[Paper Review] Constructions of indecomposable positive maps based on a new criterion for indecomposability

W. B. Hall|ArXiv.org|Jul 5, 2006

Quantum Information and Cryptography22 references3 citations

TL;DR

This paper presents a new sufficient criterion for identifying indecomposable positive maps in quantum mechanics, which are essential for detecting bound entanglement in bipartite quantum states. By applying this criterion to two families of positive maps, the author constructs separability criteria that can detect entanglement beyond the partial transpose test, offering a practical tool for identifying non-decomposable maps with potential applications in quantum information theory.

ABSTRACT

We give a criterion for a positive mapping on the space of operators on a Hilbert space to be indecomposable. We show that this criterion can be applied to two families of positive maps. These families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states.

Motivation & Objective

• To develop a practical criterion for determining whether a positive map is indecomposable, which is crucial for detecting bound entangled states.
• To extend the range of known indecomposable positive maps, which are rare and difficult to construct systematically.
• To provide a method that can be applied to specific families of maps to verify their indecomposability and thus their utility in entanglement detection.
• To address the lack of general constructions for non-decomposable maps, despite their importance in quantum information theory.
• To explore structural constraints on entanglement witnesses and positive maps that could lead to deeper characterization of indecomposable maps.

Proposed method

• Proposes a sufficient condition for a positive map to be indecomposable based on its structural form, particularly focusing on the behavior of the map on specific subspaces.
• Applies the criterion to two families of positive maps, verifying their indecomposability through trace conditions on operators under partial transposition.
• Uses the structure of entanglement witnesses and their action on maximally entangled states to derive necessary positivity constraints.
• Employs the partial transpose operation on operators in a tensor product space to analyze the positivity of the map's action.
• Relies on trace inequalities involving projectors and maximally entangled states to derive conditions under which the map cannot be decomposed into completely positive and completely co-positive components.
• Demonstrates that if certain trace expectations vanish and the full trace of the witness is zero, the witness must be zero, implying the map is indecomposable.

Experimental results

Research questions

• RQ1 Can a general, verifiable criterion be formulated to determine whether a positive map is indecomposable?
• RQ2 Do the proposed families of positive maps satisfy the new criterion for indecomposability?
• RQ3 Can the new criterion detect bound entangled states that evade the partial transpose criterion?
• RQ4 Is the condition based on subspace structure and partial transposition sufficient to guarantee indecomposability?
• RQ5 Can this criterion be generalized to include well-known maps like Choi's original example?

Key findings

• The paper establishes a new sufficient condition for a positive map to be indecomposable, which is easier to verify than Størmer's necessary and sufficient condition.
• The criterion is successfully applied to two families of positive maps, confirming their indecomposability and thus their potential for detecting bound entanglement.
• The method ensures that if the map's action on specific subspaces leads to vanishing trace expectations under partial transposition, and the total trace is zero, then the map is indecomposable.
• The construction leads to separability criteria that can detect entanglement in bound entangled states, which are invisible to the partial transpose test.
• The approach provides a systematic pathway to verify indecomposability, although it does not yet cover the original Choi map, suggesting a need for further generalization.
• The work opens the door to characterizing structural features of non-decomposable maps, potentially enabling future classification and construction of such maps.