Coexistence does not imply joint measurability

One of the hallmarks of quantum theory is the realization that distinct measurements cannot in general be performed simultaneously, in stark contrast to classical physics. In this context the notions of coexistence and joint measurability are employed to analyze the possibility of measuring together two general quantum observables, characterizing different degrees of compatibility between measurements. It is known that two jointly measurable observables are always coexistent, and that the converse holds for various classes of observables, including the case of observables with two outcomes. Here we resolve, in the negative, the open question whether this equivalence holds in general. Our resolution strengthens the notions of coexistence and joint measurability by showing that both are robust against small imperfections in the measurement setups.Comment: 3 pages, 1 figure; close to published versio

Incompatibility of unbiased qubit observables and Pauli channels

A quantum observable and a channel are considered compatible if they form parts of the same measurement device, otherwise they are incompatible. Constrains on compatibility between observables and channels can be quantified via relations highlighting the necessary trade-offs between noise and disturbance within quantum measurements. In this paper we shall discuss the general properties of these compatibility relations, and then fully characterize the compatibility conditions for an unbiased qubit observable and a Pauli channel. The implications of the characterization are demonstrated on some concrete examples.Comment: 10 pages, 6 figure

Fault-ignorant Quantum Search

We investigate the problem of quantum searching on a noisy quantum computer. Taking a 'fault-ignorant' approach, we analyze quantum algorithms that solve the task for various different noise strengths, which are possibly unknown beforehand. We prove lower bounds on the runtime of such algorithms and thereby find that the quadratic speedup is necessarily lost (in our noise models). However, for low but constant noise levels the algorithms we provide (based on Grover's algorithm) still outperform the best noiseless classical search algorithm.Comment: v1: 15+8 pages, 4 figures; v2: 19+8 pages, 4 figures, published version (Introduction section significantly expanded, presentation clarified, results and order unchanged

Searching via walking: How to find a marked subgraph of a graph using quantum walks

We show how a quantum walk can be used to find a marked edge or a marked complete subgraph of a complete graph. We employ a version of a quantum walk, the scattering walk, which lends itself to experimental implementation. The edges are marked by adding elements to them that impart a specific phase shift to the particle as it enters or leaves the edge. If the complete graph has N vertices and the subgraph has K vertices, the particle becomes localized on the subgraph in O(N/K) steps. This leads to a quantum search that is quadratically faster than a corresponding classical search. We show how to implement the quantum walk using a quantum circuit and a quantum oracle, which allows us to specify the resource needed for a quantitative comparison of the efficiency of classical and quantum searches -- the number of oracle calls.Comment: 4 pages, 2 figure

Quantum searches on highly symmetric graphs

We study scattering quantum walks on highly symmetric graphs and use the walks to solve search problems on these graphs. The particle making the walk resides on the edges of the graph, and at each time step scatters at the vertices. All of the vertices have the same scattering properties except for a subset of special vertices. The object of the search is to find a special vertex. A quantum circuit implementation of these walks is presented in which the set of special vertices is specified by a quantum oracle. We consider the complete graph, a complete bipartite graph, and an $M$-partite graph. In all cases, the dimension of the Hilbert space in which the time evolution of the walk takes place is small (between three and six), so the walks can be completely analyzed analytically. Such dimensional reduction is due to the fact that these graphs have large automorphism groups. We find the usual quadratic quantum speedups in all cases considered.Comment: 11 pages, 6 figures; major revision

Strongly Incompatible Quantum Devices

The fact that there are quantum observables without a simultaneous measurement is one of the fundamental characteristics of quantum mechanics. In this work we expand the concept of joint measurability to all kinds of possible measurement devices, and we call this relation compatibility. Two devices are incompatible if they cannot be implemented as parts of a single measurement setup. We introduce also a more stringent notion of incompatibility, strong incompatibility. Both incompatibility and strong incompatibility are rigorously characterized and their difference is demonstrated by examples.Comment: 27 pages (AMSart), 6 figure

The central limit problem for random vectors with symmetries

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein's method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry and we give a brief introduction to the classical method. The spherically symmetric case is treated by a variation of Stein's method which is adapted for continuous symmetries.Comment: AMS-LaTeX, uses xy-pic, 23 pages; v3: added new corollary to Theorem

Signature Well-being: Toward a More Precise Operationalization of Well-being at the Individual Level

The scientific study of well-being has grown exponentially in recent decades but has primarily focused on the macro level, identifying what generally contributes to well-being. As a result, schools have increased the well-being of students through character strengths and resilience curriculum; institutions have increased employee engagement through aligning interests and strengths; and governments have a new benchmark for success through the deployment of global well-being indices. While great strides have been made at this level, I propose the study of well-being is missing a vital component at the individual level: Signature Well-being. The basis for this proposal is the scientific study of character strengths and the benefits gained from working from one’s signature character strengths. Signature Well-being suggests that, like signature character strengths, there is an element (or combination of elements) of well-being that is energizing, authentic and intuitive. I propose that the elements of well-being should be weighted based on this central element(s) to take into account individual differences and more accurately represent the status a person’s subjective well-being. What is signature then is this unique operationalization of one’s well-being. While the study of well-being at the macro level is a crucial endeavor, the additional study of well-being at the micro level will provide the field a more complete picture from which to build well-being