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arXiv:1611.08962 · Physical Review A

Small sets of complementary observables

Markus Grassl, Daniel McNulty, Ladislav Mišta, Tomasz Paterek

Published Jan 17, 2017• 12 citations
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Abstract

The eigenstates of complementary observables play an elementary role in quantum physics. Here the concept of unextendible sets of complementary observables is analyzed and a practical application in witnessing physical properties, such as the degree of purity of a state, is described.
DOI
10.1103/physreva.95.012118

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Abstract From a practical perspective it is advantageous to develop methods that verify entanglement in quantum states with as few measurements as possible. In this paper we investigate the minimal number of mutually unbiased bases (MUBs) needed to detect bound entanglement in bipartite ( d × d ) -dimensional states, i.e. entangled states that are positive under partial transposition. In particular, we show that a class of entanglement witnesses (EWs) composed of MUBs can detect bound entanglement if the number of measurements is greater than d / 2 + 1 . This is a substantial improvement over other detection methods, requiring significantly fewer resources than either full quantum state tomography or measuring a complete set of d + 1 MUBs. Our approach is based on a partial characterisation of the (non-)decomposability of EWs. We show that non-decomposability is a universal property of MUBs, which holds regardless of the choice of complementary observables, and we find that both the number of measurements and the structure of the witness play an important role in the detection of bound entanglement.
The conventional conception of decoherence relies on the interaction with an external set of degrees of freedom - the 'bath' - to which the system loses quantum information. But the role of the bath can be played too by any internal degrees of freedom that are not accessible to the observer, and in this sense we can talk about 'Self-Induced Decoherence'. Simulating the exact time-evolution of the few-body Bose-Hubbard Hamiltonian, we provide numerical evidence of the decay of quantum purity in the sense of 'temporal typicality' referred in the context of pure state Quantum Statistical Mechanics. We analyse the causes of such purity loss in terms of the structural differences of the interacting many-body Hamiltonian in comparison with its non-interacting counterpart, finding that the predominant role in the long term behaviour is played by the shifts in the many-body energy spectrum.
Unextendible product basis (UPB), a set of incomplete orthonormal product states whose complementary space has no product state, is very useful for constructing bound entangled states. Naturally, instead of considering the set of product states, Bravyi and Smolin considered the set of maximally entangled states. They introduced the concept of unextendible maximally entangled basis (UMEB), a set of incomplete orthonormal maximally entangled states whose complementary space contains no maximally entangled state [Phys. Rev. A 84, 042,306 (2011)]. An entangled state whose nonzero Schmidt coefficients are all equal to 1/k is called a special entangled state of “type k”. In this paper, we introduce a concept named special unextendible entangled basis of “type k” which generalizes both UPB and UMEB. A special unextendible entangled basis of “type k” (SUEBk) is a set of incomplete orthonormal special entangled states of “type k” whose complementary space has no special entangled state of “type k”. We present an efficient method to construct sets of SUEBk. The main strategy here is to decompose the whole space into two subspaces such that the rank of each element in one subspace can be easily upper bounded by k while the other one can be generated by two kinds of the special entangled states of “type k”. This method is very effective when k = pm ≥ 3 where p is a prime number. For these cases, we can obtain sets of SUEBk with continuous integer cardinality when the local dimensions are large.