Entangle

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<title>Entanglement Reference Sheet — E(Δ) = −V·cos Δ</title>
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</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>