Chapter 13. Leggett-Garg Inequality: Verifying the ‘Classical History’ in Time

We believe that the moon remains suspended in the sky even when we cannot see it. Furthermore, we think that merely looking at the moon does not alter its orbit. In this way, our common-sense worldview that “the world exists independently of observation, and ideal measurements do not disturb the future” is called Macrorealism.

The Leggett-Garg Inequality (LGI) represents the mathematical limitations that this classical intuition must satisfy.

Surprisingly, quantum systems can violate this inequality. This demonstrates that the peculiar nature of quantum mechanics is not confined to space (Bell inequality), but also exists within time (LGI). LGI provides an experimental answer to the question: “Does the system indeed have a clear ‘history’ even when we are not observing it?”

1. Fundamental Concepts

  • Macrorealism (MR): Two key assumptions that constitute our classical intuition.

    1. Macrorealism-per-se: Any physical quantity (e.g., the position of a pendulum) always has a clearly defined value (\(+1\) or \(-1\)) regardless of whether we measure it or not.
    2. Non-Invasive Measurability (NIM): In principle, it is possible to measure the value of a physical quantity without disturbing the system’s state after the measurement.

  • Leggett-Garg Inequality (LGI): If the two classical assumptions above are correct, a theorem states that the correlation between physical quantities measured at different times cannot exceed a specific mathematical boundary (inequality). The most representative form is as follows. \(K_3 = C_{12} + C_{23} - C_{13} \le 1\)

    • Here, \(C_{ij} = \langle Q(t_i) Q(t_j) \rangle\) represents the correlation between measurement values at time \(t_i\) and \(t_j\).

  • Temporal Non-Locality: The phenomenon where quantum systems violate LGI. This implies that quantum systems do not follow a clearly defined ‘history’ (violating MR), or that measurement actions fundamentally disturb future states (violating NIM). This reveals a ‘ghostly connection’ in the temporal history of a single system. ⏰

    💡 Connection Between LGI and Consistent Histories (Chapter 6)

    As learned in Chapter 6, ‘consistent histories’ can only assign classical probabilities to histories when \(D(\alpha, \beta)=0\).

    The ‘Macrorealism (MR)’ assumed by LGI claims that “the system already has the values \((Q_1, Q_2, Q_3)\) at times \(t_1, t_2, t_3\)”. This is a very strong assumption that all eight (\(2^3\)) classical histories are consistent (i.e., have zero interference).

    The violation of LGI experimentally shows that these classical histories do not possess ‘consistency’ in quantum mechanics (\(D(\alpha, \beta) \neq 0\)), thus demonstrating that the concept of a “pre-determined history” does not hold.

  • Clumsiness Loophole: The greatest challenge when experimentally testing LGI violations. When an LGI violation is observed, it is difficult to distinguish whether it is due to a genuine quantum phenomenon or because we have ‘clumsily’ measured, thereby disturbing the system (i.e., violating the NIM assumption). > 🍳 Detailed Explanation: What is the “Sutton’s Trap”? > > Let’s assume that the egg on the pan is in the state where the yolk has not burst (\(Q=+1\)). > > 1. \(t_1\): When pressed slightly with a fork, it did not burst (\(Q(t_1)=+1\)). (First measurement) > 2. \(t_2\): However, the act of pressing caused a crack in the yolk. (Disturbance occurred) > 3. \(t_3\): When pressed again, the yolk burst (\(Q(t_3)=-1\)). > > This experimenter concludes, “At \(t_1\) it was \(+1\), but at \(t_3\) it became \(-1\).” However, this result is not because the egg itself changed, but because the “clumsy measurement” at \(t_1\) contaminated the result at \(t_3\). LGI violation experiments must exclude the possibility of this “clumsy measurement” to truly prove quantum behavior.

  • Trap Avoidance Strategies: To avoid this Sutton’s trap, clever methods that ensure “measured but undisturbed” have been devised.

    • Ideal Negative Result Measurement: A detector that only clicks when the system is in the \(|+1\rangle\) state is used. If no sound is heard, we learn that the system is in the \(|-1\rangle\) state without any interaction.
    • Weak Measurement: The system is measured very weakly, causing minimal disturbance, and instead, the signal is obtained by repeating the measurement countless times statistically.
    • NSIT (No-Signaling-In-Time): It is statistically proven through separate control experiments that the measurement at \(t_1\) does not affect the statistics at \(t_2\).

2. Symbols and Key Relations

  • Observables and Correlation Functions:
    • Measurement value: \(Q(t_k) \in \{+1, -1\}\) (e.g., spin up/down)
    • Two-time correlation function: \(C_{ij} = \langle Q(t_i) Q(t_j) \rangle = \sum_{Q_i, Q_j} Q_i Q_j P(Q_i, Q_j)\)
  • LGI Inequality (K3 Form):
    • Classical assumption (MR + NIM) \(\implies\) \(K_3 = C_{12} + C_{23} - C_{13} \le 1\)

    💡 Why should this inequality hold? (Simple Proof)

    Under the classical assumption (MR), the particle has a value \((Q_1, Q_2, Q_3)\) simultaneously at the three times \(t_1, t_2, t_3\). Each \(Q_k\) is either \(+1\) or \(-1\).

    Let’s calculate \((Q_1 Q_2 + Q_2 Q_3 - Q_1 Q_3)\) for any combination. (e.g., \((+1,+1,+1) \to 1+1-1=1\); \((+1,-1,+1) \to -1-1-1=-3 \le 1\))

    Grouping by \(Q_2\), it becomes \(Q_2(Q_1+Q_3) - Q_1 Q_3\).
    • If \(Q_1=Q_3\) (both \(+1\) or both \(-1\)): \(Q_2(\pm 2) - (\pm 1) = \pm 2 Q_2 - 1\). Whether \(Q_2\) is \(+1\) or \(-1\), this value is either \(1\) or \(-3\).
    • If \(Q_1 \neq Q_3\) (one is \(+1\), one is \(-1\)): \(Q_2(0) - (-1) = 1\).

    That is, for any individual history, \(Q_1 Q_2 + Q_2 Q_3 - Q_1 Q_3 \le 1\) always holds.

    Therefore, the average of these values, \(C_{12} + C_{23} - C_{13}\), must also not exceed \(1\).

  • Quantum Violation:
    • In quantum mechanics, \(Q(t_1)\) and \(Q(t_2)\) may not simultaneously have definite values (MR violation).
    • Additionally, the measurement of \(Q(t_1)\) (e.g., \(\sigma_z\) measurement) can fundamentally change the state of \(Q(t_2)\) (NIM violation).
    • A quantum system can reach up to 1.5 (Lüders Bound), which clearly violates the classical limit of 1.

3. Examples with Deeper Insight

  • Example 1: Classical System (LGI Satisfied)
    • Situation: A classical Markov process randomly oscillating between \(+1\) and \(-1\) (e.g., a process where coin flip results gradually fade).
    • Correlation: \(C_{ij} \approx e^{-\gamma|t_i-t_j|}\) (correlation decreases exponentially as time separates).
    • Result: \(K_3 = e^{-\gamma \tau} + e^{-\gamma \tau} - e^{-2\gamma \tau} = 2e^{-\gamma \tau} - e^{-2\gamma \tau}\). (Here, \(\tau = t_2-t_1 = t_3-t_2\))
    • Interpretation: Let \(x = e^{-\gamma \tau}\), then \(2x - x^2 = 1 - (1-x)^2\). Since \(x\) is between 0 and 1, \((1-x)^2 \ge 0\), and therefore \(K_3 \le 1\). A classical system never violates LGI.
  • Example 2: Quantum Qubit (LGI Violation)
    • Situation: A spin undergoing precession in a magnetic field along the \(x\)-axis.
      • Hamiltonian: \(H = \frac{\hbar\Omega}{2} \sigma_x\) (spin rotates in the \(y-z\) plane at speed \(\Omega\))
      • Observable: \(Q = \sigma_z\) (spin measurement along the \(z\)-axis)
    • Correlation: (Assuming ideal measurements) the expected value of measuring \(\sigma_z\) at \(t_i\), evolving until \(t_j\), and measuring \(\sigma_z\) again is \(C_{ij} = \cos(\Omega(t_j-t_i))\).
    • Result: When measured at equal intervals \(\tau\), \(K_3 = \cos(\Omega\tau) + \cos(\Omega\tau) - \cos(2\Omega\tau) = 2\cos(\Omega\tau) - \cos(2\Omega\tau)\).
    • Violation: If \(\Omega\tau = \pi/3\) (\(60^\circ\) rotation) is set, \(K_3 = 2\cos(60^\circ) - \cos(120^\circ) = 2(0.5) - (-0.5) = 1 + 0.5 = \mathbf{1.5}\). This clearly violates the classical limit of 1.
  • Example 3: Effect of Decoherence (Process of Losing Quantumness)
    • Situation: The qubit from Example 2 interacting with the environment and losing quantum coherence.
    • Correlation: \(C_{ij} \approx e^{-\Gamma|t_i-t_j|} \cos(\Omega(t_j-t_i))\) (correlation additionally damps).
    • Result: The \(K_3\) value becomes smaller than 1.5. If decoherence (\(\Gamma\)) is sufficiently strong, the \(K_3\) value drops below 1, and the system behaves classically again. This illustrates how “decoherence,” learned in Chapter 5 (Open Systems) and Chapter 8 (Quantum Darwinism), suppresses LGI violation (quantumness) and “emerges” classical realism.

4. Exercises

  1. (Proof of LGI \(K_3 \le 1\)): As explained in the 💡 box of Section 2 of the main text, directly verify that for all 8 combinations of \((\pm 1, \pm 1, \pm 1)\) of \((Q_1, Q_2, Q_3)\), the inequality \(Q_1 Q_2 + Q_2 Q_3 - Q_1 Q_3 \le 1\) holds.
  2. (Derivation of Quantum Bit \(K_3\)): Show how \(C_{12} = \cos(\Omega\tau)\) is derived in the quantum bit system of Example 2 using the Heisenberg picture (operator evolution). (Hint: You need to compute \(Q(t) = \sigma_z(t) = e^{iHt/\hbar} \sigma_z(0) e^{-iHt/\hbar}\).)
  3. (Finding Maximum Violation): Differentiate \(K_3(\tau) = 2\cos(\Omega\tau) - \cos(2\Omega\tau)\) with respect to \(\tau\), and show that the condition for \(K_3\) to be maximum is \(\Omega\tau = \pi/3\).
  4. (\(K_4\) Inequality): Prove that under classical assumptions (MR+NIM), for four time points \(t_1 < t_2 < t_3 < t_4\), the inequality \(K_4 = C_{12} + C_{23} + C_{34} - C_{14} \le 2\) holds.
  5. (Design of NSIT Protocol): Explain how to design an NSIT test that compares an experimental protocol obtaining the measurement statistics \(P(Q_2)\) at \(t_2\) with a protocol obtaining the measurement statistics \(P'(Q_2)\) at \(t_2\) after performing a measurement at \(t_1\).

5. Explanation

  1. (Example) \(Q_1=1, Q_2=-1, Q_3=-1\) then \(Q_1Q_2 + Q_2Q_3 - Q_1Q_3 = (1)(-1) + (-1)(-1) - (1)(-1) = -1 + 1 + 1 = 1 \le 1\). The other 7 cases also yield 1 or -1.
  2. (Heisenberg description) \(H=\frac{\hbar\Omega}{2}\sigma_x\). The exponential function \(e^{i\frac{\Omega t}{2}\sigma_x} = \cos(\frac{\Omega t}{2})\mathbf{1} + i\sin(\frac{\Omega t}{2})\sigma_x\).
    \(\sigma_z(t) = e^{iHt/\hbar} \sigma_z(0) e^{-iHt/\hbar} = (\cos + i\sin\sigma_x) \sigma_z (\cos - i\sin\sigma_x)\)
    \(= (\cos\sigma_z - i\sin\sigma_z\sigma_x) (\cos - i\sin\sigma_x) = (\cos\sigma_z + \sin\sigma_y) (\cos - i\sin\sigma_x)\)
    \(= \cos^2\sigma_z + \sin\cos\sigma_y - i\cos\sin\sigma_z\sigma_x - i\sin^2\sigma_y\sigma_x\)
    \(= \cos^2\sigma_z + \sin\cos\sigma_y - i\sin\cos(i\sigma_y) - \sin^2(-i\sigma_z) = \cos^2\sigma_z + 2\sin\cos\sigma_y + \sin^2\sigma_z = \cos(2\frac{\Omega t}{2})\sigma_z + \sin(2\frac{\Omega t}{2})\sigma_y = \cos(\Omega t)\sigma_z + \sin(\Omega t)\sigma_y\).
    \(C_{12} = \langle \sigma_z(\tau) \sigma_z(0) \rangle = \langle (\cos(\Omega\tau)\sigma_z + \sin(\Omega\tau)\sigma_y) \sigma_z \rangle = \cos(\Omega\tau) \langle \sigma_z^2 \rangle + \sin(\Omega\tau) \langle \sigma_y \sigma_z \rangle\).
    \(\langle \sigma_z^2 \rangle = \langle \mathbf{1} \rangle = 1\) and (assuming the initial state \(\rho_0\) is symmetric about the \(z\)-axis) \(\langle \sigma_y \sigma_z \rangle = \langle i\sigma_x \rangle = 0\). Therefore, \(C_{12} = \cos(\Omega\tau)\). \(C_{13}\) is identical by replacing \(\tau \to 2\tau\).
  3. \(dK_3/d\tau = -2\Omega\sin(\Omega\tau) + 2\Omega\sin(2\Omega\tau) = 0\). \(\sin(2\Omega\tau) = \sin(\Omega\tau) \implies 2\sin(\Omega\tau)\cos(\Omega\tau) = \sin(\Omega\tau)\). \(\sin(\Omega\tau)=0\) (minimum) or \(\cos(\Omega\tau)=1/2\). When \(\cos(\Omega\tau)=1/2\), \(\Omega\tau = \pi/3\), and at this point \(K_3=1.5\) is maximum.
  4. \(K_4 = Q_1Q_2 + Q_2Q_3 + Q_3Q_4 - Q_1Q_4\).
    It can be grouped as \(Q_2(Q_1+Q_3) + Q_4(Q_3-Q_1)\). Checking all possible combinations (16 cases), the maximum is 2. (Example: \(1,1,1,-1 \to 1+1-1-(-1)=2\)).
  5. (Protocol A) Prepare many identical systems and perform measurements only at \(t_2\), obtaining \(P(Q_2)\). (Protocol B) Prepare identical systems and perform a measurement at \(t_1\) (whether recording the result or not), and also perform a measurement at \(t_2\) to obtain \(P'(Q_2)\). If \(P(Q_2)\) and \(P'(Q_2)\) show a statistically significant difference, the measurement at \(t_1\) has disturbed the statistics at \(t_2\), indicating that NSIT has failed (i.e., Sutum’s trap remains open).
  6. (Example) If \(Q_1=1, Q_2=-1, Q_3=-1\), then \(Q_1Q_2 + Q_2Q_3 - Q_1Q_3 = (1)(-1) + (-1)(-1) - (1)(-1) = -1 + 1 + 1 = 1 \le 1\). All other 7 cases also result in 1 or -1. 2. (Heisenberg description) \(H=\frac{\hbar\Omega}{2}\sigma_x\). Exponential function \(e^{i\frac{\Omega t}{2}\sigma_x} = \cos(\frac{\Omega t}{2})\mathbf{1} + i\sin(\frac{\Omega t}{2})\sigma_x\). \(\sigma_z(t) = e^{iHt/\hbar} \sigma_z(0) e^{-iHt/\hbar} = (\cos + i\sin\sigma_x) \sigma_z (\cos - i\sin\sigma_x)\) \(= (\cos\sigma_z - i\sin\sigma_z\sigma_x) (\cos - i\sin\sigma_x) = (\cos\sigma_z + \sin\sigma_y) (\cos - i\sin\sigma_x)\) \(= \cos^2\sigma_z + \sin\cos\sigma_y - i\cos\sin\sigma_z\sigma_x - i\sin^2\sigma_y\sigma_x\) \(= \cos^2\sigma_z + \sin\cos\sigma_y - i\sin\cos(i\sigma_y) - \sin^2(-i\sigma_z) = \cos^2\sigma_z + 2\sin\cos\sigma_y + \sin^2\sigma_z = \cos(2\frac{\Omega t}{2})\sigma_z + \sin(2\frac{\Omega t}{2})\sigma_y = \cos(\Omega t)\sigma_z + \sin(\Omega t)\sigma_y\). \(C_{12} = \langle \sigma_z(\tau) \sigma_z(0) \rangle = \langle (\cos(\Omega\tau)\sigma_z + \sin(\Omega\tau)\sigma_y) \sigma_z \rangle = \cos(\Omega\tau) \langle \sigma_z^2 \rangle + \sin(\Omega\tau) \langle \sigma_y \sigma_z \rangle\). \(\langle \sigma_z^2 \rangle = \langle \mathbf{1} \rangle = 1\) and, (assuming the initial state \(\rho_0\) is symmetric around the z-axis) \(\langle \sigma_y \sigma_z \rangle = \langle i\sigma_x \rangle = 0\). Therefore \(C_{12} = \cos(\Omega\tau)\). \(C_{13}\) is the same with \(\tau \to 2\tau\). 3. \(dK_3/d\tau = -2\Omega\sin(\Omega\tau) + 2\Omega\sin(2\Omega\tau) = 0\). \(\sin(2\Omega\tau) = \sin(\Omega\tau) \implies 2\sin(\Omega\tau)\cos(\Omega\tau) = \sin(\Omega\tau)\). \(\sin(\Omega\tau)=0\) (minimum) or \(\cos(\Omega\tau)=1/2\). When \(\cos(\Omega\tau)=1/2\), \(\Omega\tau = \pi/3\), and at this point \(K_3=1.5\) is maximum. 4. \(K_4 = Q_1Q_2 + Q_2Q_3 + Q_3Q_4 - Q_1Q_4\). It can be grouped as \(Q_2(Q_1+Q_3) + Q_4(Q_3-Q_1)\). Checking all possible combinations (16 cases), the maximum is 2. (Example: \(1,1,1,-1 \to 1+1-1-(-1)=2\)). 5. (Protocol A) Prepare many identical systems and perform measurements only at \(t_2\) to obtain \(P(Q_2)\). (Protocol B) Prepare the same system and perform measurements at \(t_1\) (recording the results or not), and perform measurements at \(t_2\) to obtain \(P'(Q_2)\). If \(P(Q_2)\) and \(P'(Q_2)\) show a statistically significant difference, it means that the measurement at \(t_1\) disturbed the statistics at \(t_2\), indicating that NSIT failed (i.e., the trap was not closed).