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Quantum Information Theory

Von Neumann Entropy

The quantum analogue of Shannon entropy. For a density matrix rho:

S(rho) = -Tr(rho log_2 rho) = -sum_i lambda_i log_2 lambda_i

where {lambda_i} are the eigenvalues of rho.

Properties:

  • S(rho) >= 0, with equality iff rho is a pure state
  • S(rho) <= log_2 d (d = dimension), with equality iff rho = I/d (maximally mixed)
  • Unitary invariance: S(U rho U^dagger) = S(rho)
  • For a pure bipartite state |psi>_{AB}: S(rho_A) = S(rho_B) (entanglement entropy)

Conditional Entropy and Mutual Information

  • Quantum conditional entropy: S(A|B) = S(AB) - S(B). Can be negative (impossible classically), indicating entanglement. For a Bell state, S(A|B) = -1.
  • Quantum mutual information: I(A:B) = S(A) + S(B) - S(AB). Always non-negative.
  • Coherent information: I_c(A>B) = S(B) - S(AB) = -S(A|B). Relevant for quantum channel capacity.

Key Inequalities

  • Subadditivity: S(AB) <= S(A) + S(B), with equality iff rho_{AB} = rho_A tensor rho_B
  • Strong subadditivity (Lieb-Ruskai): S(ABC) + S(B) <= S(AB) + S(BC). The most fundamental inequality in quantum information.
  • Araki-Lieb: |S(A) - S(B)| <= S(AB)
  • Concavity: S(sum_i p_i rho_i) >= sum_i p_i S(rho_i)

Other Entropy Measures

  • Quantum relative entropy: S(rho || sigma) = Tr(rho(log rho - log sigma)). Non-negative (Klein's inequality), not symmetric, serves as a parent quantity for many entropic quantities.
  • Renyi entropies: S_alpha(rho) = (1/(1-alpha)) log_2 Tr(rho^alpha). S_0 = log rank, S_1 = von Neumann (limit), S_infty = -log lambda_max.
  • Min-entropy and max-entropy: Operational significance in one-shot information theory.

Quantum Channel Capacity

Classical Capacity of Quantum Channels

The maximum rate at which classical information can be reliably transmitted through a quantum channel E.

Holevo-Schumacher-Westmoreland (HSW) theorem: C(E) = lim_{n->infty} (1/n) chi(E^{tensor n})

where chi(E) is the Holevo quantity (see below). The regularization (limit over n uses) is necessary because the Holevo quantity is not always additive -- a surprising result proven by Hastings (2009).

Quantum Capacity

The maximum rate for reliable quantum information transmission: Q(E) = lim_{n->infty} (1/n) max_{rho} I_c(rho, E^{tensor n})

where I_c is the coherent information. The quantum capacity is also regularized. Key results:

  • No-cloning bound: Q(E) <= 1 - S(E(|Phi>)) for the maximally entangled input
  • Degradable channels: Q = max I_c (single-letter formula)
  • Anti-degradable channels: Q = 0 (too noisy for quantum communication)
  • Superactivation: Two channels each with Q = 0 can have Q > 0 when used jointly (Smith-Yard 2008)

Private Capacity

The maximum rate of secret classical communication. P(E) >= Q(E), with equality for degradable channels. Connects to quantum key distribution.

Holevo Bound

Theorem: Given an ensemble {p_i, rho_i} transmitted through a channel with no additional encoding, the accessible information is bounded by:

I_acc <= chi({p_i, rho_i}) = S(sum_i p_i rho_i) - sum_i p_i S(rho_i)

The Holevo quantity chi represents the maximum classical information extractable from the ensemble via any measurement. For a single qubit, chi <= 1 bit, formalizing that one qubit cannot carry more than one classical bit without entanglement assistance (contrasting with superdense coding, which achieves 2 bits with pre-shared entanglement).

The bound is achievable asymptotically with collective measurements on many copies (HSW theorem), but may not be achievable with individual measurements.

Quantum Key Distribution (QKD)

BB84 Protocol

Bennett-Brassard (1984), the first QKD protocol:

  1. Preparation: Alice randomly chooses bits and bases (rectilinear {|0>, |1>} or diagonal {|+>, |->}), sends corresponding qubits to Bob
  2. Measurement: Bob randomly measures in rectilinear or diagonal basis
  3. Sifting: Alice and Bob publicly compare bases, keep only matching-basis results (~50%)
  4. Parameter estimation: Sacrifice a random subset to estimate error rate (QBER)
  5. Error correction: Classical error correction (e.g., Cascade, LDPC) on remaining bits
  6. Privacy amplification: Hash the corrected key to eliminate Eve's information, using universal hashing

Security: If QBER < ~11% (for BB84), a secure key can be extracted. Security proofs reduce to the no-cloning theorem and the monogamy of entanglement. Unconditional security holds against arbitrary quantum attacks.

Key rate: r >= 1 - h(QBER) - h(QBER), where h is the binary entropy function (Shor-Preskill proof via CSS codes).

E91 Protocol

Ekert (1991): Entanglement-based QKD. Alice and Bob share Bell pairs |Phi+> and measure in random bases. Security is guaranteed by Bell inequality violation -- any eavesdropping reduces the observed CHSH violation.

The equivalence between BB84 and E91 (via the entanglement-based view of BB84) unifies prepare-and-measure and entanglement-based QKD.

Device-Independent QKD

Security based solely on observed Bell inequality violations, without trusting the internal workings of the devices. Requires near-perfect Bell inequality violation and very low noise. Proven secure but extremely demanding experimentally.

Practical Considerations

  • Decoy state method: Uses different intensity pulses to detect photon-number splitting attacks on weak coherent sources
  • Continuous-variable QKD: Uses homodyne/heterodyne detection of coherent states (Gaussian modulation), compatible with standard telecom infrastructure
  • Finite-key analysis: Security with finite block lengths (not just asymptotic)
  • Quantum repeaters: Extend QKD range beyond direct transmission limits using entanglement swapping and purification
  • Current distances: ~400 km fiber (with twin-field QKD), ~1200 km satellite (Micius)

Quantum Random Number Generation

Quantum mechanics provides fundamentally unpredictable randomness, unlike classical pseudo-random or hardware-based generators.

Approaches:

  • Trusted-device: Measure qubit in superposition. Simple but security depends on device model. Examples: photon beam-splitter, vacuum fluctuation measurement.
  • Semi-device-independent: Partial assumptions (e.g., bounded dimension). The min-entropy of outputs is certified under weaker assumptions than fully trusted devices.
  • Device-independent (DIQRNG): Randomness certified by Bell inequality violation alone. A loophole-free Bell test generates certified private randomness from any no-signaling adversary. Rate: H_min(A|E) >= 1 - log_2(1 + sqrt(2 - S^2/4)) for CHSH value S.

Randomness expansion: Generate more random bits than the initial seed using quantum devices. Unbounded expansion is possible with two or more devices.

Quantum Money

Proposed by Wiesner (~1970s, published 1983): Banknotes containing quantum states that cannot be counterfeited (by no-cloning).

Private-key quantum money: The bank must verify each note. Wiesner's scheme: each note has a serial number and a sequence of qubits in random BB84 states. The bank stores the basis choices; verification measures in the correct bases. Counterfeiting requires cloning unknown quantum states -- impossible.

Public-key quantum money: Anyone can verify without the bank. Much harder to construct. Candidates based on:

  • Knot invariants (Farhi et al.)
  • Lattice-based constructions
  • Hidden subspace states (Aaronson-Christiano) -- broken for their specific proposal, but the framework persists

Quantum lightning: Stronger primitive where even the minter cannot produce two identical notes. Based on collision-resistant hash functions and conjectured computational hardness assumptions. Recent progress connects quantum money to cryptographic assumptions and lattice problems.

Entanglement Theory

Entanglement Measures

  • Entanglement entropy: S(rho_A) for pure bipartite states. Unique measure up to normalization.
  • Entanglement of formation: E_F(rho) = min_{decompositions} sum_i p_i S(rho_A^i). Minimum average entropy over all pure-state decompositions.
  • Distillable entanglement: E_D(rho) = maximum rate of Bell pairs extractable via LOCC.
  • Entanglement cost: E_C(rho) = minimum rate of Bell pairs needed to create rho via LOCC.
  • Negativity: N(rho) = (||rho^{T_A}||_1 - 1)/2. Computable but not a faithful measure.
  • Squashed entanglement: E_{sq}(rho) = (1/2) inf_{E} I(A:B|E). Additive and faithful.

Entanglement Distillation and Bound Entanglement

LOCC (local operations and classical communication) cannot create entanglement. Given n copies of a noisy entangled state, entanglement distillation extracts m < n near-perfect Bell pairs.

Bound entanglement: States that are entangled but have zero distillable entanglement (E_D = 0). Exist for all dimensions >= 2x3. PPT (positive partial transpose) entangled states are bound entangled. The existence of NPT bound entanglement remains an open question.

Monogamy of Entanglement

CKW inequality: For three qubits, C^2(A:BC) >= C^2(A:B) + C^2(A:C), where C is the concurrence. Entanglement cannot be freely shared -- a key resource for QKD security.