Mutually Unbiased Bases and Complementary Spin 1 Observables

The two observables are complementary if they cannot be measured simultaneously, however they become maximally complementary if their eigenstates are mutually unbiased. Only then the measurement of one observable gives no information about the other observable. The spin projection operators onto three mutually orthogonal directions are maximally complementary only for the spin 1/2. For the higher spin numbers they are no longer unbiased. In this work we examine the properties of spin 1 Mutually Unbiased Bases (MUBs) and look for the physical meaning of the corresponding operators. We show that if the computational basis is chosen to be the eigenbasis of the spin projection operator onto some direction z, the states of the other MUBs have to be squeezed. Then, we introduce the analogs of momentum and position operators and interpret what information about the spin vector the observer gains while measuring them. Finally, we study the generation and the measurement of MUBs states by introducing the Fourier like transform through spin squeezing. The higher spin numbers are also considered.Comment: 7 pages, 3 figures, comments welcom

Dextral shear along the eastern margin of the Colorado Plateau- a kinematic link between Laramide contraction and Rio Grande rifting (ca 75 Ma to 13 Ma)

Kinematic data associated with both Laramide‐age and ‐style and Rio Grande rift‐related structures show that the latest Cretaceous to Neogene interaction between the Colorado Plateau and the North American craton was dominantly coupled with a component of dextral shear. Consistent with earlier studies, minor‐fault data in this study yielded results of varied kinematics. Inverted to a common northeast‐oriented hemisphere, the mean trend of kinematic shortening associated with Laramide‐age structures is 056∘±6 role= presentation style= box-sizing: border-box; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; position: relative; \u3e056∘±6056∘±6 °. Inverted to a common west‐oriented hemisphere, the mean trend of kinematic extension associated with Neogene rifting is 300∘±34 role= presentation style= box-sizing: border-box; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; position: relative; \u3e300∘±34300∘±34 °. The observed dispersion in these directions suggests multiphase deformation, particularly during rifting, along the margin of the plateau since the latest Cretaceous. These data were evaluated using a simple two‐dimensional transcurrent kinematic model; assuming a minimal importance of strain partitioning, a mean trend of convergence between the Colorado Plateau and the North American craton was estimated to be 055∘±5 role= presentation style= box-sizing: border-box; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; position: relative; \u3e055∘±5055∘±5 °. Subsequent Rio Grande rifting, which separated the plateau from the craton, was associated with a mean divergence trend of 307∘±5.8 role= presentation style= box-sizing: border-box; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; position: relative; \u3e307∘±5.8307∘±5.8 °. Analysis of paleomagnetic data from Pennsylvanian to Triassic red beds along the eastern margin of the plateau and from rocks within the rift indicate clockwise rotations of uplifted blocks. Given the lack of regional strike‐slip and dip‐slip faults of common trends, the consistent clockwise rotations support an absence of strain partitioning. Correspondingly, for the north‐south‐trending eastern margin of the plateau, the apparently clockwise‐rotated paleomagnetic data are consistent with dextral transpressive shear between the plateau and the craton. Previous data indicating counterclockwise rotations of crust within parts of the Española rift basin are, if reliable, consistent with dextral transtensive shear. Overall, the transition from latest Cretaceous/Early Cenozoic shortening to Cenozoic extension seems characterized by a quasi‐continuous change from dextral transpressive to dextral transtensive deformation. This interpretation for the kinematic history of the eastern margin of the plateau demonstrates the importance of a dextral shear coupling between the craton and the Farallon plate system—a conclusion rarely implied by previous models of Cenozoic multistress field tectonics during deformation of the Cordilleran foreland

Paleomagnetic dating of fault slip in the southern Rocky Mountains, USA, and itsimportance to an integrated Laramide foreland strain field

The Laramide orogen of the U.S. Cordillera formed in the latest Cretaceous, and deformation lasted into the earliest Oligocene. Along and proximal to the eastern and northern margins of the Colorado Plateau, deformation associated with this event mainly took place along reactivated structures. Related tectonic models invoke some role for the plateau either as a stress guide transmitting compression to the foreland or as a freely rotating microplate. Models dominated by northward displacements of the Colorado Plateau also require covariance between timing and magnitude of dextral strike-slip deformation in the eastern domain and thrust deformation in the northern domain. Here we show that fault-zone materials that are exposed in a major, large-magnitude-displacement strike-slip fault zone east of the plateau contain a well-defined magnetization of late Paleozoic age, suggesting that the fault zone has not been strongly modified since the late Paleozoic. Given that these fault-zone materials include indurated metagranitic crush breccias that must have been at or near the surface at the onset of Carboniferous sedimentation, it is likely that the observed large-magnitude displacements are the result of a poorly understood Precambrian tectonic event. Large-magnitude dextral-slip estimates along this and similar structures may be incorrectly assigned to younger tectonic events. In this context, Laramide strain estimates north of the plateau should not be linked with these older displacements and may instead have resulted from a complex combination of Laramide plateau rotation and general east-directed shortening associated with the formation of the Sevier fold-and-thrust belt salient

The 3D interplay between folding and faulting in a syn-orogenic extensional system: the Simplon Fault Zone in the Central Alps (Switzerland and Italy)

International audienceExtensional low-angle detachments developed in convergent or post-collisional settings are often associated with upright folding of the exhumed footwall. TheSimplon Fault Zone (SFZ) is a Miocene low-angle detachment that developed during convergence in the Central Alps (Switzerland and Italy), accommodating a large component of orogen-parallel extension. Its footwall shows complex structural relationships between large-scale backfolds, mylonites and a discrete brittle detachment and forms a 3D gneiss dome reflecting upright folding with fold axes oriented both parallel and perpendicular to the extension direction. We present a regional study that investigates the interplay between folding and faulting and its implications for the resulting exhumation pattern of the gneiss dome using 3D geometric modelling (computer software GeoModeller), together with a consideration of the chronological relationships from field relationships and 40Ar/39Ar dating. The early Simplon mylonitic fabric is clearly folded by both extension-parallel and extensionperpendicular folds, forming a doubly plunging antiform, whereas the later ductile-to-brittle fabric and the cataclastic detachment are only affected by wavy extension-parallel folds. This observation, together with the interpreted cooling pattern across the SFZ, suggests that updoming of the footwall initiated at the onset of faulting during ductile shearing around 18.5 Ma, due to coeval extension and perpendicular convergence. New 40Ar/39Ar dating on micas (biotite and muscovite) from a sample affected by a strong crenulation cleavage parallel to the axial plane of the Glishorn and Berisal parasitic folds establishes that these folds formed at ca. 10 Ma, broadly coeval with late movement along the more discrete detachment of the SFZ. These extension-parallel folds in the footwall of the SFZ developed due to continued convergence across the Alps, accelerating ongoing exhumation of the western Lepontine dome and promoting coeval uplift of the crystalline Aar and Gotthard massifs in the late Miocene