Construction of extremal local positive operator-valued measures under symmetry

Abstract

We study the local implementation of POVMs when we require only the faithful reproduction of the statistics of the measurement outcomes for all initial states. We first demonstrate that any POVM with separable elements can be implemented by a separable super-operator, and develop techniques for calculating the extreme points of POVMs under a certain class of constraint that includes separability and PPT-ness. As examples we consider measurements that are invariant under various symmetry groups (Werner, Isotropic, Bell-diagonal, Local Orthogonal), and demonstrate that in these cases separability of the POVM elements is equivalent to implementability via LOCC. We also calculate the extrema of these classes of measurement under the groups that we consider, and give explicit LOCC protocols for attaining them. These protocols are hence optimal methods for locally discriminating between states of these symmetries. One of many interesting consequences is that the best way to locally discriminate Bell diagonal mixed states is to perform a 2-outcome POVM using local von Neumann projections. This is true regardless of the cost function, the number of states being discriminated, or the prior probabilities. Our results give the first cases of local mixed state discrimination that can be analysed quantitatively in full, and may have application to other problems such as demonstrations of non-locality, experimental entanglement witnesses, and perhaps even entanglement distillation.

15 pages, 1 figure. Title changed, and explanation improved in accordance with published verion (Phys. Rev. A.)

References in corpus (36)

• Entanglement of Formation of an Arbitrary State of Two Qubits
• Mixed State Entanglement and Quantum Error Correction
• Separability Criterion for Density Matrices
• Concentrating Partial Entanglement by Local Operations
• Quantifying Entanglement
• Entanglement Measures and Purification Procedures
• Quantum Nonlocality without Entanglement
• Unextendible Product Bases and Bound Entanglement
• Entanglement Measures under Symmetry
• Local Distinguishability of Multipartite Orthogonal Quantum States
• Optimal local implementation of non-local quantum gates
• Quantum Data Hiding
• An improved bound on distillable entanglement
• Entanglement in Quantum Information Theory
• The entanglement cost under operations preserving the positivity of partial transpose
• Nonlocality, Asymmetry, and Distinguishing Bipartite States
• Distinguishability of the Bell states
• Direct detection of quantum entanglement
• Hiding bits in Bell states
• Entangling operations and their implementation using a small amount of entanglement
• A rigorous treatment of distillable entanglement
• Local indistinguishability: more nonlocality with less entanglement
• Local indistinguishability and LOCC monotones
• Causal and localizable quantum operations
• Hiding classical data in multi-partite quantum states
• Optimal local discrimination of two multipartite pure states
• The non-local content of quantum operations
• The asymptotic relative entropy of entanglement
• Asymptotic Relative Entropy of Entanglement for Orthogonally Invariant States
• Optimally Conclusive Discrimination of Non-orthogonal Entangled States Locally
• Operator monotones, the reduction criterion and the relative entropy
• Measuring the entanglement of bipartite pure states
• Optimal Conclusive Discrimination of Two Non-orthogonal Pure Product Multipartite States Locally
• On local indistinguishability of orthogonal pure states by using a bound on distillable entanglement
• Measurements of semi-local and non-maximally entangled states
• Efficient distillation beyond qubits

Cited by in corpus (15)

• Faithful Squashed Entanglement
• Bounds on Multipartite Entangled Orthogonal State Discrimination Using Local Operations and Classical Communication
• Finite blocklength converse bounds for quantum channels
• A study of LOCC-detection of a maximally entangled state using hypothesis testing
• Entanglement and local information access for graph states
• Local copying and local discrimination as a study for non-locality of a set
• Group theoretical study of LOCC-detection of maximally entangled state using hypothesis testing
• Robustness of Entanglement as a Resource
• Minimizing back-action through entangled measurements
• Distillation Protocols that Involve Local Distinguishing: Composing Upper and Lower Bounds on Locally Accessible Information
• On multipartite invariant states II. Orthogonal symmetry
• Local hypothesis testing between a pure bipartite state and the white noise state
• Quantifying the intrinsic randomness in sequential measurements
• On the Physical Explanation for Quantum Computational Speedup
• Activation of entanglement in generalized entanglement swapping