quantum-positive-maps

name: quantum-positive-maps description: "Analysis of positive trace-preserving (PTP) maps in quantum information theory. Petz recovery map construction, sufficiency conditions, and Jordan algebra generalizations. Use when: (1) Analyzing quantum state interconversion via positive maps, (2) Implementing Petz recovery for quantum channel inversion, (3) Studying minimal sufficient algebras in quantum systems, (4) Generalizing Koashi-Imoto decomposition to PTP setting."

Quantum Positive Maps

Analysis of positive trace-preserving (PTP) maps in quantum information theory.

Positive Trace-Preserving (PTP) Maps

Definition

A map Φ: B(H) → B(H') is:

• Positive: Φ(A) ≥ 0 whenever A ≥ 0
• Trace-preserving: Tr[Φ(A)] = Tr[A]
• NOT necessarily completely positive

Importance

PTP maps describe:

• Physical processes beyond quantum channels
• Partial measurements
• Non-CP operations (e.g., transpose)

Petz Recovery Map

Standard Petz Recovery

For completely positive trace-preserving (CPTP) map Φ and state σ:

R_σ^Φ(A) = σ Φ†(Φ(σ)^{-1/2} A Φ(σ)^{-1/2})

Properties:

• Approximate inversion: R_σ^Φ∘Φ ≈ Id when Φ is sufficient for σ
• Monotonicity: Petz recovery saturates data processing inequality

Generalization to PTP

For PTP map Φ, Petz recovery construction extends:

R_σ^Φ(A) = σ Φ†(J(Φ(σ))^{-1/2} A J(Φ(σ))^{-1/2})

Where J is Jordan product operation.

Sufficiency and Minimal Algebras

Sufficiency Definition

A map Φ is sufficient for family of states {σ_i} if:

Φ(σ_i) distinct for each i

And information about which σ_i is present can be recovered from Φ(σ_i).

Koashi-Imoto Decomposition

For CPTP maps, minimal sufficient *-algebra decomposition:

H = ⊕_α H_α ⊗ K_α

Φ acts trivially on K_α subspaces.

Generalization to PTP

Minimal sufficient Jordan algebra decomposition:

H = ⊕_α J_α

Where J_α are Jordan subalgebras (not necessarily *-subalgebras).

Neyman-Pearson Tests

Statistical Tests and Jordan Algebras

Key result: Neyman-Pearson tests generate minimal sufficient Jordan algebra

Connection:

1. Hypothesis testing → optimal discrimination
2. Neyman-Pearson lemma → optimal test structure
3. Optimal tests → Jordan algebra elements

Implications

• Statistical tests determine algebraic structure
• Minimal sufficient algebra has statistical interpretation
• Quantum hypothesis testing connects to algebraic decomposition

Mathematical Framework

Jordan Algebras

Jordan product:

A ◦ B = (AB + BA)/2

Jordan algebra: closed under ◦ product (not necessarily associative).

Minimal Sufficient Jordan Algebra

For PTP map Φ and family {σ_i}:

J_min = {A : Φ(A) = Φ(A ◦ σ_i) for some i}

Generated by Neyman-Pearson test operators.

Recovery Map Construction

def construct_petz_recovery_ptp(Phi, sigma):
    """
    Construct Petz recovery for PTP map.
    
    Steps:
    1. Compute Phi(sigma)
    2. Find Jordan inverse
    3. Apply recovery formula
    """
    
    # Output state
    sigma_out = Phi(sigma)
    
    # Jordan inverse (generalized)
    J_inv = jordan_inverse(sigma_out)
    
    # Recovery map
    def recovery(A):
        # Apply Phi† (dual map)
        temp = Phi.dual(J_inv.sqrt() @ A @ J_inv.sqrt())
        # Jordan product with sigma
        return sigma @ temp
    
    return recovery

Implementation Patterns

Pattern 1: Sufficiency Test

def test_sufficiency(Phi, states):
    """
    Test if PTP map Φ is sufficient for state family.
    
    Check:
    1. Output states distinct
    2. Information recoverable
    3. Minimal algebra condition
    """
    
    # Apply map to all states
    outputs = [Phi(sigma) for sigma in states]
    
    # Check distinctness
    for i, j in combinations(outputs, 2):
        if np.allclose(outputs[i], outputs[j]):
            return f"Not sufficient: σ_{i} and σ_{j} indistinguishable"
    
    # Construct minimal Jordan algebra
    J_min = construct_minimal_jordan_algebra(Phi, states)
    
    return f"Sufficient, minimal algebra dimension: {J_min.dim}"

Pattern 2: Koashi-Imoto Generalization

def decompose_ptp_map(Phi, sigma):
    """
    Decompose PTP map similar to Koashi-Imoto.
    
    Find Jordan subalgebras where Φ acts trivially.
    """
    
    # Find minimal sufficient Jordan algebra
    J_min = find_minimal_jordan_algebra(Phi, sigma)
    
    # Decompose Hilbert space
    decomposition = jordan_decompose(J_min)
    
    return decomposition

Applications

1. Quantum Channel Inversion: Petz recovery for approximate channel reversal
2. Quantum Error Correction: Sufficiency for error-correcting codes
3. Quantum Hypothesis Testing: Neyman-Pearson connection
4. Resource Theory: Monotonicity and recovery

References

See jordan_algebras.md for Jordan algebra theory.

Source

Based on arxiv:2604.08380 - "Sufficiency and Petz recovery for positive maps" by Lauritz van Luijk & Henrik Wilming.