearRect(0,0,w,h); | |
| ctx.fillStyle="#0a1220"; ctx.fillRect(0,0,w,h); | |
| ctx.strokeStyle="#1f2937"; ctx.strokeRect(0,0,w,h); | |
| const labels = ["++","+−","−+","−−"]; | |
| const maxH = h*0.7, barW = w*0.15, gap = w*0.06; let x = w*0.12; | |
| for(let i=0;i<4;i++){ | |
| const ph = Math.max(0,probs[i])*maxH; | |
| ctx.fillStyle = i%2? "#22d3ee" : "#38bdf8"; | |
| ctx.fillRect(x, h-40-ph, barW, ph); | |
| ctx.fillStyle="#cbd5e1"; ctx.font="28px system-ui"; ctx.fillText(labels[i], x+barW*0.22, h-10); | |
| ctx.fillStyle="#94a3b8"; ctx.font="22px system-ui"; ctx.fillText(probs[i].toFixed(4), x+8, h-50-ph); | |
| x += barW + gap; | |
| } | |
| } | |
| // Fisher information (Δ in radians physically, but we input degrees; derivative uses sin(Δ) correctly) | |
| function fisher(V, ddeg){ | |
| const mu = -V * cosd(ddeg); | |
| const dmu = V * sind(ddeg); // dμ/dΔ(rad) = V sin(Δ) | |
| const denom = 1 - mu*mu; | |
| if(denom <= 1e-12) return NaN; | |
| return (dmu*dmu)/denom; | |
| } | |
| function fisher_max(V){ | |
| let bestF = 0, bestD = 90; | |
| for(let d=0; d<=180; d+=0.1){ | |
| const F = fisher(V, d); | |
| if(!isNaN(F) && F > bestF){ bestF = F; bestD = d; } | |
| } | |
| return {F:bestF, D:bestD}; | |
| } | |
| function classify_regime(S){ | |
| const eps=1e-6, ts = 2*Math.SQRT2; | |
| if(S <= 2+eps) return {txt:"local (≤2)", cls:"ok"}; | |
| if(S <= ts+eps) return {txt:"quantum-admissible (≤2√2)", cls:"warn"}; | |
| return {txt:"super-quantum (violates IC)", cls:"err"}; | |
| } | |
| function updateBadge(S, Vv){ | |
| // Use S for current geometry and also V-threshold info | |
| if(S > 2 + 1e-6){ violBadge.textContent = "violates CHSH"; violBadge.className = "badge ok"; return; } | |
| if(Math.abs(S-2) <= 1e-6){ violBadge.textContent = "saturates local bound"; violBadge.className = "badge warn"; return; } | |
| violBadge.textContent = "no violation"; violBadge.className = "badge err"; | |
| } | |
| function updateMarginals(E){ | |
| // From joint_from_E | |
| const [Ppp,Ppm,Pmp,Pmm] = joint_from_E(E); | |
| const Pa = Ppp + Ppm; // a=+1 | |
| const Pb = Ppp + Pmp; // b=+1 | |
| PaPlus.textContent = Pa.toFixed(4); | |
| PbPlus.textContent = Pb.toFixed(4); | |
| } | |
| function updateAll(){ | |
| Vlab.textContent = Number(V.value).toFixed(2); | |
| A0lab.textContent = Number(A0.value).toFixed(1); | |
| A1lab.textContent = Number(A1.value).toFixed(1); | |
| B0lab.textContent = Number(B0.value).toFixed(1); | |
| B1lab.textContent = Number(B1.value).toFixed(1); | |
| const Vv = Number(V.value); | |
| const a0 = Number(A0.value), a1 = Number(A1.value), b0 = Number(B0.value), b1 = Number(B1.value); | |
| DeltaLab.textContent = `A0–B0 = ${wrap180(a0-b0).toFixed(1)}°`; | |
| const {S,E00:EE00,E01,E10,E11} = S_from(Vv, a0,a1,b0,b1); | |
| E00.textContent = EE00.toFixed(4); | |
| Sval.textContent = S.toFixed(4); | |
| pwin.textContent = pwin_from_S(S).toFixed(4); | |
| updateBadge(S, Vv); | |
| const reg = classify_regime(S); | |
| regime.textContent = reg.txt; | |
| regime.className = reg.cls; | |
| // single Δ block | |
| const Dv = Number(D.value); | |
| Dlab.textContent = D.value; | |
| Dnum.value = D.value; | |
| const EΔ = E_of(Vv, Dv); | |
| Edelta.textContent = EΔ.toFixed(6); | |
| updateMarginals(EΔ); | |
| const FΔ = fisher(Vv, Dv); | |
| Fdelta.textContent = isNaN(FΔ)? "undefined" : FΔ.toFixed(6); | |
| const Fm = fisher_max(Vv); | |
| Fmax.textContent = `${Fm.F.toFixed(6)} @ Δ≈${Fm.D.toFixed(1)}°`; | |
| drawBars(joint_from_E(EΔ)); | |
| SABpill.textContent = S.toFixed(4); | |
| } | |
| function monoCheck(){ | |
| const SAB = Number(Sval.textContent); | |
| const SAC = Number(SACin.value); | |
| SACpill.textContent = SAC.toFixed(4); | |
| const sum = SAB*SAB + SAC*SAC; | |
| sumMono.textContent = sum.toFixed(4); | |
| const ok = sum <= 8 + 1e-6; | |
| monoStatus.textContent = ok ? "satisfied" : "violated"; | |
| monoStatus.className = ok ? "ok" : "err"; | |
| } | |
| function setCHSHopt(){ | |
| A0.value = 0; A1.value = 45; B0.value = 22.5; B1.value = 67.5; | |
| updateAll(); | |
| } | |
| // Sync Δ number/range | |
| D.addEventListener('input', ()=>{ Dlab.textContent=D.value; Dnum.value=D.value; updateAll(); }); | |
| Dnum.addEventListener('input', ()=>{ let v=Number(Dnum.value); if(isNaN(v)) v=0; v=Math.max(0,Math.min(180,v)); D.value=v; Dlab.textContent=D.value; updateAll(); }); | |
| [V,A0,A1,B0,B1].forEach(el=>el.addEventListener('input', updateAll)); | |
| document.getElementById('recalc').addEventListener('click', updateAll); | |
| document.getElementById('monoCheck').addEventListener('click', monoCheck); | |
| document.getElementById('setOpt').addEventListener('click', setCHSHopt); | |
| // init | |
| updateAll(); | |
| </script> | |
| </body> | |
| </html> |