Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations

The appearance of negative terms in quasiprobability representations of quantum theory is known to be inevitable, and, due to its equivalence with the onset of contextuality, of central interest in quantum computation and information. Until recently, however, nothing has been known about how much negativity is necessary in a quasiprobability representation. Zhu proved that the upper and lower bounds with respect to one type of negativity measure are saturated by quasiprobability representations which are in one-to-one correspondence with the elusive symmetric informationally complete quantum measurements (SICs). We define a family of negativity measures which includes Zhu's as a special case and consider another member of the family which we call "sum negativity." We prove a sufficient condition for local maxima in sum negativity and find exact global maxima in dimensions $3$ and $4$. Notably, we find that Zhu's result on the SICs does not generally extend to sum negativity, although the analogous result does hold in dimension $4$. Finally, the Hoggar lines in dimension $8$ make an appearance in a conjecture on sum negativity.Comment: 21 pages. v2: journal version, added reference

The expression pattern of MUC1 (EMA) is related to tumour characteristics and clinical outcome of invasive ductal breast carcinoma

Aims: To clarify MUC1 patterns in invasive ductal breast carcinoma and to relate them to clinicopathological parameters, coexpression of other biological markers and prognosis. Methods and results: Samples from 243 consecutive patients with primary ductal carcinoma were incorporated into tissue microarrays (TMAs). Slides were stained for MUC1, oestrogen receptor (ER), progesterone receptor (PR), Her2/neu, p53 and cyclin D1. Apical membrane MUC1 expression was associated with smaller tumours (P = 0.001), lower tumour grades (P < 0.001), PR positivity (P = 0.003) and increased overall survival (OS; P = 0.030). Diffuse cytoplasmic MUC1 expression was associated with cyclin D1 positivity (P = 0.009) and increased relapse-free survival (RFS; P = 0.034). Negativity for MUC1 was associated with ER negativity (P = 0.004), PR negativity (P = 0.001) and cyclin D1 negativity (P = 0.009). In stepwise multivariate analysis MUC1 negativity was an independent predictor of both RFS [hazard ratio (HR) 3.5, 95% confidence interval (CI) 1.5, 8.5; P = 0.005] and OS (HR 14.7, 9 5% Cl 4.9, 44. 1; P < 0.001). Conclusions: The expression pattern of MUC1 in invasive ductal breast carcinoma is related to tumour characteristics and clinical outcome. In addition, negative MUC1 expression is an independent risk factor for poor RFS and OS, besides 'classical' prognostic indicators

It's worse than you thought : the feedback negativity and violations of reward prediction in gambling tasks

The reinforcement learning theory suggests that the feedback negativity should be larger when feedback is unexpected. Two recent studies found, however, that the feedback negativity was unaffected by outcome probability. To further examine this issue, participants in the present studies made reward predictions on each trial of a gambling task where objective reward probability was indicated by a cue. In Study 1, participants made reward predictions following the cue, but prior to their gambling choice; in Study 2, predictions were made following their gambling choice. Predicted and unpredicted outcomes were associated with equivalent feedback negativities in Study 1. In Study 2, however, the feedback negativity was larger for unpredicted outcomes. These data suggest that the magnitude of the feedback negativity is sensitive to violations of reward prediction, but that this effect may depend on the close coupling of prediction and outcome