Questions: Quantum Shannon Theory

Question 1

The Holevo bound states that the maximum classical information extractable from an ensemble of quantum states {p_i, rho_i} is bounded by:

• A. The number of qubits n
• B. The Shannon entropy of the probability distribution p_i
• C. The Holevo quantity chi = S(sum_i p_i * rho_i) - sum_i p_i * S(rho_i), where S is the von Neumann entropy
• D. The entanglement entropy of the ensemble

Answer: (C)

The Holevo bound limits the accessible information — the mutual information between the classical input (which state was prepared) and the classical output (measurement result) — to at most chi = S(rho) - sum_i p_i S(rho_i), where rho = sum_i p_i rho_i is the average state. For a single qubit, chi <= 1, consistent with the Holevo bound limiting one qubit to carrying at most one classical bit (without entanglement assistance). The bound is achievable asymptotically using appropriate collective measurements.

Question 2

The quantum channel capacity (for transmitting quantum information) has a simple single-letter formula analogous to Shannon's classical channel capacity formula.

• T. True
• F. False

Answer: False

Unlike classical channel capacity, which has a clean single-letter formula C = max I(X;Y), the quantum channel capacity Q is given by a regularized formula: Q = lim_{n->infinity} (1/n) max Q^(1)(channel^{tensor n}), where Q^(1) is the coherent information. This regularization means the capacity of n uses of the channel can be superadditive — the capacity per use can increase when channels are used jointly. Computing the quantum channel capacity is in general undecidable. This is one of the deepest surprises in quantum information theory.

Question 3

How does entanglement assistance change the classical capacity of a quantum channel?

Model answer: When sender and receiver share pre-distributed entanglement, the classical capacity of a quantum channel can increase beyond the unassisted (Holevo) capacity. The entanglement-assisted classical capacity C_E is given by the quantum mutual information: C_E = max S(rho) + S(channel(rho)) - S((id tensor channel)(Phi)), where Phi is a purification. For some channels, C_E can be up to twice the unassisted capacity — superdense coding over a noiseless qubit channel being the extreme example (1 qubit + 1 ebit transmits 2 classical bits). The entanglement-assisted capacity has a simple single-letter formula, unlike the unassisted quantum capacity.

The fact that entanglement-assisted capacity has a clean formula while unassisted capacities do not highlights a recurring theme: entanglement simplifies the theory. The Bennett-Shor-Smolin-Thapliyal theorem provides the formula, and it is always at least as large as the unassisted classical capacity. This demonstrates that entanglement is a genuine resource for communication, not just for cryptography or computation.