04 - Quantum Simulation of Causal Diamonds

Introduction

Imagine standing at a crossroads: the future events you can influence form a “light cone”—information propagates at light speed, forming a conical region. Now think in reverse: all past events that can influence you also form a backward light cone. The intersection of these two light cones is a Causal Diamond.

graph TB
    A["Past Light Cone"] --> C["Your<br/>Spacetime Point"]
    B["Future Light Cone"] --> C
    C --> D["Causal Diamond<br/>Diamond = Past ∩ Future"]

    style C fill:#fff4e1
    style D fill:#e8f5e8

In unified time scale theory, causal diamonds are not only fundamental units of spacetime geometry, but also natural stages for information processing and quantum entanglement. This chapter will show how to quantum simulate causal diamonds in the laboratory, verifying their topological structure and zero-mode double cover.

Source theory:

• euler-gls-extend/null-modular-double-cover-causal-diamond-chain.md
• euler-gls-info/14-causal-diamond-chain-null-modular-double-cover.md

Basic Concepts of Causal Diamonds

Geometric Definition

In Minkowski spacetime $(t,x)$ (for simplicity, take 1+1 dimensions), a causal diamond $D$ is bounded by four null boundaries:

$$D=\{(t,x):|t-t_0|+|x-x_0|\le R\}$$

where $(t_0,x_0)$ is the center point, $R$ is the “radius.”

Metaphor:

Like a diamond (rhombus), with four vertices:

• Top vertex (future): $(t_0+R,x_0)$
• Bottom vertex (past): $(t_0-R,x_0)$
• Left/right points: $(t_0,x_0\pm R)$

The four edges are light rays (45-degree lines), information propagates along these edges at light speed.

Double Cover Structure of Null Boundaries

Each boundary can be decomposed into two layers (double cover):

$$\tilde{E}=E^{+}\bigsqcup E^{-}$$

• $E^{+}$: right-moving light rays ($v$ coordinate, $v=t+x$)
• $E^{-}$: left-moving light rays ($u$ coordinate, $u=t-x$)

Physical meaning:

This is not a simple geometric decomposition, but reflects the deep structure of modular theory:

• Modular involution $J$: exchanges $E^{+}\leftrightarrow E^{-}$ and reverses direction
• Modular group ${\Delta }^{it}$: “advancement” along light ray direction

Formula:

The modular Hamiltonian $K_D$ can be written as an integral over two layers:

$$K_D=2\pi \sum _{\sigma =\pm }{\int }_{E^{\sigma }}{g}_{\sigma }(\lambda ,x_{\perp })T_{\sigma \sigma }(\lambda ,x_{\perp })\mathrm{d}\lambda {\mathrm{d}}^{d-2}{x}_{\perp }$$

where:

• $\lambda$: affine parameter (“distance” along light ray)
• $T_{\sigma \sigma }$: null components of stress-energy tensor ($T_{++}=T_{vv}$, $T_{--}=T_{uu}$)
• $g_{\sigma }$: geometric weight function

In CFT spherical diamond, $g_{\sigma }(\lambda )=\lambda (1-\lambda )$ (exact formula!)

Why Quantum Simulation?

Laboratory Cannot Create True Causal Diamonds

Reasons are simple:

1. Spacetime geometry fixed: We live in flat Minkowski spacetime (or weak gravitational field), cannot arbitrarily “sculpt” spacetime shape
2. Light speed limit: Information propagation speed $c$ is fixed, cannot adjust
3. Scale problem: Cosmological-scale causal diamonds (e.g., cosmic horizon) cannot be realized in laboratory

Quantum Simulation Approach

Core idea: Use controllable quantum systems to simulate the algebraic structure and entanglement properties of causal diamonds, rather than directly copying spacetime geometry.

Analogy:

Like using circuits to simulate water flow:

• Voltage $\leftrightarrow$ water pressure
• Current $\leftrightarrow$ water flow rate
• Resistance $\leftrightarrow$ pipe friction

Although physical implementations are completely different, the mathematical relations are the same!

Simulation Goals

Verify the following theoretical predictions:

1. Double-layer entanglement structure: entanglement entropies of $E^{+}$ and $E^{-}$ layers satisfy specific relations
2. Markov property: conditional independence of diamond chain $I(D_{j-1}:D_{j+1}|D_j)=0$
3. ${\mathbb{Z}}_2$ parity invariant: topological index remains stable under parameter changes
4. Zero-mode lifetime: exponential decay of boundary zero modes $\sim e^{-L/\xi }$

Quantum Simulation Platforms

Platform One: Cold Atom Optical Lattice

System: Ultracold atoms (e.g., ${}^{87}$Rb) trapped in optical lattice

How to simulate causal diamond?

graph LR
    A["1D Optical Lattice<br/>L Sites"] --> B["Time Evolution<br/>Hamiltonian H"]
    B --> C["Measure<br/>Local Density ρ_i"]
    C --> D["Reconstruct<br/>Entanglement Entropy S"]

    style A fill:#e1f5ff
    style D fill:#e8f5e8

Mapping relation:

Hamiltonian:

Bose-Hubbard model:

$$H=-J\sum _{⟨i,j⟩}(a_i^{†}{a}_j+\text{h.c.})+\frac{U}{2}\sum _i{n}_i(n_i-1)-\mu \sum _i{n}_i$$

Parameters:

• $J$: hopping strength (simulates “light speed”)
• $U$: on-site interaction
• $\mu$: chemical potential

Protocol:

1. Prepare initial state: Mott insulator state (1 atom per site)
2. Quench evolution: suddenly change $J/U$ ratio, release particles
3. Time slicing: freeze system at different times $t_j$
4. Measure: local density $⟨n_i⟩$, correlation functions $⟨n_i{n}_j⟩$
5. Reconstruct: compute reduced density matrix ${\rho }_A$, extract entanglement entropy $S(A)=-\text{Tr}({\rho }_A\log {\rho }_A)$

Platform Two: Ion Trap Quantum Computer

System: Trapped ions (e.g., ${}^{171}$Yb${}^{+}$) linear array

Advantages:

• High-fidelity gate operations ($>99.9\%$)
• Long coherence time ($\sim$ minutes)
• Arbitrary long-range interactions (via resonant lasers)

How to implement?

graph TB
    A["Ion Chain<br/>N ~ 10-50"] --> B["Laser Control<br/>Single/Dual Ion Gates"]
    B --> C["Time Evolution<br/>Digital Trotter Decomposition"]
    C --> D["Projective Measurement<br/>Fluorescence Detection"]

    style A fill:#e1f5ff
    style D fill:#e8f5e8

Digitalized Hamiltonian:

Decompose continuous evolution $e^{-iHt}$ into discrete gate sequence:

$$e^{-iHt}\approx \prod _{k=1}^M{e}^{-i{H}_k\delta t}$$

Each $H_k$ implemented with 1-2 ion gates (e.g., MS gate, single-ion rotation).

Causal diamond encoding:

Use spatial distribution of ion chain to encode diamond structure:

• Center ion $i_0$: diamond center
• Left/right neighbors $i_0\pm 1,i_0\pm 2,\dots$: diamond interior
• Evolution time $t$: corresponds to diamond “expansion”

Entanglement measurement:

Quantum state tomography reconstructs $\rho$, compute:

$$S(A)=-\text{Tr}({\rho }_A\log {\rho }_A)$$

Or use SWAP test to directly estimate entanglement negativity.

Platform Three: Superconducting Qubits

System: Josephson junction superconducting circuits (e.g., transmon qubits)

Architecture:

2D grid or 1D chain, nearest-neighbor coupling or programmable full connectivity.

Simulation strategy:

Similar to ion trap, but:

• Faster gates ($\sim$ ns)
• Shorter coherence time ($\sim \mu$s)
• Readout via microwave cavity

Special advantage:

Can implement time-reversal operations, verify properties of modular group ${\Delta }^{it}$ and modular involution $J$.

Loschmidt echo:

$$L(t)=|⟨{\psi }_0|e^{+iHt}{e}^{-iHt}|{\psi }_0⟩{|}^2$$

Ideal case $L=1$, decoherence causes $L<1$. Can be used to test Markov property.

Verifying Double-Layer Entanglement Structure

Theoretical Prediction

Causal diamond boundary $E=E^{+}\cup E^{-}$, entanglement entropy satisfies:

$$S(E)=S(E^{+})+S(E^{-})-I(E^{+}:E^{-})$$

where $I(E^{+}:E^{-})$ is mutual information:

$$I(E^{+}:E^{-})=S(E^{+})+S(E^{-})-S(E^{+}\cup E^{-})$$

Double-layer property:

If modular involution $J$ is perfectly symmetric, then $S(E^{+})=S(E^{-})$, and:

$$I(E^{+}:E^{-})=2S(E^{+})-S(E)$$

Experimental Measurement

Cold atom scheme:

1. Define subsystems:

   • $A^{+}=\{i:i\in E^{+}\}$ (sites occupied by right-propagating modes)
   • $A^{-}=\{i:i\in E^{-}\}$ (sites occupied by left-propagating modes)

2. Measure entanglement entropy:

   • Use replica trick or tensor network methods
   • For 1D systems, can efficiently compute using matrix product states (MPS)

3. Extract mutual information:

   • Separately measure $S(A^{+})$, $S(A^{-})$, $S(A^{+}\cup A^{-})$
   • Compute $I=S(A^{+})+S(A^{-})-S(A^{+}\cup A^{-})$

Expected result (CFT):

Near critical point, entanglement entropy scales as:

$$S(A)=\frac{c}{3}\log \left(\frac{L}{ϵ}\right)+s_0$$

where:

• $c$: central charge
• $L$: subsystem size
• $ϵ$: short-distance cutoff (lattice constant)
• $s_0$: non-universal constant

Verify double-layer symmetry:

Test $|S(E^{+})-S(E^{-})|<{\delta }_{\text{tol}}$ (typical $\sim 5\%$)

Conditional Independence of Markov Chain

Theory: Markov Property of Diamond Chain

Consider three adjacent causal diamonds $D_{j-1},D_j,D_{j+1}$:

graph LR
    A["D_j-1"] --> B["D_j"]
    B --> C["D_j+1"]

    style B fill:#fff4e1

Markov property:

$$I(D_{j-1}:D_{j+1}|D_j)=0$$

That is: given the “middle” diamond $D_j$, the “past” $D_{j-1}$ and “future” $D_{j+1}$ are conditionally independent.

Modular Hamiltonian identity:

$$K_{D_{j-1}\cup D_j}+K_{D_j\cup D_{j+1}}-K_{D_j}-K_{D_{j-1}\cup D_j\cup D_{j+1}}=0$$

This is the operator form of conditional independence.

Experimental Verification

Protocol:

1. Construct diamond chain:

   In cold atom or ion trap systems, use time evolution to naturally generate chain structure:

   • $t=0$: prepare initial state at site $i_0$
   • $t=\tau$: evolution forms diamond $D_1$ (radius $\sim J\tau$)
   • $t=2\tau$: expands to $D_2$
   • $t=3\tau$: forms $D_3$

2. Measure three-body conditional mutual information:

$$I(A:C|B)=S(AB)+S(BC)-S(B)-S(ABC)$$

Requires measuring 4 entanglement entropies!

3. Simplified scheme: Petz recovery map

   Theoretical guarantee: Markov property $\iff$ perfect recovery

   $$({\text{id}}_A\otimes {\mathcal{R}}_{B\to BC})({\rho }_{AB})={\rho }_{ABC}$$

   Can indirectly verify via fidelity $F$:

   $$F({\rho }_{ABC},{\rho }_{ABC}^{\text{rec}})>1-\epsilon$$

   where ${\rho }_{ABC}^{\text{rec}}$ is the recovered state.

Numerical simulation expectation:

For free fermion systems (exactly solvable), Markov property strictly holds.

For interacting systems, small deviation $I(A:C|B)\sim e^{-L/\xi }$ ($\xi$: correlation length).

Measurement of ${\mathbb{Z}}_2$ Parity Invariant

Theory: Parity Topological Index

For diamond chain, define ${\mathbb{Z}}_2$ index:

$$\nu (\gamma )=\lfloor \frac{\Theta (\gamma )}{\pi }\rfloor \mathrm{mod}2\in \{0,1\}$$

where $\Theta (\gamma )$ is the chain’s phase accumulation:

$$\Theta (\gamma )=\frac{1}{2}{\int }_{\mathcal{I}(\gamma )}\text{tr }Q(E)h_{\ell }(E-E_0)\mathrm{d}E$$

• $Q(E)=-i{S}^{†}{\partial }_{E}S$: Wigner-Smith matrix
• $h_{\ell }$: window function (e.g., Gaussian)
• $\mathcal{I}(\gamma )$: energy window

Parity property:

When system parameters continuously change, $\nu$ may flip $0\leftrightarrow 1$, but only at critical points (similar to topological phase transition).

${\mathbb{Z}}_2$ robustness:

For small perturbations $\delta \Theta <\pi /2$, $\nu$ remains unchanged!

Experimental Scheme

Ion trap implementation:

1. Prepare diamond chain:

   Use programmable Hamiltonian:

   $$H(\gamma )=\sum _i({\sigma }_i^x{\sigma }_{i+1}^x+\gamma {\sigma }_i^z)$$

   Parameter $\gamma$ tunable (e.g., via magnetic field).

2. Measure scattering matrix:

   Connect chain ends to “leads” (continuous degrees of freedom), measure transmission/reflection.

   In ion trap, use boundary ions as detectors.

3. Extract phase:

   $$\phi (\gamma )=\arg \det S(\gamma )$$

   Use interferometry or quantum state tomography.

4. Scan $\gamma$:

   From ${\gamma }_0$ to ${\gamma }_1$, record change in $\phi (\gamma )$:

   $$\Delta \phi =\phi ({\gamma }_1)-\phi ({\gamma }_0)$$

5. Determine parity:

   $$\nu ({\gamma }_1)-\nu ({\gamma }_0)=\lfloor \frac{\Delta \phi }{\pi }\rfloor \mathrm{mod}2$$

   If $\Delta \phi \approx \pi$ (odd multiple), then $\nu$ flips!

Expected observation:

At topological phase transition point ${\gamma }_c$, $\Delta \phi$ jumps $\pm \pi$, $\nu$ flips.

Example:

Kitaev chain (topological superconductor):

• $\gamma <{\gamma }_c$: trivial phase, $\nu =0$
• $\gamma >{\gamma }_c$: topological phase, $\nu =1$

Majorana zero modes appear on boundaries!

Measurement of Zero-Mode Lifetime

Theory: Zero Modes of Double Cover

Causal diamond boundaries support zero modes—localized states with zero energy.

Double cover structure:

Each zero mode has a “copy” in both $E^{+}$ and $E^{-}$ layers, forming an entangled pair.

Lifetime formula:

Spatial distribution of zero mode:

$$|\psi (x){|}^2\sim e^{-|x-x_0|/\xi }$$

where $\xi$ is the localization length.

For finite-size system $L$, energy splitting of zero mode:

$$\Delta E\sim e^{-L/\xi }$$

Cold Atom Measurement

System: 1D Bose gas, potential barrier applied at boundary

Protocol:

1. Prepare boundary state:

   Use local laser to create potential well at lattice edge:

   $$V(x)=V_0{e}^{-x^2/{\sigma }^2}$$

2. Evolution measurement:

   Monitor oscillation of boundary atom number $N_{\text{edge}}(t)$:

   $$N_{\text{edge}}(t)\approx N_0+A\cos (\Delta E\cdot t/\hslash )$$

3. Extract splitting:

   Fourier transform $N_{\text{edge}}(t)$, peak frequency ${\omega }_0=\Delta E/\hslash$.

4. Fit localization length:

   Change system size $L$, repeat measurement of $\Delta E(L)$.

   Fit:

   $$\log \Delta E=-\frac{L}{\xi }+\text{const}$$

   Slope gives $\xi$!

Expected result:

For topological boundary states (e.g., SSH model), $\xi$ diverges $\Rightarrow$ zero modes perfectly localized.

For non-topological states, $\xi$ is finite, zero modes “leak” into bulk.

Technical Challenges and Solutions

Challenge 1: Decoherence

Problem:

Environmental noise (thermal fluctuations, laser jitter) destroys entanglement, time scale ${\tau }_{\text{dec}}\sim 10-100\mu$s (cold atoms).

Solution:

• Dynamic decoupling: periodic pulses cancel noise
• Quantum error correction codes: encoding protection (e.g., surface code)
• Low temperature environment: $<1\mu$K (optical lattice), $<10$ mK (ion trap)

Challenge 2: Finite-Size Effects

Problem:

Diamond chain length $L_{\text{chain}}$ limited (cold atoms $\sim 100$ sites, ion trap $\sim 50$), far smaller than theoretical infinite chain.

Solution:

• Finite-size scaling: measure multiple $L$, extrapolate to $L\to \infty$
• Periodic boundary conditions: eliminate boundary effects (sacrifice open system properties)
• Matrix product states: numerical simulation guides experimental parameter selection

Challenge 3: Measurement Destruction

Problem:

Projective measurements collapse wavefunction, cannot repeatedly measure same state.

Solution:

• Weak measurement: reduce measurement strength, minimize perturbation
• Quantum non-demolition measurement (QND): measure only conserved quantities (e.g., particle number)
• Integrated imaging: single-shot imaging obtains spatial distribution, repeat preparation for statistics

Challenge 4: Calibration and Systematics

Problem:

Hamiltonian parameters ($J,U,\mu$) drift, causing evolution deviation.

Solution:

• In-situ calibration: use known energy spectrum (e.g., Mott state) to calibrate parameters
• Real-time feedback: monitor characteristic signals (e.g., interference fringes), adjust lasers
• Blind analysis: hide parameter true values, avoid confirmation bias

Summary

This chapter shows how to simulate causal diamonds and their topological properties on quantum platforms:

Key Concepts

1. Causal diamond: intersection of past/future light cones in spacetime
2. Double-layer boundary: $E=E^{+}\cup E^{-}$, carrying modular theory structure
3. Markov chain: conditional independence of adjacent diamonds
4. ${\mathbb{Z}}_2$ index: topological invariant, robust to perturbations

Experimental Platforms

• Cold atom optical lattice: large systems, long evolution time
• Ion trap: high fidelity, arbitrary connectivity
• Superconducting qubits: fast gates, easy integration

Measurement Targets

Expected Precision

• Entanglement entropy: relative error $\sim 10\%$
• Conditional mutual information: absolute error $\sim 0.1$ bits
• ${\mathbb{Z}}_2$ index: completely robust (integer quantity)
• Localization length: $\Delta \xi /\xi \sim 20\%$

The next chapter will turn to cosmological scales, exploring how Fast Radio Burst (FRB) observations verify vacuum polarization effects of the unified time scale.

References

[1] Casini, H., Huerta, M., “Entanglement entropy in free quantum field theory,” J. Phys. A 42, 504007 (2009).

[2] Blanco, D. D., et al., “Relative entropy and holography,” JHEP 08, 060 (2013).

[3] Jafferis, D., et al., “Relative entropy equals bulk relative entropy,” JHEP 06, 004 (2016).

[4] Schollwöck, U., “The density-matrix renormalization group in the age of matrix product states,” Ann. Phys. 326, 96 (2011).

[5] Bloch, I., et al., “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885 (2008).

[6] euler-gls-extend/null-modular-double-cover-causal-diamond-chain.md [7] euler-gls-info/14-causal-diamond-chain-null-modular-double-cover.md