Some Nearly Quantum Theories

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras. Subject to some reasonable constraints, we show that no such composite exists having the exceptional Jordan algebra as a direct summand. We then construct several dagger compact categories of such Jordan-algebraic models. One of these neatly unifies real, complex and quaternionic mixed-state quantum mechanics, with the exception of the quaternionic "bit". Another is similar, except in that (i) it excludes the quaternionic bit, and (ii) the composite of two complex quantum systems comes with an extra classical bit. In both of these categories, states are morphisms from systems to the tensor unit, which helps give the categorical structure a clear operational interpretation. A no-go result shows that the first of these categories, at least, cannot be extended to include spin factors other than the (real, complex, and quaternionic) quantum bits, while preserving the representation of states as morphisms. The same is true for attempts to extend the second category to even-dimensional spin-factors. Interesting phenomena exhibited by some composites in these categories include failure of local tomography, supermultiplicativity of the maximal number of mutually distinguishable states, and mixed states whose marginals are pure.Comment: In Proceedings QPL 2015, arXiv:1511.0118

Composites and Categories of Euclidean Jordan Algebras

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras (EJAs), satisfying some reasonable additional constraints motivated by the desire to construct dagger-compact categories of such models. We show that no such composite has the exceptional Jordan algebra as a direct summand, nor does any such composite exist if one factor has an exceptional summand, unless the other factor is a direct sum of one-dimensional Jordan algebras (representing essentially a classical system). Moreover, we show that any composite of simple, non-exceptional EJAs is a direct summand of their universal tensor product, sharply limiting the possibilities. These results warrant our focussing on concrete Jordan algebras of hermitian matrices, i.e., euclidean Jordan algebras with a preferred embedding in a complex matrix algebra}. We show that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category InvQM of embedded euclidean Jordan algebras, having fewer objects but more morphisms, that is not only compact closed but dagger-compact. This category unifies finite-dimensional real, complex and quaternionic mixed-state quantum mechanics, except that the composite of two complex quantum systems comes with an extra classical bit. Our notion of composite requires neither tomographic locality, nor preservation of purity under tensor product. The categories we construct include examples in which both of these conditions fail. In such cases, the information capacity (the maximum number of mutually distinguishable states) of a composite is greater than the product of the capacities of its constituents.Comment: 60 pages, 3 tables. Substantially revised, with some new result

Locally Tomographic Shadows (Extended Abstract)

Given a monoidal probabilistic theory -- a symmetric monoidal category $\mathcal{C}$ of systems and processes, together with a functor $\mathbf{V}$ assigning concrete probabilistic models to objects of $\mathcal{C}$ -- we construct a locally tomographic probabilistic theory LT$(\mathcal{C},\mathbf{V})$ -- the locally tomographic shadow of $(\mathcal{C},\mathbf{V})$ -- describing phenomena observable by local agents controlling systems in $\mathcal{C}$, and able to pool information about joint measurements made on those systems. Some globally distinct states become locally indistinguishable in LT$(\mathcal{C},\mathbf{V})$, and we restrict the set of processes to those that respect this indistinguishability. This construction is investigated in some detail for real quantum theory.Comment: In Proceedings QPL 2023, arXiv:2308.1548

Stochastic errors in quantum instruments

Fault-tolerant quantum computation requires non-destructive quantum measurements with classical feed-forward. Many experimental groups are actively working towards implementing such capabilities and so they need to be accurately evaluated. As with unitary channels, an arbitrary imperfect implementation of a quantum instrument is difficult to analyze. In this paper, we define a class of quantum instruments that correspond to stochastic errors and thus are amenable to standard analysis methods. We derive efficiently computable upper- and lower-bounds on the diamond distance between two quantum instruments. Furthermore, we show that, for the special case of uniform stochastic instruments, the diamond distance and the natural generalization of the process infidelity to quantum instruments coincide and are equal to a well-defined probability of an error occurring during the measurement

Seasonal variation in activity and nearshore habitat use of Lake Trout in a subarctic lake

Abstract Background In lake ecosystems, predatory fish can move and forage across both nearshore and offshore habitats. This coupling of sub-habitats, which is important in stabilizing lake food webs, has largely been assessed from a dietary perspective and has not included movement data. As such, empirical estimates of the seasonal dynamics of these coupling movements by fish are rarely quantified, especially for northern lakes. Here we collect fine-scale fish movement data on Lake Trout (Salvelinus namaycush), a predatory cold-water fish known to link nearshore and offshore habitats, to test for seasonal drivers of activity, habitat use and diet in a subarctic lake. Methods We used an acoustic telemetry positioning array to track the depth and spatial movements of 43 Lake Trout in a subarctic lake over two years. From these data we estimated seasonal 50% home ranges, movements rates, tail beat activity, depth use, and nearshore habitat use. Additionally, we examined stomach contents to quantify seasonal diet. Data from water temperature and light loggers were used to monitor abiotic lake conditions and compare to telemetry data. Results Lake Trout showed repeatable seasonal patterns of nearshore habitat use that peaked each spring and fall, were lower throughout the long winter, and least in summer when this habitat was above preferred temperatures. Stomach content data showed that Lake Trout acquired the most nearshore prey during the brief spring season, followed by fall, and winter, supporting telemetry results. Activity rates were highest in spring when feeding on invertebrates and least in summer when foraging offshore, presumably on large-bodied prey fish. High rates of nearshore activity in fall were associated with spawning. Nearshore habitat use was widespread and not localized to specific regions of the lake, although there was high overlap of winter nearshore core areas between years. Conclusions We provide empirical demonstrations of the seasonal extent to which a mobile top predator links nearshore and offshore habitats in a subarctic lake. Our findings suggest that the nearshore is an important foraging area for Lake Trout for much of the year, and the role of this zone for feeding should be considered in addition to its traditional importance as spawning habitat