Negativity (quantum mechanics)

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In quantum mechanics, negativity is a measure of quantum entanglement which is easy to compute. It is a measure deriving from the PPT criterion for separability.[1] It has shown to be an entanglement monotone [2][3] and hence a proper measure of entanglement.

Definition

The negativity of a subsystem A can be defined in terms of a density matrix \rho as:

\mathcal{N}(\rho) \equiv \frac{||\rho^{\Gamma_A}||_1-1}{2}

where:

  •  \rho^{\Gamma_A} is the partial transpose of  \rho with respect to subsystem  A
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is the trace norm or the sum of the singular values of the operator  X .

An alternative and equivalent definition is the absolute sum of the negative eigenvalues of \rho^{\Gamma_A}:

 \mathcal{N}(\rho) = \sum_i \frac{|\lambda_{i}|-\lambda_{i}}{2}

where \lambda_i are all of the eigenvalues.

Properties

\mathcal{N}(\sum_{i}p_{i}\rho_{i}) \le \sum_{i}p_{i}\mathcal{N}(\rho_{i})
\mathcal{N}(P(\rho_{i})) \le \mathcal{N}(\rho_{i})

where P(\rho) is an arbitrary LOCC operation over \rho

Logarithmic negativity

The logarithmic negativity is an entanglement measure which is easily computable and an upper bound to the distillable entanglement.[4] It is defined as

E_N(\rho) \equiv \log_2 ||\rho^{\Gamma_A}||_1

where \Gamma_A is the partial transpose operation and || \cdot ||_1 denotes the trace norm.

It relates to the negativity as follows:[1]

E_N(\rho) := \log_2( 2 \mathcal{N} +1)

Properties

The logarithmic negativity

  • can be zero even if the state is entangled (if the state is PPT entangled).
  • does not reduce to the entropy of entanglement on pure states like most other entanglement measures.
  • is additive on tensor products: E_N(\rho \otimes \sigma) = E_N(\rho) + E_N(\sigma)
  • is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces H_1, H_2, \ldots (typically with increasing dimension) we can have a sequence of quantum states \rho_1, \rho_2, \ldots which converges to \rho^{\otimes n_1}, \rho^{\otimes n_2}, \ldots (typically with increasing n_i) in the trace distance, but the sequence E_N(\rho_1)/n_1, E_N(\rho_2)/n_2, \ldots does not converge to E_N(\rho).
  • is an upper bound to the distillable entanglement

References

  • This page uses material from Quantwiki licensed under GNU Free Documentation License 1.2
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