Starting Point: Entangled Quantum Clocks
Consider two qubits in the Bell state
\[\frac{|00\rangle+|11\rangle}{\sqrt{2}}.\]
If Alice and Bob measure using the same orthonormal basis, their answers match. They both obtain 0 or they both obtain 1, with the two joint outcomes equally likely.
The Natural Classical Explanation
A natural question is whether the matching answers were already decided when the particles became entangled.
This can be compared with placing matching cards into two envelopes. Alice opens her envelope and discovers that her card is red. She immediately knows that Bob’s card is red too. Nothing had to travel from Alice to Bob. Alice merely discovered a fact that was already true.
If entangled particles behaved this way, their correlation would have a simple explanation: both particles received matching instructions before they separated.
What Does Realism Mean?
In this discussion, realism does not merely mean that the physical world exists. It refers to the idea that the relevant physical properties have definite values independent of whether we measure them.
For example, suppose a red ball is hidden inside a closed box. Before opening the box, we do not know its color, but the ball is nevertheless already red. Opening the box reveals the value; it does not create the value.
Applied to a quantum clock, a realist description could say that before we ask whether the hand points toward twelve or six, there is already a definite answer. We simply do not know it.
This creates an important distinction:
Classical ignorance: the value already exists, but we do not know it.
The standard quantum model used in the book: a superposition should not simply be interpreted as one definite but unknown measurement result.
What Does Locality Mean?
Locality is the idea that a physical system is influenced by what happens in its vicinity rather than by an instantaneous physical influence from an arbitrarily distant location.
Suppose Alice and Bob separate while sharing
\[\frac{|00\rangle+|11\rangle}{\sqrt{2}}.\]
If Alice measures and obtains 0, Bob will also obtain 0 when measuring in the same basis. This correlation raises the question of whether Alice’s measurement somehow affects Bob’s distant system.
A local explanation avoids such an influence: perhaps both particles already carried correlated answers when they separated.
Local Realism
The two ideas can therefore be combined:
Realism: the relevant measurement answers already have definite values.
Locality: Alice’s later measurement does not need to instantaneously change Bob’s distant particle.
This provides an intuitive classical explanation for entanglement.
Hidden Variables
Einstein’s concern was that quantum mechanics might be incomplete. Perhaps its probabilities arise because some additional information is missing from the quantum description.
Suppose quantum mechanics predicts
\[P(0)=\frac12,\qquad P(1)=\frac12.\]
One possibility is that nature fundamentally chooses between these outcomes probabilistically. Another possibility is that some additional variable, conventionally represented by \(\lambda\), determines the answer:
\[\text{state}+\lambda\longrightarrow\text{determined measurement result}.\]
If we do not know \(\lambda\), the results can appear random to us even though the deeper theory is deterministic. Such additional unknown quantities are called hidden variables.
Three Possible Measurement Directions
Bernhardt prepares Bell’s argument using three measurement directions: 0 degrees, 120 degrees, and 240 degrees. In the quantum-clock analogy these correspond to asking about twelve, four, and eight.
A classical hidden-variable explanation could imagine that every entangled pair already carries an answer for all three possible questions.
For example, one pair might carry the instruction pattern:
Question Alice Bob 12? 0 0 4? 1 1 8? 0 0
If both Alice and Bob ask the same question, they obtain the same answer because their instructions were correlated when the pair was created.
No later signal between Alice and Bob is required.
The Critical Question
This classical explanation is extremely attractive because it appears to explain the mysterious correlation without requiring a distant measurement to change another particle.
But John Bell discovered that this idea does not have to remain a philosophical interpretation. The predictions of a local predetermined-answer model can be compared mathematically with the predictions of quantum mechanics.
This is the point at which Bell’s inequality begins.
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- Note type: learning-note
- Subject: learning-notes
- Source: Quantum Computing for Everyone
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