We examine the problem of finding the minimum number of Pauli measurements needed to uniquely determine an arbitrary n-qubit pure state among all quantum states. We show that only 11 Pauli measurements are needed to determine an arbitrary two-qubit pure state compared to the full quantum state tomography with 16 measurements, and only 31 Pauli measurements are needed to determine an arbitrary three-qubit pure state compared to the full quantum state tomography with 64 measurements. We demonstrate that our protocol is robust under depolarizing error with simulated random pure states. We experimentally test the protocol on two- and three-qubit systems with nuclear magnetic resonance techniques. We show that the pure state tomography protocol saves us a number of measurements without considerable loss of fidelity. We compare our protocol with same-size sets of randomly selected Pauli operators and find that our selected set of Pauli measurements significantly outperforms those random sampling sets. As a direct application, our scheme can also be used to reduce the number of settings needed for pure-state tomography in quantum optical systems.

}, doi = {http://dx.doi.org/10.1103/PhysRevA.93.032140}, url = {http://arxiv.org/abs/1601.05379}, author = {Xian Ma and Tyler Jackson and Hui Zhou and Jianxin Chen and Dawei Lu and Michael D. Mazurek and Kent A.G. Fisher and Xinhua Peng and David Kribs and Kevin J. Resch and Zhengfeng Ji and Bei Zeng and Raymond Laflamme} } @article {1451, title = {Symmetric Extension of Two-Qubit States}, journal = {Physical Review A}, volume = {90}, year = {2014}, month = {2014/9/17}, abstract = { Quantum key distribution uses public discussion protocols to establish shared secret keys. In the exploration of ultimate limits to such protocols, the property of symmetric extendibility of underlying bipartite states $\rho_{AB}$ plays an important role. A bipartite state $\rho_{AB}$ is symmetric extendible if there exits a tripartite state $\rho_{ABB{\textquoteright}}$, such that the $AB$ marginal state is identical to the $AB{\textquoteright}$ marginal state, i.e. $\rho_{AB{\textquoteright}}=\rho_{AB}$. For a symmetric extendible state $\rho_{AB}$, the first task of the public discussion protocol is to break this symmetric extendibility. Therefore to characterize all bi-partite quantum states that possess symmetric extensions is of vital importance. We prove a simple analytical formula that a two-qubit state $\rho_{AB}$ admits a symmetric extension if and only if $\tr(\rho_B^2)\geq \tr(\rho_{AB}^2)-4\sqrt{\det{\rho_{AB}}}$. Given the intimate relationship between the symmetric extension problem and the quantum marginal problem, our result also provides the first analytical necessary and sufficient condition for the quantum marginal problem with overlapping marginals. }, doi = {10.1103/PhysRevA.90.032318}, url = {http://arxiv.org/abs/1310.3530v2}, author = {Jianxin Chen and Zhengfeng Ji and David Kribs and Norbert L{\"u}tkenhaus and Bei Zeng} } @article {1459, title = {Uniqueness of Quantum States Compatible with Given Measurement Results}, journal = {Physical Review A}, volume = {88}, year = {2013}, month = {2013/7/11}, abstract = { We discuss the uniqueness of quantum states compatible with given results for measuring a set of observables. For a given pure state, we consider two different types of uniqueness: (1) no other pure state is compatible with the same measurement results and (2) no other state, pure or mixed, is compatible with the same measurement results. For case (1), it is known that for a d-dimensional Hilbert space, there exists a set of 4d-5 observables that uniquely determines any pure state. We show that for case (2), 5d-7 observables suffice to uniquely determine any pure state. Thus there is a gap between the results for (1) and (2), and we give some examples to illustrate this. The case of observables corresponding to reduced density matrices (RDMs) of a multipartite system is also discussed, where we improve known bounds on local dimensions for case (2) in which almost all pure states are uniquely determined by their RDMs. We further discuss circumstances where (1) can imply (2). We use convexity of the numerical range of operators to show that when only two observables are measured, (1) always implies (2). More generally, if there is a compact group of symmetries of the state space which has the span of the observables measured as the set of fixed points, then (1) implies (2). We analyze the possible dimensions for the span of such observables. Our results extend naturally to the case of low rank quantum states. }, doi = {10.1103/PhysRevA.88.012109}, url = {http://arxiv.org/abs/1212.3503v2}, author = {Jianxin Chen and Hillary Dawkins and Zhengfeng Ji and Nathaniel Johnston and David Kribs and Frederic Shultz and Bei Zeng} } @article {1454, title = {Ground-State Spaces of Frustration-Free Hamiltonians}, journal = {Journal of Mathematical Physics}, volume = {53}, year = {2012}, month = {2012/01/01}, pages = {102201}, abstract = { We study the ground-state space properties for frustration-free Hamiltonians. We introduce a concept of {\textquoteleft}reduced spaces{\textquoteright} to characterize local structures of ground-state spaces. For a many-body system, we characterize mathematical structures for the set $\Theta_k$ of all the $k$-particle reduced spaces, which with a binary operation called join forms a semilattice that can be interpreted as an abstract convex structure. The smallest nonzero elements in $\Theta_k$, called atoms, are analogs of extreme points. We study the properties of atoms in $\Theta_k$ and discuss its relationship with ground states of $k$-local frustration-free Hamiltonians. For spin-1/2 systems, we show that all the atoms in $\Theta_2$ are unique ground states of some 2-local frustration-free Hamiltonians. Moreover, we show that the elements in $\Theta_k$ may not be the join of atoms, indicating a richer structure for $\Theta_k$ beyond the convex structure. Our study of $\Theta_k$ deepens the understanding of ground-state space properties for frustration-free Hamiltonians, from a new angle of reduced spaces. }, doi = {10.1063/1.4748527}, url = {http://arxiv.org/abs/1112.0762v1}, author = {Jianxin Chen and Zhengfeng Ji and David Kribs and Zhaohui Wei and Bei Zeng} }