Categories: Quantum information, Quantum mechanics.

Bell state

In quantum information, the Bell states are a set of four two-qubit states which are simple and useful examples of quantum entanglement. They are given by:

∣Φ±⟩=12(∣0⟩A∣0⟩B±∣1⟩A∣1⟩B)∣Ψ±⟩=12(∣0⟩A∣1⟩B±∣1⟩A∣0⟩B)\begin{aligned} \boxed{ \begin{aligned} \ket{\Phi^{\pm}} &= \frac{1}{\sqrt{2}} \Big( \ket{0}_A \ket{0}_B \pm \ket{1}_A \ket{1}_B \Big) \\ \ket{\Psi^{\pm}} &= \frac{1}{\sqrt{2}} \Big( \ket{0}_A \ket{1}_B \pm \ket{1}_A \ket{0}_B \Big) \end{aligned} } \end{aligned}

Where e.g. ∣0⟩A∣1⟩B≡∣0⟩A⊗∣1⟩B\ket{0}_A \ket{1}_B \equiv \ket{0}_A \otimes \ket{1}_B is the tensor product of qubit AA in state ∣0⟩\ket{0} and BB in ∣1⟩\ket{1}. These states form an orthonormal basis for the two-qubit Hilbert space.

More importantly, however, is that all four Bell states are maximally entangled. For brevity, we will only show this for ∣Φ+⟩\ket{\Phi^{+}} here. Consider the following pure density operator:

ρ^=∣Φ+⟩⟨Φ+∣=12(∣0⟩A∣0⟩B+∣1⟩A∣1⟩B)(⟨0∣A⟨0∣B+⟨1∣A⟨1∣B)\begin{aligned} \hat{\rho} = \ket{\Phi^{+}} \bra{\Phi^{+}} &= \frac{1}{2} \Big( \ket{0}_A \ket{0}_B + \ket{1}_A \ket{1}_B \Big) \Big( \bra{0}_A \bra{0}_B + \bra{1}_A \bra{1}_B \Big) \end{aligned}

The reduced density operator ρ^A\hat{\rho}_A of qubit AA is then calculated like so, using a partial trace:

ρ^A=TrB(ρ^)=∑b=0,1⟨b∣B(∣Φ+⟩⟨Φ+∣)∣b⟩B=∑b=0,1(∣0⟩A⟨b∣0⟩B+∣1⟩A⟨b∣1⟩B)(⟨0∣A⟨0∣b⟩B+⟨1∣A⟨1∣b⟩B)=12(∣0⟩A⟨0∣A+∣1⟩A⟨1∣A)=12I^\begin{aligned} \hat{\rho}_A &= \Tr_B(\hat{\rho}) = \sum_{b = 0, 1} \bra{b}_B \Big( \ket{\Phi^{+}} \bra{\Phi^{+}} \Big) \ket{b}_B \\ &= \sum_{b = 0, 1} \Big( \ket{0}_A \inprod{b}{0}_B + \ket{1}_A \inprod{b}{1}_B \Big) \Big( \bra{0}_A \inprod{0}{b}_B + \bra{1}_A \inprod{1}{b}_B \Big) \\ &= \frac{1}{2} \Big( \ket{0}_A \bra{0}_A + \ket{1}_A \bra{1}_A \Big) = \frac{1}{2} \hat{I} \end{aligned}

The same holds for qubit BB. This result is maximally mixed, therefore ∣Φ+⟩\ket{\Phi^{+}} is maximally entangled. This means that a measurement of qubit AA has a 50-50 chance to yield ∣0⟩\ket{0} or ∣1⟩\ket{1}, or in other words, no useful information can be gathered from measuring just one of the qubits.

However, due to the entanglement, measuring AA also has consequences for qubit BB:

∣⟨0∣A ⁣⟨0∣B⋅∣Φ+⟩∣2=12(⟨0∣0⟩A⟨0∣0⟩B+⟨0∣1⟩A⟨0∣1⟩B)2=12∣⟨0∣A ⁣⟨1∣B⋅∣Φ+⟩∣2=12(⟨0∣0⟩A⟨1∣0⟩B+⟨0∣1⟩A⟨1∣1⟩B)2=0∣⟨1∣A ⁣⟨0∣B⋅∣Φ+⟩∣2=12(⟨1∣0⟩A⟨0∣0⟩B+⟨1∣1⟩A⟨0∣1⟩B)2=0∣⟨1∣A ⁣⟨1∣B⋅∣Φ+⟩∣2=12(⟨1∣0⟩A⟨1∣0⟩B+⟨1∣1⟩A⟨1∣1⟩B)2=12\begin{aligned} \big| \bra{0}_A \! \bra{0}_B \cdot \ket{\Phi^{+}} \big|^2 &= \frac{1}{2} \Big( \inprod{0}{0}_A \inprod{0}{0}_B + \inprod{0}{1}_A \inprod{0}{1}_B \Big)^2 = \frac{1}{2} \\ \big| \bra{0}_A \! \bra{1}_B \cdot \ket{\Phi^{+}} \big|^2 &= \frac{1}{2} \Big( \inprod{0}{0}_A \inprod{1}{0}_B + \inprod{0}{1}_A \inprod{1}{1}_B \Big)^2 = 0 \\ \big| \bra{1}_A \! \bra{0}_B \cdot \ket{\Phi^{+}} \big|^2 &= \frac{1}{2} \Big( \inprod{1}{0}_A \inprod{0}{0}_B + \inprod{1}{1}_A \inprod{0}{1}_B \Big)^2 = 0 \\ \big| \bra{1}_A \! \bra{1}_B \cdot \ket{\Phi^{+}} \big|^2 &= \frac{1}{2} \Big( \inprod{1}{0}_A \inprod{1}{0}_B + \inprod{1}{1}_A \inprod{1}{1}_B \Big)^2 = \frac{1}{2} \end{aligned}

As an example, if AA collapses into ∣0⟩\ket{0} due to a measurement, then BB instantly also collapses into ∣0⟩\ket{0}, never ∣1⟩\ket{1}, even if it was not measured. This was a specific example for ∣Φ+⟩\ket{\Phi^{+}}, but analogous results can be found for the other Bell states.

References

  1. J.B. Brask, Quantum information: lecture notes, 2021, unpublished.