Alice and Bob share the Bell state |Φ⁺⟩ = (|↑↑⟩ + |↓↓⟩)/√2. Alice measures her particle and finds spin-up. What is the state of Bob's particle immediately after Alice's measurement, and can Alice use this to send Bob a message?
ABob's particle is still in superposition (|↑⟩ + |↓⟩)/√2, unaffected by Alice's distant measurement
BBob's particle is spin-up, and this was predetermined by a hidden variable set at the moment of entanglement
CBob's particle is spin-up, but Alice cannot use this to send a message because her own outcome was random and uncontrollable
DBob's particle is spin-down, since particles in a Bell state must have opposite spins
Measuring Alice's particle as spin-up projects |Φ⁺⟩ onto |↑↑⟩, instantly placing Bob's particle in |↑⟩. The correlation is real and nonlocal. But Alice cannot use this to send information: she got +½ randomly with 50% probability and had no control over the outcome — she cannot choose to send a '1' by getting spin-up. Option B (hidden variables) is what Einstein proposed but Bell's theorem rules out for local theories. Option D confuses |Φ⁺⟩ with the singlet state |Ψ⁻⟩ = (|↑↓⟩ − |↓↑⟩)/√2, which does have anti-correlated spins.
Question 2 Multiple Choice
Alice and Bob share a maximally entangled pair. Alice measures spin-up. She immediately texts Bob: 'I got spin-up, so you must have spin-up too — I just transmitted information faster than light!' What is fundamentally wrong with this claim?
ANothing — she did transmit one bit of information faster than light, but only probabilistically
BBob's individual outcomes are random and indistinguishable from a world where no entanglement exists; the correlation only appears when they compare results through a classical channel
CThe claim fails only because text messages travel slower than light; quantum signaling itself would be instantaneous
DAlice cannot know Bob's result without measuring her particle first, introducing a delay
Bob's marginal distribution (the probabilities he observes for his own measurements) is exactly 50/50 regardless of whether Alice has measured, hasn't measured, or doesn't exist. There is no observable difference for Bob. The correlation between their outcomes — that they always agree — only becomes visible when they *compare* results via a classical channel, which is bounded by the speed of light. Information transfer requires the sender to control what the receiver observes; Alice cannot choose her outcome, so no message can be encoded.
Question 3 True / False
Bell's theorem demonstrates that no theory based on local hidden variables can reproduce all the statistical predictions of quantum mechanics for entangled states.
TTrue
FFalse
Answer: True
Bell's theorem (1964) derives an inequality that any local hidden variable theory must satisfy, but which quantum mechanics violates. The key insight is that the correlations in entangled states are too strong — they exceed what is possible if each particle carried predetermined values set at the moment of entanglement. Loophole-free experiments have confirmed quantum mechanics' predictions with overwhelming statistical confidence. Any explanation of entanglement via 'the particles agreed in advance' must therefore be either nonlocal (signaling faster than light) or non-realist (outcomes don't exist before measurement) — both abandoning core classical intuitions.
Question 4 True / False
Quantum entanglement enables faster-than-light communication because measuring one particle of an entangled pair instantaneously determines the state of the other particle, regardless of the distance between them.
TTrue
FFalse
Answer: False
While the correlation between measurement outcomes is instantaneous (nonlocal), it cannot be used to transmit information. Each particle's individual outcome is fundamentally random — Alice gets +½ or −½ with equal probability, regardless of Bob's situation, and cannot control which outcome she gets. Bob's outcomes are likewise random. Neither party can distinguish their situation from a world without entanglement by looking only at their own results. The correlation only emerges when both parties compare results through a classical (light-speed-limited) channel. No superluminal signal is transmitted.
Question 5 Short Answer
Why can't Alice and Bob use quantum entanglement to send information faster than light, even though measuring Alice's particle instantaneously determines the state of Bob's particle?
Think about your answer, then reveal below.
Model answer: Entanglement produces correlations between measurement outcomes, but each individual outcome is fundamentally random and uncontrollable. Alice cannot choose to get spin-up; she gets a random result with 50% probability. Bob similarly gets a random result with 50/50 distribution — the same distribution he would see whether or not Alice has measured, and whether or not they share entanglement. There is no observable difference for Bob. The correlation between their outcomes — only visible when they compare results — requires a classical channel, which is bounded by the speed of light. Since information transfer requires the sender to control what the receiver observes, and entanglement provides no such control, no FTL signaling is possible.
This is why entanglement is useful for quantum key distribution (sharing random bits securely) and quantum teleportation (transmitting quantum states, not classical information), but never for faster-than-light communication. The randomness of individual outcomes is not a limitation to overcome — it is fundamental to why quantum mechanics remains consistent with special relativity.