Bell diagonal state
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Bell diagonal states are a class of bipartite qubit states that are frequently used in quantum information and quantum computation theory.[1]
Definition
The Bell diagonal state is defined as the probabilistic mixture of Bell states:
- [math]\displaystyle{ |\phi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B) }[/math]
- [math]\displaystyle{ |\phi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B - |1\rangle_A \otimes |1\rangle_B) }[/math]
- [math]\displaystyle{ |\psi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B + |1\rangle_A \otimes |0\rangle_B) }[/math]
- [math]\displaystyle{ |\psi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B) }[/math]
In density operator form, a Bell diagonal state is defined as
[math]\displaystyle{ \varrho^{Bell}=p_1|\phi^+\rangle \langle \phi^+|+p_2|\phi^-\rangle\langle \phi^-|+p_3|\psi^+\rangle\langle \psi^+|+p_4|\psi^-\rangle\langle\psi^-| }[/math]
where [math]\displaystyle{ p_1,p_2,p_3,p_4 }[/math]is a probability distribution. Since [math]\displaystyle{ p_1+p_2+p_3+p_4=1 }[/math], a Bell diagonal state is determined by three real parameters. The maximum probability of a Bell diagonal state is defined as [math]\displaystyle{ p_{max}=\max\{p_1,p_2,p_3,p_4\} }[/math].
Properties
1. A Bell-diagonal state is separable if all the probabilities are less or equal to 1/2, i.e., [math]\displaystyle{ p_\text{max}\leq 1/2 }[/math].[2]
2. Many entanglement measures have a simple formulas for entangled Bell-diagonal states:[1]
Relative entropy of entanglement: [math]\displaystyle{ S_r=1-h(p_\text{max}) }[/math],[3] where [math]\displaystyle{ h }[/math] is the binary entropy function.
Entanglement of formation: [math]\displaystyle{ E_f=h(\frac{1}{2}+\sqrt{p_\text{max}(1-p_\text{max})}) }[/math],where [math]\displaystyle{ h }[/math] is the binary entropy function.
Negativity: [math]\displaystyle{ N=p_\text{max}-1/2 }[/math]
Log-negativity: [math]\displaystyle{ E_N=\log(2 p_\text{max} ) }[/math]
3. Any 2-qubit state where the reduced density matrices are maximally mixed, [math]\displaystyle{ \rho_A=\rho_B=I/2 }[/math], is Bell-diagonal in some local basis. Viz., there exist local unitaries [math]\displaystyle{ U=U_1\otimes U_2 }[/math] such that [math]\displaystyle{ U\rho U^{\dagger} }[/math]is Bell-diagonal.[2]
References
- ↑ 1.0 1.1 Horodecki, Ryszard; Horodecki, Paweł; Horodecki, Michał; Horodecki, Karol (2009-06-17). "Quantum entanglement". Reviews of Modern Physics 81 (2): 865–942. doi:10.1103/RevModPhys.81.865. Bibcode: 2009RvMP...81..865H. https://link.aps.org/doi/10.1103/RevModPhys.81.865.
- ↑ 2.0 2.1 Horodecki, Ryszard; Horodecki, Michal/ (1996-09-01). "Information-theoretic aspects of inseparability of mixed states". Physical Review A 54 (3): 1838–1843. doi:10.1103/PhysRevA.54.1838. PMID 9913669. Bibcode: 1996PhRvA..54.1838H. https://link.aps.org/doi/10.1103/PhysRevA.54.1838.
- ↑ Vedral, V.; Plenio, M. B.; Rippin, M. A.; Knight, P. L. (1997-03-24). "Quantifying Entanglement". Physical Review Letters 78 (12): 2275–2279. doi:10.1103/PhysRevLett.78.2275. Bibcode: 1997PhRvL..78.2275V. https://link.aps.org/doi/10.1103/PhysRevLett.78.2275.
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