Self-Assessment Quiz: Multiple Qubits and Entanglement

Twenty questions on tensor products, Bell states, entanglement, correlations, and what entanglement does and does not let you do. Aim for 16+.

Question 1

A two-qubit system has a state space of dimension:

A) 2 B) 3 C) 4 D) 8

Question 2

The state $\frac{1}{\sqrt2}(|00\rangle + |11\rangle)$ is:

A) A product state B) Entangled C) Mixed D) Unnormalized

Question 3

Which of these two-qubit states is not entangled?

A) $\frac{1}{\sqrt2}(|00\rangle+|11\rangle)$ B) $\frac{1}{\sqrt2}(|01\rangle-|10\rangle)$ C) $\frac{1}{2}(|00\rangle+|01\rangle+|10\rangle+|11\rangle)$ D) $\frac{1}{\sqrt2}(|00\rangle-|11\rangle)$

Question 4

The four Bell states are distinguished from one another by:

A) Their entanglement (some are more entangled) B) Two bits: a relative phase and a parity C) Their normalization D) The number of qubits

Question 5

The Bell state $|\Phi^+\rangle$ is created from $|00\rangle$ by:

A) $H$ on both qubits B) $H$ on qubit 0, then CNOT(0→1) C) CNOT(0→1) then $H$ on qubit 1 D) $X$ on qubit 1

Question 6

Measuring qubit 0 of $|\Phi^+\rangle$ and getting $0$ leaves qubit 1 in:

A) $|0\rangle$ B) $|1\rangle$ C) $|+\rangle$ D) A mixed state

Question 7

The reduced state of one qubit of a Bell pair is:

A) $|0\rangle$ B) $|+\rangle$ C) Maximally mixed D) Undefined

Question 8

That reduced state tells us that a Bell pair's information is:

A) Stored redundantly in each qubit B) Stored entirely in the correlations, not in either qubit locally C) Lost D) Classical

Question 9

Entanglement allows:

A) Faster-than-light signalling B) Correlations stronger than any classical model, but no signalling C) Cloning of unknown states D) Deterministic measurement outcomes

Question 10

The CHSH inequality bounds classical correlations by $|S| \le 2$. Quantum mechanics allows up to:

A) $2$ B) $2\sqrt2 \approx 2.83$ C) $4$ D) Unbounded

Question 11

An experimental CHSH value of $S = 2.4 \pm 0.05$ indicates:

A) An experimental error B) Violation of local realism, with imperfect (noisy) entanglement C) A product state D) Perfect entanglement

Question 12

The GHZ state on three qubits is:

A) $\frac{1}{\sqrt2}(|000\rangle + |111\rangle)$ B) $\frac{1}{\sqrt3}(|001\rangle+|010\rangle+|100\rangle)$ C) $|+\rangle^{\otimes 3}$ D) $\frac{1}{2}(|00\rangle+|11\rangle)^{\otimes 2}$

Question 13

Measuring one qubit of a GHZ state:

A) Leaves the others entangled B) Collapses all three to the same value C) Has no effect D) Produces a Bell state on the remaining two

Question 14

The Schmidt rank of a bipartite pure state is 1 exactly when:

A) The state is maximally entangled B) The state is a product state C) The state is mixed D) Both qubits are measured

Question 15

$n$ qubits require how many complex amplitudes to describe?

A) $2n$ B) $n^2$ C) $2^n$ D) $2^{2n}$

Question 16

True or false: Entanglement can be used to transmit a message faster than light.

Question 17

True or false: Applying a local unitary to one qubit of a Bell pair can destroy the entanglement.

Question 18

True or false: A maximally entangled pair, viewed one qubit at a time, looks exactly like random noise.

Question 19

Short answer. Explain why the reduced state of a Bell-pair qubit being maximally mixed is required by the no-signalling principle.

Question 20

Short answer. Two labs share Bell pairs and observe perfectly correlated outcomes. Why is this not enough to conclude they have beaten classical physics, and what would be?

Answer Key