Dataset Viewer
id
string | domain
string | title
string | physics_level
string | problem_type
string | natural_language_problem
string | lean_problem_statement
string | lean_answer_key
string | solution_approach
string | key_concepts
sequence | has_error
bool | error_info
dict | format
string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
quantum_state_normalization_001_original
|
quantum_states
|
Pure Quantum State Normalization
|
graduate
|
original
|
**Problem 1: Pure Quantum State Normalization**
Consider a qubit in the pure state |ψ⟩ = α|0⟩ + β|1⟩.
**Part (a)**: What constraint must the amplitudes α and β satisfy for this to be a valid quantum state?
**Part (b)**: Prove that for any complex numbers α and β satisfying this constraint, we have |α|² ≤ 1.
**Part (c)**: Explain the physical interpretation of |α|² and |β|².
|
-- Quantum state normalization
theorem quantum_state_constraint (α β : ℂ)
(h : Complex.normSq α + Complex.normSq β = 1) :
Complex.normSq α ≤ 1 := by
sorry
|
import Mathlib.Data.Complex.Basic
theorem complex_norm_nonneg (z : ℂ) : 0 ≤ Complex.normSq z := by
exact Complex.normSq_nonneg z
|
1. Use the normalization condition |α|² + |β|² = 1
2. Apply Complex.normSq_nonneg to show |β|² ≥ 0
3. Use linear arithmetic to conclude |α|² ≤ 1
|
[
"quantum states",
"normalization",
"probability amplitudes",
"complex numbers"
] | false | null |
complete_with_answer_keys
|
quantum_state_normalization_001_normalization_violation
|
quantum_states
|
Pure Quantum State Normalization
|
graduate
|
corrupted
|
**Problem 1: Pure Quantum State Normalization**
Consider a qubit in the pure state |ψ⟩ = α|0⟩ + β|1⟩.
**Part (a)**: What constraint must the amplitudes α and β satisfy for this to be a valid quantum state?
**Part (b)**: Prove that for any complex numbers α and β satisfying this constraint, we have |α|² ≤ 1.
**Part (c)**: Explain the physical interpretation of |α|² and |β|².
|
-- Quantum state normalization
theorem quantum_state_constraint (α β : ℂ)
(h : Complex.normSq α + Complex.normSq β = 1) :
Complex.normSq α ≤ 1 := by
sorry
|
import Mathlib.Data.Complex.Basic
theorem complex_norm_nonneg (z : ℂ) : 0 ≤ Complex.normSq z := by
exact Complex.normSq_nonneg z
|
1. Use the normalization condition |α|² + |β|² = 1
2. Apply Complex.normSq_nonneg to show |β|² ≥ 0
3. Use linear arithmetic to conclude |α|² ≤ 1
|
[
"quantum states",
"normalization",
"probability amplitudes",
"complex numbers"
] | true |
{
"corrupted_token": "|α|² + |β|² = 2",
"error_explanation": "Quantum states must be normalized: |α|² + |β|² = 1, not 2. This violates the fundamental requirement that probability amplitudes sum to unity.",
"error_subtype": "normalization_violation",
"error_type": "physics_error",
"original_token": "|α|² + |β|² = 1"
}
|
complete_with_answer_keys
|
pauli_matrix_algebra_002_original
|
quantum_operators
|
Pauli Matrix Algebraic Properties
|
graduate
|
original
|
**Problem 2: Pauli Matrix Algebra**
The Pauli X matrix is σₓ = |0⟩⟨1| + |1⟩⟨0| = [[0,1],[1,0]].
**Part (a)**: Prove that σₓ² = I (Pauli X is involutory).
**Part (b)**: Show that σₓ is Hermitian: σₓ† = σₓ.
**Part (c)**: Explain why Hermiticity is required for quantum observables.
|
-- Pauli X matrix properties
def pauli_x : Matrix (Fin 2) (Fin 2) ℂ := !![0, 1; 1, 0]
theorem pauli_x_involutory : pauli_x * pauli_x = 1 := by
sorry
|
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Matrix.Basic
def pauli_x : Matrix (Fin 2) (Fin 2) ℂ := !![0, 1; 1, 0]
theorem pauli_x_involutory : pauli_x * pauli_x = 1 := by
ext i j
simp [pauli_x, Matrix.mul_apply]
fin_cases i <;> fin_cases j <;> simp
|
1. Define the Pauli X matrix explicitly
2. Compute the matrix product using ext and fin_cases
3. Verify each matrix element equals the identity
|
[
"Pauli matrices",
"involutory operators",
"Hermitian operators",
"quantum observables"
] | false | null |
complete_with_answer_keys
|
pauli_matrix_algebra_002_eigenvalue_error
|
quantum_operators
|
Pauli Matrix Algebraic Properties
|
graduate
|
corrupted
|
**Problem 2: Pauli Matrix Algebra**
The Pauli X matrix is σₓ = |0⟩⟨1| + |1⟩⟨0| = [[0,1],[1,0]].
**Part (a)**: Prove that σₓ² = 0 (Pauli X is involutory).
**Part (b)**: Show that σₓ is Hermitian: σₓ† = σₓ.
**Part (c)**: Explain why Hermiticity is required for quantum observables.
|
-- Pauli X matrix properties
def pauli_x : Matrix (Fin 2) (Fin 2) ℂ := !![0, 1; 1, 0]
theorem pauli_x_involutory : pauli_x * pauli_x = 1 := by
sorry
|
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Matrix.Basic
def pauli_x : Matrix (Fin 2) (Fin 2) ℂ := !![0, 1; 1, 0]
theorem pauli_x_involutory : pauli_x * pauli_x = 1 := by
ext i j
simp [pauli_x, Matrix.mul_apply]
fin_cases i <;> fin_cases j <;> simp
|
1. Define the Pauli X matrix explicitly
2. Compute the matrix product using ext and fin_cases
3. Verify each matrix element equals the identity
|
[
"Pauli matrices",
"involutory operators",
"Hermitian operators",
"quantum observables"
] | true |
{
"corrupted_token": "σₓ² = 0",
"error_explanation": "Pauli matrices are involutory (σᵢ² = I), not nilpotent (σᵢ² = 0). This confuses fundamental matrix properties.",
"error_subtype": "eigenvalue_error",
"error_type": "algebra_error",
"original_token": "σₓ² = I"
}
|
complete_with_answer_keys
|
measurement_probability_003_original
|
quantum_measurements
|
Quantum Measurement Probabilities
|
graduate
|
original
|
**Problem 3: Born Rule and Measurement Probabilities**
For a quantum state ρ and measurement operator M, the Born rule gives the probability P = Tr(ρM).
**Part (a)**: Prove that measurement probabilities are always non-negative: P ≥ 0.
**Part (b)**: For a complete set of measurement operators {M₁, M₂, ..., Mₙ} with Σᵢ Mᵢ = I, show that total probability equals 1: Σᵢ Tr(ρMᵢ) = 1.
**Part (c)**: Explain the physical significance of this probability conservation.
|
-- Measurement probability properties
theorem measurement_probability_conservation (ρ : Matrix (Fin 2) (Fin 2) ℂ)
(M₁ M₂ : Matrix (Fin 2) (Fin 2) ℂ) (h : M₁ + M₂ = 1) :
Matrix.trace (ρ * M₁) + Matrix.trace (ρ * M₂) = Matrix.trace ρ := by
sorry
|
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
theorem trace_linearity (A B : Matrix (Fin 2) (Fin 2) ℂ) :
Matrix.trace (A + B) = Matrix.trace A + Matrix.trace B := by
simp [Matrix.trace, Finset.sum_add_distrib]
|
1. Use trace linearity: Tr(A + B) = Tr(A) + Tr(B)
2. Apply the completeness relation Σᵢ Mᵢ = I
3. Use the fact that Tr(ρI) = Tr(ρ)
|
[
"Born rule",
"measurement probability",
"POVM",
"probability conservation",
"trace"
] | false | null |
complete_with_answer_keys
|
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