Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 48 (2015)

Linearity of quantum probability measure and Hardy’s model†
K. Fujikawa,∗1 C.H. Oh,∗2 and C. Zhang∗2Hardy proposed a characterization of entanglementthat does not use inequalities by EPR-type arguments.
It is however disturbing that his scheme, which is intended as a measure of entanglement, completely fails
for the maximally entangled case.1)
The local hidden-variables model in d = 4 = 2 × 2
dimensions of the Hilbert space is defined by2)∫⟨ψ|a · σ ⊗ b · σ|ψ⟩ =ρ(λ)dλa(ψ, λ)b(ψ, λ), (1)Λwhere a and b are 3-dimensional unit vectors, σ stands
for the Pauli matrix, and a(ψ, λ) and b(ψ, λ) are dichotomic variables assuming the eigenvalues ±1 of a·σ
and b · σ, respectively. One can show that this local hidden-variables model does not satisfy the linearity of the quantum mechanical probability measure in
the sense ⟨ψ|a · σ ⊗ b · σ|ψ⟩ + ⟨ψ|a · σ ⊗ b′ · σ|ψ⟩ =⟨ψ|a · σ ⊗ (b + b′ ) · σ|ψ⟩ for non-collinear b and b′ .
If the linearity of the probability measure is strictly
imposed, which is tantamount to asking that the noncontextual hidden-variables model in d = 4 gives theCHSH inequality |⟨B⟩| ≤ 2 uniquely,3) it is shown thatthe hidden-variables model can describe only separable
quantum mechanical states4)∫⟨ψ|a · σ ⊗ b · σ|ψ⟩ =ρ1 (λ1 )dλ1 a(ψ, λ1 )∫(2)
× ρ2 (λ2 )dλ2 b(ψ, λ2 ).
In this case, it is shown that Hardy’s model becomestrivial. Although Hardy’s paradox is interesting as anexperimental test of local realism, its mathematical basis is less solid than hitherto assumed.
Hardy’s model
Hardy defines the projection operators1)Ûi = |ui ⟩⟨ui |, D̂i = |di ⟩⟨di |,(3)

with i = 1, 2, and1[β 1/2 |+⟩i + α1/2 |−⟩i ],α+β1[β 3/2 |+⟩i − α3/2 |−⟩i ]|di ⟩ = √3
α + β3|ui ⟩ = √(4)

for the entangled state |ψ⟩ = α|+⟩1 |+⟩2 − β|−⟩1 |−⟩2
with α2 + β 2 = 1.
†∗1∗2

Condensed from the article in Int. J. Mod. Phys. A29,1450017 (2014).
RIKEN Nishina Center
Center for Quantum Technologies, National University ofSingapore

He then shows the relations⟨ψ|D1 U2 D1 |ψ⟩= 1,⟨ψ|D1 |ψ⟩⟨ψ|D2 U1 D2 |ψ⟩= 1,⟨ψ|D2 |ψ⟩⟨ψ|D1 D2 D1 |ψ⟩αβ= 1−,⟨ψ|D1 |ψ⟩(1 − αβ)⟨ψ|U1 U2 |ψ⟩ = 0,(5)
(6)
(7)
(8)

with 0 < αβ ≤ 1/2.
In the hidden-variables model, the projection operators are assigned their eigenvalues; for example,
D1 (ψ, λ) = 1 or 0, dependingon the hidden-variable∫
λ. Relation (7) implies dλρ(λ)D1 (ψ, λ)D2 (ψ, λ) ̸= 0
for 0 < αβ < 1/2 and thus

D1 (ψ, λ) = 1 and D2 (ψ, λ) = 1(9)∫for some λ, while (8) implies dλρ(λ)U1 (ψ, λ)U2 (ψ, λ) =0 and thus
U1 (ψ, λ)U2 (ψ, λ) = 0(10)for all λ. On the other hand relations (5) and (6) imply
D1 (ψ, λ) = 1 ⇒ U2 (ψ, λ) = 1,D2 (ψ, λ) = 1 ⇒ U1 (ψ, λ) = 1,(11)⟨U1 ⟩ = ⟨U2 ⟩ = ⟨D1 ⟩ = ⟨D2 ⟩ = 0,(12)respectively, where ⇒ means ”inevitably implies”.
For the entangled state with 0 < αβ < 1/2 except
for αβ = 1/2, which implies the maximum entanglement, the relations (9)-(11) are inconsistent.1) This iscalled Hardy’s paradox, which shows the inconsistencyof local realism with entanglement except for the maximally entangled case without referring to inequality.
On the other hand, for a pure state, Bell’s theorem|⟨B⟩| ≤ 2 with B = a·σ⊗(b+b′ )·σ+a′ ·σ⊗(b−b′ )·σ forany a, a′ , b, and b′ implies5) relation (2), namely, theseparable state. The separable state in Hardy’s model,
which is consistent with local realism, imlies α = 1 and
β = 0, for example, for which |ψ⟩ = |+⟩1 |+⟩2 whileÛi = |−⟩ii ⟨−| and D̂i = |−⟩ii ⟨−|. In this case,and all the correlations vanish; thus, Hardy’s model
becomes mathematically trivial. Hardy’s model is inconsistent with local realism by construction.
References
1) L. Hardy: Phys. Rev. Lett. 71, 1665 (1993).
2) J. S. Bell: Physics 1, 195 (1965).
3) J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt:Phys. Rev. Lett. 23, 888 (1969).
4) K. Fujikawa: Prog. Theor. Phys. 127, 975 (2012).
5) N. Gisin: Phys. Lett. A154, 201 (1991).

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