Quantum Information and Ternary Qubits
Quantum information theory provides a natural framework for ternary computing. Where classical bits are binary and qubits span a 2-dimensional Hilbert space, qutrits inhabit a 3-dimensional Hilbert space -- the quantum analog of Trinity's balanced ternary. This page documents Bell inequalities, qutrits, entanglement entropy, and their implementation in Trinity.
Source: src/sacred/const.zig (physics, quantum, groups sections)
Bell Inequalities
CHSH Inequality (1969)
For any local hidden-variable theory:
|S| ≤ 2 (classical bound)
For quantum mechanics:
|S| ≤ 2*sqrt(2) = 2.8284... (Tsirelson bound)
Background
Bell's theorem (1964) proves that no local hidden-variable theory can reproduce all predictions of quantum mechanics. The CHSH inequality (Clauser, Horne, Shimony, Holt, 1969) provides an experimentally testable form.
Trinity Implementation
Trinity stores both bounds as compile-time constants:
CHSH_CLASSICAL = 2.0
CHSH_QUANTUM = 2*sqrt(2) = 2.8284271247461903
The quantum violation (2.828 > 2.0) is experimentally confirmed by Aspect et al. (1982) and forms the basis of quantum key distribution and device-independent cryptography.
The Ternary Connection
The violation factor 2*sqrt(2)/2 = sqrt(2) = 1.414... is related to the quantum advantage. For ternary (qutrit) systems, the corresponding violation is even larger, governed by the CGLMP inequality (see below).
References:
- Bell, J. S. "On the Einstein Podolsky Rosen Paradox." Physics 1(3), pp. 195--200, 1964.
- Clauser, J. F. et al. "Proposed Experiment to Test Local Hidden-Variable Theories." Physical Review Letters 23(15), pp. 880--884, 1969.
- Aspect, A. et al. "Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment." Physical Review Letters 49(2), pp. 91--94, 1982.
Qutrits: Ternary Quantum States
Qutrit Definition
A qutrit is a quantum system with three basis states:
|psi> = alpha|0> + beta|1> + gamma|2>
where |alpha|^2 + |beta|^2 + |gamma|^2 = 1.
Comparison with Qubits
| Property | Qubit (d=2) | Qutrit (d=3) |
|---|---|---|
| Basis states | |0>, |1> | |0>, |1>, |2> |
| Hilbert space dimension | 2 | 3 |
| Information per unit | 1 bit | log2(3) = 1.585 bits |
| Maximally mixed state | I/2 | I/3 |
| Entanglement dimension | 2x2 = 4 | 3x3 = 9 |
| Max Bell violation | 2*sqrt(2) = 2.828 | I_3 > 2 (CGLMP) |
Trinity's Balanced Ternary as Qutrit Encoding
Trinity maps the balanced ternary alphabet to qutrit states:
-1 --> |0> (negative trit)
0 --> |1> (zero trit)
+1 --> |2> (positive trit)
A uniform superposition of all three basis states has amplitude 1/sqrt(3) per state:
|psi_uniform> = (1/sqrt(3)) * (|0> + |1> + |2>)
The factor 1/sqrt(3) connects to the Trinity Identity: phi^2 + 1/phi^2 = 3, and the probability per state is 1/3.
Reference: Muthukrishnan, A. and Stroud, C. R. "Multivalued Logic Gates for Quantum Computation." Physical Review A 62(5), 052309, 2000.
CGLMP Inequality
CGLMP Inequality (Collins et al., 2002)
For d-dimensional quantum systems, the generalized Bell inequality is:
I_d ≤ 2 (classical bound)
For d = 3 (qutrits), quantum mechanics predicts:
I_3 = 2.9149... (maximal quantum violation)
The CGLMP inequality generalizes CHSH to higher-dimensional systems. The key result is that qutrits violate Bell inequalities more strongly than qubits:
| Dimension | Max Quantum Value | Classical Bound | Violation Ratio |
|---|---|---|---|
| d = 2 (qubit) | 2.828 | 2 | 1.414 |
| d = 3 (qutrit) | 2.915 | 2 | 1.457 |
| d → infinity | 3.0 | 2 | 1.5 |
The limit value 3.0 as d → infinity connects to the Trinity constant.
Trinity's FPGA implementation computes CGLMP violation values for qutrit Bell tests:
CGLMP I_3 = 2.4277 > 2.0 (classical bound violated)
Note: The theoretical maximum for d = 3 qutrits is I_3^max = 2.9149 (Collins et al., 2002). The value 2.4277 computed here represents the violation achievable with a specific entangled state (not the maximal violation), demonstrating that even non-optimal qutrit states exceed the classical bound.
References:
- Collins, D. et al. "Bell Inequalities for Arbitrarily High-Dimensional Systems." Physical Review Letters 88(4), 040404, 2002.
- Kaszlikowski, D. et al. "Violations of Local Realism by Two Entangled N-Dimensional Systems Are Stronger than for Two Qubits." Physical Review Letters 85(21), pp. 4418--4421, 2000.
Von Neumann Entropy
Von Neumann Entropy
For a quantum state with density matrix rho:
S(rho) = -Tr(rho * ln(rho))
Properties
S(rho) >= 0 (non-negativity)
S(rho) = 0 iff rho is pure (pure states have zero entropy)
S(rho) <= ln(d) (maximized for maximally mixed state)
Entanglement Entropy for Qutrits
For a maximally entangled qutrit pair:
S_max = ln(3) = 1.0986...
Compare with a qubit pair: S_max = ln(2) = 0.6931. The qutrit pair carries 58.5% more entanglement entropy -- the same 58.5% information advantage that appears in Theorem 3 (Ternary Information Density).
Connection to VSA
The Von Neumann entropy of a quantum state is analogous to the Shannon entropy of a ternary vector's component distribution. For a random ternary vector v in {-1, 0, +1}^n with uniform distribution:
H(v_i) = log2(3) = 1.585 bits
This is the classical analog of the quantum information capacity per qutrit.
Reference: Nielsen, M. A. and Chuang, I. L. Quantum Computation and Quantum Information. Cambridge University Press, 10th anniversary edition, 2010.
Ternary Quantum Gates
Phase Gates for Qutrits
The qutrit phase gate applies phase factors to basis states:
Z_3 = diag(1, omega, omega^2)
where omega = exp(2pii/3) is a primitive cube root of unity.
SU(3) Symmetry
The group of qutrit unitary operations is SU(3) -- the same gauge group governing the strong nuclear force (quantum chromodynamics). This triple connection is remarkable:
| Domain | SU(3) Role |
|---|---|
| Particle physics | Color charge symmetry (red, green, blue) |
| Quantum computing | Qutrit gate group |
| Trinity VSA | Symmetry of ternary alphabet transformations |
Trinity implements SU(3) properties:
SU3_CASIMIR = 4/3 = 1.333...
SU3_GOLDEN = 3/(2*phi) = 0.927...
QUARK_COLORS = 3
GENERATIONS = 3
Berry Phase and Geometric Quantum Computation
The Berry phase arises when a quantum state is adiabatically transported around a closed loop in parameter space. For qutrit systems, the geometric phase provides a natural mechanism for fault-tolerant computation because:
- Geometric phases depend only on the path geometry, not the speed
- Ternary geometric gates are inherently more robust than binary ones
- The SU(3) structure provides richer interference patterns
Reference: Muthukrishnan, A. and Stroud, C. R. "Multivalued Logic Gates for Quantum Computation." Physical Review A 62(5), 052309, 2000.
Quantum Advantage of Ternary
Information-Theoretic
| Metric | Binary (d=2) | Ternary (d=3) | Advantage |
|---|---|---|---|
| Information per symbol | 1 bit | 1.585 bits | +58.5% |
| Max entanglement (ln(d)) | 0.693 | 1.099 | +58.5% |
| Bell violation (CGLMP) | 2.828 | 2.915 | +3.1% |
| Gate group dimension | dim(SU(2))=3 | dim(SU(3))=8 | +167% |
Computational
- Grover search: For N items, a qutrit Grover search uses log3(N) qutrits vs log2(N) qubits, achieving the same sqrt(N) speedup with fewer physical units
- Quantum error correction: Ternary codes can correct more errors per physical unit due to higher information density
- Magic state distillation: Qutrit magic states have better distillation rates than qubit ones (Campbell, 2012)
Try It with TRI CLI
tri math quantum # Berry phase gates + geometric phase
tri math qutrit # Ternary phase gates + qutrit state demo
tri math quantum-sim # Quantum simulation with Bell violation
tri math su3 # Full SU(3) simulation with color charges
tri math holo-render # Holographic ASCII renderer (ads|spin|penrose|entropy|hawking)
tri math qg-sim # Quantum gravity time-evolution simulation (spin foam, Regge)
References
- Bell, J. S. "On the Einstein Podolsky Rosen Paradox." Physics 1(3), pp. 195--200, 1964.
- Clauser, J. F., Horne, M. A., Shimony, A., and Holt, R. A. "Proposed Experiment to Test Local Hidden-Variable Theories." Physical Review Letters 23(15), pp. 880--884, 1969.
- Aspect, A., Dalibard, J., and Roger, G. "Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment." Physical Review Letters 49(2), pp. 91--94, 1982.
- Nielsen, M. A. and Chuang, I. L. Quantum Computation and Quantum Information. Cambridge University Press, 10th anniversary edition, 2010.
- Muthukrishnan, A. and Stroud, C. R. "Multivalued Logic Gates for Quantum Computation." Physical Review A 62(5), 052309, 2000.
- Collins, D., Gisin, N., Linden, N., Massar, S., and Popescu, S. "Bell Inequalities for Arbitrarily High-Dimensional Systems." Physical Review Letters 88(4), 040404, 2002.
- Campbell, E. T. "Enhanced Fault-Tolerant Quantum Computing in d-Level Systems." Physical Review Letters 113, 230501, 2014.
- Caves, C. M., Milburn, G. J. "Qutrit Entanglement." Optics Communications 179, pp. 439--446, 2000.
phi^2 + 1/phi^2 = 3 = TRINITY