Established physics · Quantum Mechanics 301 · Lesson 11 of 15

Composite systems and entanglement

A joint state can contain relationships that cannot be replaced by independently prepared states of its parts, even when some local or joint probability tables match.

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Picture it

Two laboratories receive qubits labeled A and B. A preparation gives matching 0/1 results with equal odds for 00 and 11. That table alone could come from a coherent entangled pair or from a machine that randomly prepares 00 or 11. Ask a second, complementary measurement question. The two preparations can now predict different correlations.

Build a joint description

Two distinguishable two-state subsystems have four product-basis alternatives: 00, 01, 10, and 11. Their joint amplitudes describe the composite system. A product state is built from a state of A and a state of B, so its joint amplitudes factor. A general joint state need not have that form. For many position variables, the wavefunction is defined over joint configurations rather than being a separate ordinary three-dimensional wave for each part.

Entanglement is more than ordinary correlation

For pure states, entanglement means that the joint state cannot be written as one product of independent subsystem states. For general mixed states, the stronger wording matters: an entangled state cannot be represented as any statistical mixture of product states. Ordinary shared randomness can produce correlations while still fitting such a mixture. Correlation by itself is therefore not the definition of entanglement.

A matching basis does not tell the whole story

Compare the pure state (|00⟩ + |11⟩)/√2 with an equal mixture of 00 and 11. Their 0/1 tables match. In the complementary ± basis, the pure state still has perfectly matching results, while this mixture produces all four outcomes equally. These chosen preparations can be distinguished this way. One matching table for an unknown preparation is not a general proof of entanglement.

The part need not have a pure state of its own

Either subsystem of this maximally entangled pair has a mixed reduced state, giving 50/50 local results in the displayed bases. The complete pair can nevertheless have a pure state. Reduced does not mean a smaller copy of the full wavefunction. The examples here use labeled subsystems; identical particles bring additional exchange rules, revisited in the final connection to the Standard Model.

One joint state, several observable tables
  1. Prepare the pair
  2. Choose both measurement bases
  3. Compare joint and local statistics

The pure entangled preparation and the specified mixture agree in one basis and disagree in another.

Try the example on paper

In the 0/1 basis, |Φ+⟩ and an equal mixture of 00/11 both give 1/2 for 00 and 11. In the ± basis, |Φ+⟩ gives 1/2 for ++ and −−, while the mixture gives 1/4 for every pair. Each part of the entangled pair alone remains 50/50.

Make it concrete

Measure both qubits in the 0/1 basis: the entangled pair and the specified mixture both give P(00) = P(11) = 1/2. In the ± basis, the entangled pair gives P(++) = P(−−) = 1/2; the mixture gives 1/4 for every pair. A product preparation |+⟩|+⟩ instead gives ++ with certainty in that basis.

The precise version

The joint space is H_A ⊗ H_B. A pure product has amplitudes cᵢⱼ = aᵢbⱼ. For |Φ+⟩ = (|00⟩ + |11⟩)/√2, ρ_A = Tr_B(|Φ+⟩⟨Φ+|) = I/2. Compare it with ρ_mix = (|00⟩⟨00| + |11⟩⟨11|)/2. The off-diagonal coherence distinguishes their joint states and complementary-basis predictions.

Watch the trap

Perfectly matching or opposite outcomes in just one setting do not by themselves establish entanglement. Also, entanglement does not mean each part secretly carries a full independent copy of the pair’s state.

Keep this

Describe the joint state and the local states separately. Compare enough measurement settings to distinguish coherence from shared randomness.

Words for this lesson: tensor product · product state · entanglement · mixed state · density matrix · reduced state · basis

Check your understanding

Reason through each question before opening its answer and explanation.

1. How many product-basis alternatives do two distinguishable two-state subsystems have?

  1. 4
  2. 8
  3. 2
Show answer and explanation

4

The pairs are 00, 01, 10, and 11: two choices for A times two for B.

2. Does a perfectly matching 0/1 table alone prove entanglement?

  1. Only when the laboratories are far apart
  2. Yes, all correlation is entanglement
  3. No; an ordinary mixture of 00 and 11 also matches
Show answer and explanation

No; an ordinary mixture of 00 and 11 also matches

Shared randomness can create that same table. Other settings can distinguish the specified preparations.

3. In the ± basis, what does the entangled |Φ+⟩ example predict?

  1. Only matching results, with ++ and −− equally likely
  2. Every pair equally likely
  3. Only opposite results
Show answer and explanation

Only matching results, with ++ and −− equally likely

This Bell state has matching correlations in both displayed bases. The singlet used next has a different sign pattern.

4. Can a pure joint state have mixed reduced states for its parts?

  1. No, purity must be copied to each part
  2. Only if total probability is lost
  3. Yes, as the maximally entangled pair illustrates
Show answer and explanation

Yes, as the maximally entangled pair illustrates

Tracing out one part can leave a mixed local state even when the pair’s state is pure.

5. What characterizes an entangled mixed state?

  1. Its local outcomes can be controlled remotely
  2. It cannot be expressed as any mixture of product states
  3. Its outcomes must always be opposite
Show answer and explanation

It cannot be expressed as any mixture of product states

Nonseparability of the joint density matrix is the relevant criterion, rather than one particular outcome pattern.

6. What does the complementary-basis comparison below establish?

  1. A way to distinguish the particular displayed preparations
  2. A universal entanglement test using one table for every unknown system
  3. A controllable distant message
Show answer and explanation

A way to distinguish the particular displayed preparations

The calculation compares known candidate states. General state verification can require additional settings and assumptions.

Further reading