arXiv:1008.0994 · Physical Review A

Local unitary invariants for N -qubit pure states

S. Shelly Sharma, Naresh Kumar Sharma

Published Nov 30, 2010• 16 citations

Abstract

The concept of negativity font, a basic unit of multipartite entanglement, is introduced. Transformation properties of determinants of negativity fonts under local unitary (LU) transformations are exploited to obtain relevant $N$-qubit polynomial invariants and construct entanglement monotones from first principles. It is shown that entanglement monotones that detect the entanglement of specific parts of the composite system may be constructed to distinguish between states with distinct types of entanglement. The structural difference between entanglement monotones for an odd and even number of qubits is brought out.

DOI: 10.1103/physreva.82.052340

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arXiv:1008.0994 · Physical Review A

Local unitary invariants for N -qubit pure states

S. Shelly Sharma, Naresh Kumar Sharma

Published Nov 30, 2010• 16 citations

arXiv HTML PDF

Abstract

N

-qubit polynomial invariants and construct entanglement monotones from first principles. It is shown that entanglement monotones that detect the entanglement of specific parts of the composite system may be constructed to distinguish between states with distinct types of entanglement. The structural difference between entanglement monotones for an odd and even number of qubits is brought out.

DOI: 10.1103/physreva.82.052340

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Classification of multipartite entanglement via negativity fonts

S. Shelly Sharma, Naresh Kumar Sharma

Partial transposition of state operator is a well-known tool to detect quantum correlations between two parts of a composite system. In this paper, the global partial transpose (GPT) is linked to conceptually multipartite underlying structures in a state---the negativity fonts. If $K$ -way negativity fonts with nonzero determinants exist, then selective partial transposition of a pure state, involving $K$ of the $N$ qubits ($K\ensuremath{\leqslant}N$), yields an operator with negative eigenvalues, identifying $K$ -body correlations in the state. Expansion of GPT in terms of $K$ -way partially transposed (KPT) operators reveals the nature of intricate intrinsic correlations in the state. Classification criteria for multipartite entangled states based on the underlying structure of global partial transpose of canonical state are proposed. The number of $N$ -partite entanglement types for an $N$ -qubit system is found to be ${2}^{N\ensuremath{-}1}\ensuremath{-}N+2$, while the number of major entanglement classes is ${2}^{N\ensuremath{-}1}\ensuremath{-}1$. Major classes for three- and four-qubit states are listed. Subclasses are determined by the number and type of negativity fonts in canonical states.Partial transposition of state operator is a well-known tool to detect quantum correlations between two parts of a composite system. In this paper, the global partial transpose (GPT) is linked to conceptually multipartite underlying structures in a state---the negativity fonts. If $K$-way negativity fonts with nonzero determinants exist, then selective partial transposition of a pure state, involving $K$ of the $N$ qubits ($K\ensuremath{\leqslant}N$), yields an operator with negative eigenvalues, identifying $K$-body correlations in the state. Expansion of GPT in terms of $K$-way partially transposed (KPT) operators reveals the nature of intricate intrinsic correlations in the state. Classification criteria for multipartite entangled states based on the underlying structure of global partial transpose of canonical state are proposed. The number of $N$-partite entanglement types for an $N$-qubit system is found to be ${2}^{N\ensuremath{-}1}\ensuremath{-}N+2$, while the number of major entanglement classes is ${2}^{N\ensuremath{-}1}\ensuremath{-}1$. Major classes for three- and four-qubit states are listed. Subclasses are determined by the number and type of negativity fonts in canonical states.

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N

K

-way negativity fonts with nonzero determinants exist, then selective partial transposition of a pure state, involving

K

of the

N

qubits ($K\ensuremath{\leqslant}N$), yields an operator with negative eigenvalues, identifying

K

-body correlations in the state. Expansion of GPT in terms of

K

-way partially transposed (KPT) operators reveals the nature of intricate intrinsic correlations in the state. Classification criteria for multipartite entangled states based on the underlying structure of global partial transpose of canonical state are proposed. The number of

N

-partite entanglement types for an

N

-qubit system is found to be ${2}^{N\ensuremath{-}1}\ensuremath{-}N+2$, while the number of major entanglement classes is ${2}^{N\ensuremath{-}1}\ensuremath{-}1$. Major classes for three- and four-qubit states are listed. Subclasses are determined by the number and type of negativity fonts in canonical states.

We show that a positive homogeneous function that is invariant under determinant-1 stochastic local operations and classical communication (SLOCC) transformations defines an

N

-qubit entanglement monotone if and only if the homogeneous degree is not larger than 4. We then describe a common basis and formalism for the

N

-tangle and other known invariant polynomials of degree 4. This allows us to elucidate the relation of the four-qubit invariants defined by Luque and Thibon [Phys. Rev. A 67, 042303 (2003)] and the reduced two-qubit density matrices of the states under consideration, thus giving a physical interpretation for those invariants. We demonstrate that this is a special case of a completely general law that holds for any multipartite system with bipartitions of equal dimension, e.g., for an even number of qudits.