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Tuesday, September 15, 2026

Bell State: Two Qubits, One Shared State

Two qubitsThe basic unit of a quantum computer. Like a 'bit' in a normal computer, but instead of being only 0 or 1 it can be 0, 1, or a blend of both at once. can look like a pair of perfectly correlated classical bits. A Bell stateThe simplest entangled pair: two qubits sharing one joint state, such as an equal blend of 00 and 11. Each qubit alone looks random, yet the pair stays correlated whichever way you measure. reveals the difference only when you ask the pair a different question.

Perfect correlation is not enough

I flip a coin and prepare two ordinary bits: heads gives 00, tails gives 11. Either bit alone is random, yet knowing the first tells you the second. That sounds like quantum entanglementA quantum link where two qubits' states become tied together, so acting on or measuring one affects the other. It is a key resource for quantum computing., and it is not. Now prepare two qubits in the Bell state1

|Φ+⟩  =  (|00⟩ + |11⟩) / √2

Measure each qubit as 0 or 1 and half the runs give 00, half give 11, so those measurementsA physical process that produces a classical outcome and updates the quantum state. In an ideal projective measurement, the state is left in an eigenstate associated with the observed outcome. alone cannot tell the machines apart. The difference lies in the plus sign, or more exactly in its having a definite sign at all. In the coin machine, one of 00 or 11 is secretly true on every run. In the Bell state, both remain parts of one coherent quantum stateThe mathematical description used to predict the probabilities of different measurement outcomes for a quantum system. with a fixed phase between them.

Ask a different question

So change the question: rotate the measurement basisThe set of alternatives a quantum measurement is designed to distinguish. and ask each qubit whether it is + or −. What should happen? The coin machine predicts all four combinations, ++, +−, −+ and −−, because a definite 0 or 1 lands on + or − at random. The Bell state gives only ++ or −−: the correlation survives the change of question.

A coin can fake agreement on one question. It cannot fake agreement on two.

Within quantum mechanics, only the Bell state agrees perfectly on both questions. That does not yet rule out every classical story, since each pair could carry preset answers. Closing that door takes measurements at intermediate angles, where Bell-state correlations violate a Bell inequalityA numerical limit on the correlations any 'local hidden variable' theory can produce. Quantum entanglement breaks it, which is how we know the link is real.,2,3 which any local scheme of preset answers must obey.2

How to make one

Start with |00⟩ and apply a Hadamard gateA single-qubit gate that puts a qubit into an even mix of 0 and 1. The standard way to create a superposition. to the first qubit, giving (|00⟩ + |10⟩)/√2, where the second qubit is still 0 in both branches, so nothing is entangled yet. A CNOT gateA two-qubit gate that flips a target qubit only when a control qubit is 1. The standard two-qubit building block. controlled by the first qubit flips the second qubit only in the 10 branch, giving (|00⟩ + |11⟩)/√2. The first gate creates two alternatives; the second ties the other qubit to them without measuring which occurred.1 Real, noisy gates only approximate this ideal.

Same outcomes, different physics. Left, a classical coin-flip pair: measured as 0 or 1 it gives 00 or 11, measured as plus or minus it gives all four combinations. Right, the Bell state (|00⟩ + |11⟩)/√2: measured as 0 or 1 it also gives 00 or 11, but measured as plus or minus it gives only ++ or −−.
Figure 1: Same outcomes, different physics. Measured as 0 or 1, a coin-flip pair and the Bell state |Φ+⟩ look identical. Change the question to + or − and the classical pair gives all four combinations, while the Bell state gives only ++ or −−.

There are four Bell states

|Φ±⟩  =  (|00⟩ ± |11⟩) / √2

|Ψ±⟩  =  (|01⟩ ± |10⟩) / √2

The Φ states give matching 0/1 outcomes and the Ψ states opposite ones, while the + and − signs encode relative phasesThe phase difference between the parts of a superposition. Invisible when you only ask 0 or 1, it changes the results when you measure another way. that show up only when the measurement basisThe set of alternatives a quantum measurement is designed to distinguish. changes. Together the four form the Bell basisThe set of all four Bell states. Any two-qubit state can be written as a blend of them. for two qubitsThe basic unit of a quantum computer. Like a 'bit' in a normal computer, but instead of being only 0 or 1 it can be 0, 1, or a blend of both at once..1

Why Bell states matter

A Bell pair is the smallest clean example of quantum entanglementA quantum link where two qubits' states become tied together, so acting on or measuring one affects the other. It is a key resource for quantum computing. and a working resource at the heart of quantum teleportationMoving an unknown qubit state to a distant qubit using a shared entangled pair plus an ordinary message. Nothing travels faster than light, and the original state is destroyed.,4 superdense codingSending two classical bits by transmitting a single qubit that is half of an entangled pair the receiver already shares.,5 entanglement swappingEntangling two qubits that never met, by jointly measuring their partners from two separate entangled pairs. The basic trick behind quantum repeaters.6 and many quantum-network protocols.

Half 00 and half 11 proves nothing. A classical mixture can fake it.

None of this sends anything faster than light. Each observer alone sees a random result, and teleportation needs an ordinary classical message before the receiver can recover the transmitted quantum stateThe mathematical description used to predict the probabilities of different measurement outcomes for a quantum system..4 Nor does half 00 and half 11 show that a Bell stateThe simplest entangled pair: two qubits sharing one joint state, such as an equal blend of 00 and 11. Each qubit alone looks random, yet the pair stays correlated whichever way you measure. was made: a classical mixtureA system that is secretly in one definite state or another, picked at random. It can mimic some quantum statistics but carries no phase relationship. gives the same statistics, so the phase must be probed in a second basis.

That is the Bell state’s real trick: not two qubits sharing the same answer, but two qubits sharing a state that cannot be reduced to separate answers.

Sources

  1. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information, 10th Anniversary Edition (Cambridge University Press, 2010). DOI: 10.1017/CBO9780511976667. ↩
  2. Bell, J. S. “On the Einstein Podolsky Rosen Paradox.” Physics Physique Fizika 1, 195–200 (1964). DOI: 10.1103/PhysicsPhysiqueFizika.1.195. ↩
  3. Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. “Proposed Experiment to Test Local Hidden-Variable Theories.” Physical Review Letters 23, 880–884 (1969). DOI: 10.1103/PhysRevLett.23.880. ↩
  4. Bennett, C. H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A. & Wootters, W. K. “Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels.” Physical Review Letters 70, 1895–1899 (1993). DOI: 10.1103/PhysRevLett.70.1895. ↩
  5. Bennett, C. H. & Wiesner, S. J. “Communication via One- and Two-Particle Operators on Einstein-Podolsky-Rosen States.” Physical Review Letters 69, 2881–2884 (1992). DOI: 10.1103/PhysRevLett.69.2881. ↩
  6. Żukowski, M., Zeilinger, A., Horne, M. A. & Ekert, A. K. “‘Event-Ready-Detectors’ Bell Experiment via Entanglement Swapping.” Physical Review Letters 71, 4287–4290 (1993). DOI: 10.1103/PhysRevLett.71.4287. ↩