Quantum Nonlocality Pt. 4: More on the CHSH Inequality

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1 Quantum Nonlocality Pt. 4: More on the CHSH Inequality PHYS Southern Illinois University May 4, 2017 PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11

2 The CHSH Inequality Correlations are called nonlocal if they do not belong to the set LHV. The CHSH Inequality If p(ab xy) are LHV correlations for a, b, x, y {0, 1}, then 1 I (1) 0 where I 1 = p(00 00) p(01 01) p(10 10) p(00 11) However, Alice and Bob can perform local relabelings of the input/output which do not change the correlations. This consists of three operations: Relabeling of inputs Relabeling of outputs Conditional relabeling of outputs PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11

3 The CHSH Inequality For example, all the different relabeling of outputs generates the expressions I 1 = p(00 00) p(01 01) p(10 10) p(00 11) I 2 = p(10 00) p(11 01) p(00 10) p(10 11) I 3 = p(01 00) p(00 01) p(11 10) p(01 11) I 4 = p(11 00) p(10 01) p(01 10) p(11 11) The CHSH Inequalities take the form for k = 1, 2, 3, 4. 1 I k 0. PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11

4 The CHSH Inequality The CHSH Inequality was originally given in terms of expectation values. Consider the function f (a, b) = ( 1) a b. The value of this function is given by whether a and b are equal: { 1 if a = b f (a, b) =. 1 if a b Let E xy denote the expectation value of f (a, b) when Alice inputs x and Bob inputs y: E xy = p(00 xy) + p(11 xy) p(01 xy) p(10 xy). PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11

5 The CHSH Inequality Consider the sum E 00 + E 01 + E 10 E 11. In terms of the probabilities: E 00 + E 01 + E 10 E 11 = p(00 00) + p(11 00) p(01 00) p(10 00) + p(00 01) + p(11 01) p(01 01) p(10 01) + p(00 10) + p(11 10) p(01 10) p(10 10) p(00 11) p(11 11) + p(01 00) + p(10 00) E 00 + E 01 + E 10 E 11 = I 1 + I 4 I 3 I 2 CHSH Inequality (Expectation Form): 2 E 00 + E 01 + E 10 E PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11

6 In d-dimensional quantum systems, von Neumann measurements have d possible outcomes: the experimenter projects into an orthonormal basis. For qubits, every orthonormal basis can be expressed as { ˆn, ˆn }, where ˆn = cos(θ/2) 0 + e iφ sin(θ/2) 1 for unit vector ˆn = (cos φ sin θ, sin φ sin θ, cos θ). The corresponding projectors: P ±ˆn = ± ˆn ±ˆn = 1 (I ± ˆn σ). 2 An observable with ±1 eigenvalues and ± ˆn eigenvectors is given by A = P +ˆn P ˆn = ˆn ˆn ˆn ˆn = ˆn σ. PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11

7 To test the CHSH Inequality in two qubits, Alice chooses to measure in one of two orthonormal bases: { â (0), â (0) } and { â (1), â (1) }; and Bob likewise chooses between two bases: { ˆb (0), ˆb (0) } and { ˆb (1), ˆb (1) }. The corresponding observables with ±1 outcomes are A (0) = â (0) â (0) â (0) â (0) A (1) = â (1) â (1) â (1) â (1) B (0) = ˆb (0) ˆb (0) ˆb (0) ˆb (0) B (1) = ˆb (1) ˆb (1) ˆb (1) ˆb (1). PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11

8 When using these observables to measure a quantum state ρ AB, the outputs are a, b {+1, 1}. { 1 if a = b The function f (a, b) = 1 if a b of Alice and Bob s outputs. is then equivalent to the product E xy = A (x) B (y) ρ AB ( = tr[ â (x), ˆb (y) â (x), b (y) ˆ + â (x), ˆb (y) â (x), b ˆ ) ρ AB ] ( ) tr[ â (x), ˆb (y) â (x), ˆb (y) + â (x), ˆb (y) â (x), ˆb (y) ρ AB ] PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11

9 We can write ± â (x), ±ˆb (y) ±â (x), ±ˆb (y) = 1 ( ) (I ± â (x) σ) (I ± ˆb (y) σ) 4 ± â (x), ˆb (y) ±â (x), ±ˆb (y) = 1 ( ) (I ± â (x) σ) (I ˆb (y) σ) 4 ( E xy = tr[ â (x), ˆb (y) â (x), b (y) ˆ + â (x), ˆb (y) â (x), b ˆ ) ρ AB ] ( ) tr[ â (x), ˆb (y) â (x), ˆb (y) + â (x), ˆb (y) â (x), ˆb (y) ρ AB ] = tr[â (x) σ ˆb (y) σρ AB ]. PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11

10 Remark This theorem only applies to one particular choice of measurements. To decide whether a state ρ AB can generate nonlocal correlations, one must consider all measurement directions. PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11 Theorem: A two-qubit quantum state ρ AB generates LHV correlations for local measurements iff Alice: { â (0), â (0) }, { â (1), â (1) } Bob: { ˆb (0), ˆb (0) }, { ˆb (1), ˆb (1) } 2 tr[â (0) σ ˆb (0) σρ AB ] + tr[â (0) σ ˆb (1) σρ AB ] + tr[â (1) σ ˆb (0) σρ AB ] tr[â (1) σ ˆb (1) σρ AB ] 2.

11 Example Consider the isotropic state ρ f = f Φ f 3 (I I Φ+ ) and the local projections given by â (0) = +, â (0) = â (1) = 0, â (1) = 1 ˆb (0) = cos π sin π 8 1, ˆb (0) = sin π 8 0 cos π 8 1 ˆb (1) = sin π 8 0 cos π 8 1, ˆb (1) = cos π 8 0 sin π 8 1 For what values of f will these measurements generate nonlocal correlations? PHYS Southern Illinois University Quantum Nonlocality Pt. 4: More on the CHSH Inequality May 4, / 11