The von Neumann entropy S(rho) = -Tr[rho log_2(rho)] is the quantum analog of Shannon entropy. For a pure quantum state |psi>, what is S(|psi>)?
AS(|psi>) = 1 bit, because the state is deterministic
BS(|psi>) = 0 bits — a pure state has zero entropy because it is completely determined
CS(|psi>) = log_2(d) where d is the Hilbert space dimension
DS(|psi>) cannot be defined for pure states
A pure state |psi> has density matrix rho = |psi><psi|, which has a single eigenvalue 1 and all others 0. Then S(rho) = -1*log_2(1) = 0. A pure state is maximally certain — if you measure the system, you get |psi> with probability 1. The maximally mixed state (equal superposition of all basis states) has S = log_2(d), where d is dimension. Pure states have zero entropy; mixed states have positive entropy. This parallels Shannon entropy: a deterministic classical variable has H = 0.
Question 2 True / False
The no-cloning theorem states that an unknown quantum state cannot be perfectly cloned. This has no classical analog and fundamentally constrains quantum communication.
TTrue
FFalse
Answer: True
In classical information, copying data is free (make a backup file). In quantum mechanics, Wiesner and Dieks proved independently that no quantum circuit can clone an arbitrary unknown state |psi>. Proof: suppose U clones |psi> to |psi>|psi>. For two different states |psi> and |phi>, U(|psi>|0>) = |psi>|psi> and U(|phi>|0>) = |phi>|phi>. But unitarity requires <psi|phi> = <psi|phi><psi|phi>, which is impossible unless <psi|phi> = 0 or 1 (orthogonal or identical states). Thus cloning fails for unknown non-orthogonal states. This explains why quantum key distribution achieves information-theoretic security: an eavesdropper cannot clone the quantum states used to communicate the key without disturbing them, allowing detection.
Question 3 Short Answer
Explain the relationship between quantum capacity, classical capacity, and entanglement-assisted capacity of a quantum channel. Why are these three different?
Think about your answer, then reveal below.
Model answer: A quantum channel describes how quantum states transform (decohere) when transmitted: rho -> N(rho). The **classical capacity** C is the maximum rate (in bits per channel use) of classical information that can be reliably sent by encoding into quantum states and decoding at the receiver — it is the quantum analog of Shannon capacity. The **quantum capacity** C_Q is the maximum rate of qubits (quantum information) that can be reliably transmitted and decoded in quantum form. The **entanglement-assisted capacity** C_E is the classical capacity when sender and receiver share pre-distributed entanglement. Remarkably, C_E can be twice C (for some channels), and C_Q can be positive even when C = 0 (superactivation). These differences arise because quantum entanglement creates correlations with no classical analog, allowing simultaneous encoding of both classical and quantum information in entangled states. Classical information always separates as individual bits; quantum information can be entangled across multiple qubits, allowing more efficient encoding.
For a depolarizing channel with high noise, classical capacity might be C ≈ 0.1 bits/use, but with shared entanglement, C_E ≈ 0.2 bits/use — entanglement doubles the classical capacity. This is possible because entanglement provides a resource (correlated states) that the sender and receiver can exploit. Quantum capacity C_Q quantifies how much quantum information (not just classical bits) can be reliably encoded and decoded, which requires more sophisticated protocols than classical communication.
Question 4 Multiple Choice
In quantum key distribution (QKD), the eavesdropper is information-theoretically secure against even if they have unlimited computational power. What physical principle prevents the eavesdropper from learning the key without detection?
AThe computational complexity of certain mathematical problems (like factoring) makes eavesdropping difficult
BThe no-cloning theorem: the eavesdropper cannot copy the quantum states used to communicate the key, and attempting to measure them disturbs the states, introducing detectable errors
CClassical encryption schemes are combined with quantum channels
DThe eavesdropper lacks the right quantum equipment
QKD (e.g., BB84 protocol) encodes the cryptographic key in quantum bits sent through a quantum channel. The legitimate parties (Alice and Bob) randomly choose bases to measure the qubits, and later publicly compare bases to establish a shared key. An eavesdropper (Eve) who tries to intercept the qubits faces a dilemma: to extract information, she must measure them, but measurement disturbs them (for non-commuting observables). The disturbance introduces errors that Alice and Bob detect via a statistical test. Since Eve cannot clone the qubits (no-cloning theorem), she cannot copy them and try multiple measurements. Either Eve learns nothing (measures in the wrong basis and causes a disturbance), or she risks detection. This is information-theoretically secure: it depends on quantum mechanics, not computational complexity.