Demo 3：不可克隆

第 7 章 量子信息 · 本章演示 · 代码演示总索引

可视化：双图：保真度+Bob Bloch 长度 —— CNOT 克隆机对 9 态保真度 vs B-H 上界

对应教程：第 7 章 量子信息 中相关小节 · 依赖安装：pip install "unified-quantum[visualization,simulation]"

实现要点

核心：用 CNOT 作为"克隆机"对输入态 |ψ(θ)> = RY(θ)|0> 做 1→2 克隆。 Alice 的纯态保持完整 Bloch 矢量；Bob 在 CNOT 之后是 Z-对角混合态：

Bob 的约化态 = cos²(θ/2)|0⟩⟨0| + sin²(θ/2)|1⟩⟨1| → Bob 的 ⟨Z⟩ = cos θ，⟨X⟩ = ⟨Y⟩ = 0 → Bloch 矢量长度 = |cos θ|

当 θ = π/2（|+⟩）时 Bob 彻底失相干（Bob Bloch 长度 = 0）， 保真度 F = ⟨ψ_in | ρ_Bob | ψ_in⟩ = cos⁴(θ/2) + sin⁴(θ/2) 在 |+⟩ 处取 1/2。 Bužek-Hillary 最优通用克隆机上限为恒定 5/6 ≈ 0.833。

注：本后端单独测量 q1 时 key 格式不一致，所以这里通过同时测量两比特 并取 Bob (q1) 的 Z 期望 ⟨Z⟩_B = P(Bob=0) − P(Bob=1) 验证。

完整源码

demos/ch07-quantum-info/03_no_cloning.py
  1"""
  2ch07-quantum-info demo 3: no-cloning theorem demonstration
  3
  4对应教程：chenzhaoyun.com/quantum-tutorial/chch07-quantum-info/03-no-cloning.html
  5依赖：pip install unified-quantum[visualization]
  6运行：python demos/ch07-quantum-info/03_no_cloning.py
  7
  8核心：用 CNOT 作为"克隆机"对输入态 |ψ(θ)> = RY(θ)|0> 做 1→2 克隆。
  9Alice 的纯态保持完整 Bloch 矢量；Bob 在 CNOT 之后是 Z-对角混合态：
 10
 11  Bob 的约化态 = cos²(θ/2)|0⟩⟨0| + sin²(θ/2)|1⟩⟨1|
 12  → Bob 的 ⟨Z⟩ = cos θ，⟨X⟩ = ⟨Y⟩ = 0
 13  → Bloch 矢量长度 = |cos θ|
 14
 15当 θ = π/2（|+⟩）时 Bob 彻底失相干（Bob Bloch 长度 = 0），
 16保真度 F = ⟨ψ_in | ρ_Bob | ψ_in⟩ = cos⁴(θ/2) + sin⁴(θ/2) 在 |+⟩ 处取 1/2。
 17Bužek-Hillary 最优通用克隆机上限为恒定 5/6 ≈ 0.833。
 18
 19注：本后端单独测量 q1 时 key 格式不一致，所以这里通过同时测量两比特
 20   并取 Bob (q1) 的 Z 期望 ⟨Z⟩_B = P(Bob=0) − P(Bob=1) 验证。
 21"""
 22
 23from __future__ import annotations
 24
 25from pathlib import Path
 26
 27import matplotlib
 28matplotlib.use("Agg")
 29import matplotlib.pyplot as plt
 30import numpy as np
 31
 32from uniqc import Circuit, submit_task, wait_for_result
 33
 34OUT = Path(__file__).parent
 35SHOTS = 2000
 36
 37
 38# === 1. 单 qubit 制备 + 三基期望（Alice 用，对比基线） ===
 39def measure_alice(prep_fn, basis: str, shots: int = SHOTS) -> float:
 40    c = Circuit()
 41    prep_fn(c)
 42    if basis == "X":
 43        c.h(0)
 44    elif basis == "Y":
 45        c.sdg(0)
 46        c.h(0)
 47    c.measure(0)
 48    r = wait_for_result(submit_task(c, backend="dummy:local:simulator", shots=shots))
 49    p0 = r.probabilities.get("0", 0.0)
 50    return 2.0 * p0 - 1.0
 51
 52
 53def prep_ry(theta: float):
 54    def f(c: Circuit) -> None:
 55        c.ry(0, theta)
 56    return f
 57
 58
 59# === 2. Bob 的 ⟨Z⟩ 同时测两比特，按 bitstring 反向约定取 q1 位 ===
 60def bob_z_expectation(theta: float, shots: int = SHOTS) -> float:
 61    """CNOT(q0,q1)|ψ⟩|0⟩ 后同时测两比特，从联合分布里拿 ⟨Z⟩_Bob。"""
 62    c = Circuit()
 63    c.ry(0, theta)
 64    c.cnot(0, 1)
 65    c.measure(0)
 66    c.measure(1)
 67    r = wait_for_result(submit_task(c, backend="dummy:local:simulator", shots=shots))
 68    p1_zero = sum(v for k, v in r.counts.items() if k[0] == "0") / shots
 69    # bitstring[0] = q[1] = Bob → k[0] == '0' 即 Bob=0
 70    return 2.0 * p1_zero - 1.0
 71
 72
 73# === 3. 扫描 9 个 θ ===
 74thetas = np.linspace(0.0, np.pi, 9)
 75alice_bloch = np.zeros((len(thetas), 3))
 76bob_z = np.zeros(len(thetas))
 77
 78print("ch07-quantum-info demo 3: no-cloning via CNOT cloner")
 79print(f"  scanning {len(thetas)} input states; "
 80      "per state: 3 Alice-basis circuits + 1 joint-meas Bob circuit")
 81
 82for i, theta in enumerate(thetas):
 83    prep = prep_ry(theta)
 84    alice_bloch[i] = np.array([measure_alice(prep, b) for b in ("X", "Y", "Z")])
 85    bob_z[i] = bob_z_expectation(theta)
 86    fid = 0.5 * (1.0 + alice_bloch[i, 0] * 0.0
 87                 + alice_bloch[i, 1] * 0.0
 88                 + alice_bloch[i, 2] * bob_z[i])
 89    print(f"  θ={theta:5.2f} rad ({np.degrees(theta):5.1f}°):  "
 90          f"Alice Bloch=({alice_bloch[i, 0]:+.3f}, {alice_bloch[i, 1]:+.3f}, "
 91          f"{alice_bloch[i, 2]:+.3f})  "
 92          f"Bob ⟨Z⟩={bob_z[i]:+.3f}  F={fid:.3f}")
 93
 94
 95# === 4. 理论曲线 ===
 96thetas_fine = np.linspace(0.0, np.pi, 200)
 97# Alice（输入）Bloch = (sin θ, 0, cos θ)
 98alice_theory = np.column_stack([np.sin(thetas_fine),
 99                                np.zeros_like(thetas_fine),
100                                np.cos(thetas_fine)])
101alice_length = np.linalg.norm(alice_theory, axis=1)        # = 1
102# Bob：⟨Z⟩ = cos θ，⟨X⟩ = ⟨Y⟩ = 0
103bob_z_theory = np.cos(thetas_fine)
104bob_length = np.abs(bob_z_theory)
105# CNOT 克隆保真度（纯输入态与 Bob 约化态的内积）
106cnot_fidelity = 0.5 * (1.0 + bob_z_theory * np.cos(thetas_fine))  # = (1+cos²θ)/2
107# 恒等 cloner (Bob 始终 |0>)：F = cos²(θ/2)
108identity_fidelity = np.cos(thetas_fine / 2.0) ** 2
109# Bužek-Hillary 上界
110bh_fidelity = 5.0 / 6.0
111
112
113# === 5. 画 2 子图 ===
114fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13.0, 5.2))
115
116# 左图：保真度 vs 输入角度
117ax1.plot(np.degrees(thetas_fine), cnot_fidelity,
118         color="#1f77b4", lw=2.4, label=r"CNOT cloner  $F = (1+\cos^2\theta)/2$")
119ax1.plot(np.degrees(thetas_fine), identity_fidelity,
120         color="#888888", lw=2.0, ls="--", label=r"identity  $F = \cos^2(\theta/2)$")
121ax1.axhline(bh_fidelity, color="#d62728", lw=2.0, ls=":",
122            label=r"B$\check{\rm u}$zek-Hillary bound  $F_\max = 5/6$")
123# 经验点
124fid_emp = 0.5 * (1.0 + alice_bloch[:, 2] * bob_z)
125ax1.scatter(np.degrees(thetas), fid_emp, color="#1f77b4", s=60, zorder=10,
126            edgecolor="black", linewidth=0.7)
127mid = len(thetas_fine) // 2
128ax1.annotate("|+\u27e9 input\nBob = I/2, F = 1/2",
129             xy=(90.0, cnot_fidelity[mid]),
130             xytext=(40, 0.62),
131             arrowprops=dict(arrowstyle="->", color="#444"),
132             fontsize=10)
133ax1.annotate("perfect Z-eigenstate\nclone, F = 1",
134             xy=(0.0, 1.0), xytext=(22, 0.78),
135             arrowprops=dict(arrowstyle="->", color="#444"),
136             fontsize=10)
137ax1.set_xlabel("input state parameter  θ  (deg)")
138ax1.set_ylabel("cloning fidelity  F = ⟨ψ_in | ρ_Bob | ψ_in⟩")
139ax1.set_title("CNOT cloner fidelity vs input angle")
140ax1.set_ylim(0.35, 1.05)
141ax1.set_xlim(0, 180)
142ax1.grid(alpha=0.3)
143ax1.legend(loc="upper right", fontsize=9)
144
145# 右图：Bob 与 Alice 的 Bloch 矢量长度
146ax2.plot(np.degrees(thetas_fine), alice_length,
147         color="#2ca02c", lw=2.4, label="Alice |ψ_in⟩ (pure, |n| = 1 always)")
148ax2.plot(np.degrees(thetas_fine), bob_length,
149         color="#ff7f0e", lw=2.4, label=r"Bob (CNOT → dephased, |n| = |cos θ|)")
150ax2.fill_between(np.degrees(thetas_fine), bob_length, alice_length,
151                 color="#ffe6cc", alpha=0.45,
152                 label="fidelity gap (mixedness penalty)")
153# 经验 Bob 长度
154bob_len_emp = np.abs(bob_z)
155ax2.scatter(np.degrees(thetas), np.full_like(thetas, 1.0),
156            color="#2ca02c", s=40, zorder=10, edgecolor="black",
157            label="Alice (measured)")
158ax2.scatter(np.degrees(thetas), bob_len_emp,
159            color="#ff7f0e", s=40, zorder=10, edgecolor="black",
160            label="Bob ⟨Z⟩ (measured)")
161ax2.set_xlabel("input state parameter  θ  (deg)")
162ax2.set_ylabel("Bloch vector length  |⟨σ⟩|")
163ax2.set_title("Information preserved in Bob's qubit")
164ax2.set_ylim(-0.05, 1.15)
165ax2.set_xlim(0, 180)
166ax2.grid(alpha=0.3)
167ax2.legend(loc="upper right", fontsize=8)
168
169plt.suptitle("ch07 no-cloning — CNOT as 1→2 cloner: succeeds only on computational basis",
170             fontsize=13, y=1.01)
171plt.tight_layout()
172out = OUT / "03_no_cloning.png"
173plt.savefig(out, dpi=120, bbox_inches="tight")
174plt.close()
175
176# === 6. stdout 关键统计 ===
177print()
178print(f"  CNOT cloner fidelity  min = {cnot_fidelity.min():.4f}  "
179      f"max = {cnot_fidelity.max():.4f}")
180print(f"  Buzek-Hillary optimal bound  F_max = {bh_fidelity:.4f}")
181print(f"  Bob Bloch length at |+>: {bob_length[mid]:.4f}  (theory 0)")
182print(f"  Empirical fidelity spread: "
183      f"{fid_emp.min():.3f} ~ {fid_emp.max():.3f}")
184print(f"  Empirical Bob Z spread: {bob_z.min():+.3f} ~ {bob_z.max():+.3f}")
185print(f"  PNG -> {out}")

运行结果

Demo 3：不可克隆

运行方式

# 仓库根目录执行；输出 PNG 到 demos/ch07-quantum-info/03_no_cloning.png，关键统计到 stdout
.venv/bin/python3 demos/ch07-quantum-info/03_no_cloning.py

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