Bell States

For reference, Bell defined these as

∣Φ+⟩≜12(∣00⟩+∣11⟩)∣Φ−⟩≜12(∣00⟩−∣11⟩)∣Ψ+⟩≜12(∣01⟩+∣10⟩)∣Ψ−⟩≜12(∣01⟩−∣10⟩)\begin{aligned} \left\lvert \Phi^+ \right\rangle & \triangleq \tfrac{1}{\sqrt{2}}\left(\left\lvert 00 \right\rangle + \left\lvert 11 \right\rangle\right) \\ \left\lvert \Phi^- \right\rangle & \triangleq \tfrac{1}{\sqrt{2}}\left(\left\lvert 00 \right\rangle - \left\lvert 11 \right\rangle\right) \\ \left\lvert \Psi^+ \right\rangle & \triangleq \tfrac{1}{\sqrt{2}}\left(\left\lvert 01 \right\rangle + \left\lvert 10 \right\rangle\right) \\ \left\lvert \Psi^- \right\rangle & \triangleq \tfrac{1}{\sqrt{2}}\left(\left\lvert 01 \right\rangle - \left\lvert 10 \right\rangle\right) \end{aligned}

See Singlet

Properties

  1. Orthonormal basis for HA⊗HB=C2⊗C2H_A\otimes H_B=\mathbb{C}^2\otimes \mathbb{C}^2
∣00⟩=12(∣Φ+⟩+∣Φ−⟩),∣11⟩=12(∣Φ+⟩−∣Φ−⟩)\ket{00} = \tfrac{1}{\sqrt{2}}\big(\ket{\Phi_+} + \ket{\Phi_-}\big), \quad \ket{11} = \tfrac{1}{\sqrt{2}}\big(\ket{\Phi_+} - \ket{\Phi_-}\big) ∣01⟩=12(∣Ψ+⟩+∣Ψ−⟩),∣10⟩=12(∣Ψ+⟩−∣Ψ−⟩)\ket{01} = \tfrac{1}{\sqrt{2}}\big(\ket{\Psi_+} + \ket{\Psi_-}\big), \quad \ket{10} = \tfrac{1}{\sqrt{2}}\big(\ket{\Psi_+} - \ket{\Psi_-}\big)
  1. Can be transformed into each other via local operations UA⊗UBU_A\otimes U_B
σX⊗I ∣Φ+⟩=I⊗σX ∣Φ+⟩=∣Ψ+⟩ \sigma^X \otimes I \,\ket{\Phi_+} = I \otimes \sigma^X \,\ket{\Phi_+} = \ket{\Psi_+} σZ⊗I ∣Φ+⟩=I⊗σZ ∣Φ+⟩=∣Φ−⟩\sigma^Z \otimes I \,\ket{\Phi_+} = I \otimes \sigma^Z \,\ket{\Phi_+} = \ket{\Phi_-}
  1. Simultaneous eigenstates of σAX,σAY,σAZ,σBX,σBY,σBZ\sigma_A^X, \sigma_A^Y, \sigma_A^Z, \sigma_B^X, \sigma_B^Y,\sigma_B^Z
[σAX, σAY]≠0but[σAXσBX, σAYσBY]=0[\sigma^X_A,\ \sigma^Y_A] \neq 0 \quad \text{but} \quad [\sigma^X_A \sigma^X_B,\ \sigma^Y_A \sigma^Y_B] = 0
  1. Locally indistinguishable for all M(A)=M⊗IM^{(A)}=M\otimes I
⟨Φ±∣M(A)∣Φ±⟩=⟨Ψ±∣M(A)∣Ψ±⟩=12 Tr(M)\bra{\Phi_\pm} M^{(A)} \ket{\Phi_\pm} = \bra{\Psi_\pm} M^{(A)} \ket{\Psi_\pm} = \tfrac{1}{2}\,\text{Tr}(M) =0if M∈{σX,σY,σZ}= 0 \quad \text{if } M \in \{\sigma^X, \sigma^Y, \sigma^Z\}