How often is a random quantum state k-entangled?

The set of trace preserving, positive maps acting on density matrices of size d forms a convex body. We investigate its nested subsets consisting of k-positive maps, where k=2,...,d. Working with the measure induced by the Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes of these sets. Our results strongly suggest that the inner set of (k+1)-positive maps forms a small fraction of the outer set of k-positive maps. These results are related to analogous bounds for the relative volume of the sets of k-entangled states describing a bipartite d X d system.Comment: 19 pages in latex, 1 figure include

Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive

We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiolkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensional probability and geometry of Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision

Phase transitions for random states and a semi-circle law for the partial transpose

For a system of N identical particles in a random pure state, there is a threshold k_0 = k_0(N) ~ N/5 such that two subsystems of k particles each typically share entanglement if k > k_0, and typically do not share entanglement if k < k_0. By "random" we mean here "uniformly distributed on the sphere of the corresponding Hilbert space." The analogous phase transition for the positive partial transpose (PPT) property can be described even more precisely. For example, for N qubits the two subsystems of size k are typically in a PPT state if k k_1. Since, for a given state of the entire system, the induced state of a subsystem is given by the partial trace, the above facts can be rephrased as properties of random induced states. An important step in the analysis depends on identifying the asymptotic spectral density of the partial transposes of such random induced states, a result which is interesting in its own right.Comment: 5 pages, 2 figures. This short note contains a high-level overview of two long and technical papers, arXiv:1011.0275 and arXiv:1106.2264. Version 2: unchanged results, editorial changes, added reference, close to the published articl

Monotone graph limits and quasimonotone graphs

The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object which may be represented by a symmetric function $W$ on $[0,1]$, i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a `quasi-monotonicity' property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of the form $W_1-W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its own right; no such inequality holds for general kernels.Comment: 38 page

A Phase 1 Trial of CNDO-109-Activated Natural Killer Cells in Patients with High-Risk Acute Myeloid Leukemia

Natural killer (NK) cells are an emerging immunotherapy approach to acute myeloid leukemia (AML); however, the optimal approach to activate NK cells before adoptive transfer remains unclear. Human NK cells that are primed with the CTV-1 leukemia cell line lysate CNDO-109 exhibit enhanced cytotoxicity against NK cell–resistant cell lines. To translate this finding to the clinic, CNDO-109–activated NK cells (CNDO-109-NK cells) isolated from related HLA-haploidentical donors were evaluated in a phase 1 dose-escalation trial at doses of 3 × 105 (n = 3), 1 × 106 (n = 3), and 3 × 106 (n = 6) cells/kg in patients with AML in first complete remission (CR1) at high risk for recurrence. Before CNDO-109-NK cell administration, patients were treated with lymphodepleting fludarabine/cyclophosphamide. CNDO-109-NK cells were well tolerated, and no dose-limiting toxicities were observed at the highest tested dose. The median relapse-free survival (RFS) by dose level was 105 (3 × 105), 156 (1 × 106), and 337 (3 × 106) days. Two patients remained relapse-free in post-trial follow-up, with RFS durations exceeding 42.5 months. Donor NK cell microchimerism was detected on day 7 in 10 of 12 patients, with 3 patients having evidence of donor cells on day 14 or later. This trial establishes that CNDO-109-NK cells generated from related HLA haploidentical donors, cryopreserved, and then safely administered to AML patients with transient persistence without exogenous cytokine support. Three durable complete remissions of 32.6 to 47.6+ months were observed, suggesting additional clinical investigation of CNDO-109-NK cells for patients with myeloid malignancies, alone or in combination with additional immunotherapy strategies, is warranted

Alirocumab in Patients With Polyvascular Disease and Recent Acute Coronary Syndrome : ODYSSEY OUTCOMES Trial

Patients with acute coronary syndrome (ACS) and concomitant noncoronary atherosclerosis have a high risk of major adverse cardiovascular events (MACEs) and death. The impact of lipid lowering by proprotein convertase subtilisin-kexin type 9 inhibition in such patients is undetermined.This pre-specified analysis from ODYSSEY OUTCOMES (Evaluation of Cardiovascular Outcomes After an Acute Coronary Syndrome During Treatment With Alirocumab) determined whether polyvascular disease influenced risks of MACEs and death and their modification by alirocumab in patients with recent ACS and dyslipidemia despite intensive statin therapy.Patients were randomized to alirocumab or placebo 1 to 12 months after ACS. The primary MACEs endpoint was the composite of coronary heart disease death, nonfatal myocardial infarction, fatal or nonfatal ischemic stroke, or unstable angina requiring hospitalization. All-cause death was a secondary endpoint.Median follow-up was 2.8 years. Of 18,924 patients, 17,370 had monovascular (coronary) disease, 1,405 had polyvascular disease in 2 beds (coronary and peripheral artery or cerebrovascular), and 149 had polyvascular disease in 3 beds (coronary, peripheral artery, cerebrovascular). With placebo, the incidence of MACEs by respective vascular categories was 10.0%, 22.2%, and 39.7%. With alirocumab, the corresponding absolute risk reduction was 1.4% (95% confidence interval [CI]: 0.6% to 2.3%), 1.9% (95% CI: -2.4% to 6.2%), and 13.0% (95% CI: -2.0% to 28.0%). With placebo, the incidence of death by respective vascular categories was 3.5%, 10.0%, and 21.8%; the absolute risk reduction with alirocumab was 0.4% (95% CI: -0.1% to 1.0%), 1.3% (95% CI: -1.8% to 4.3%), and 16.2% (95% CI: 5.5% to 26.8%).In patients with recent ACS and dyslipidemia despite intensive statin therapy, polyvascular disease is associated with high risks of MACEs and death. The large absolute reductions in those risks with alirocumab are a potential benefit for these patients. (Evaluation of Cardiovascular Outcomes After an Acute Coronary Syndrome During Treatment With Alirocumab [ODYSSEY OUTCOMES]: NCT01663402)

Alirocumab in Patients With Polyvascular Disease and Recent Acute Coronary Syndrome ODYSSEY OUTCOMES Trial

risk of major adverse cardiovascular events (MACEs) and death. The impact of lipid lowering by proprotein convertase subtilisin–kexin type 9 inhibition in such patients is undetermined. OBJECTIVES This pre-specified analysis from ODYSSEY OUTCOMES (Evaluation of Cardiovascular Outcomes After an Acute Coronary Syndrome During Treatment With Alirocumab) determined whether polyvascular disease influenced risks of MACEs and death and their modification by alirocumab in patients with recent ACS and dyslipidemia despite intensive statin therapy. METHODS Patients were randomized to alirocumab or placebo 1 to 12 months after ACS. The primary MACEs endpoint was the composite of coronary heart disease death, nonfatal myocardial infarction, fatal or nonfatal ischemic stroke, or unstable angina requiring hospitalization. All-cause death was a secondary endpoint. RESULTS Median follow-up was 2.8 years. Of 18,924 patients, 17,370 had monovascular (coronary) disease, 1,405 had polyvascular disease in 2 beds (coronary and peripheral artery or cerebrovascular), and 149 had polyvascular disease in 3 beds (coronary, peripheral artery, cerebrovascular). With placebo, the incidence of MACEs by respective vascular categories was 10.0%, 22.2%, and 39.7%. With alirocumab, the corresponding absolute risk reduction was 1.4% (95% confidence interval [CI]: 0.6% to 2.3%), 1.9% (95% CI: 2.4% to 6.2%), and 13.0% (95% CI: 2.0% to 28.0%). With placebo, the incidence of death by respective vascular categories was 3.5%, 10.0%, and 21.8%; the absolute risk reduction with alirocumab was 0.4% (95% CI: 0.1% to 1.0%), 1.3% (95% CI: 1.8% to 4.3%), and 16.2% (95% CI: 5.5% to 26.8%). CONCLUSIONS In patients with recent ACS and dyslipidemia despite intensive statin therapy, polyvascular disease is associated with high risks of MACEs and death. The large absolute reductions in those risks with alirocumab are a potential benefit for these patients. (Evaluation of Cardiovascular Outcomes After an Acute Coronary Syndrome During Treatment With Alirocumab [ODYSSEY OUTCOMES]: NCT01663402) (J Am Coll Cardiol 2019;74:1167–76) © 2019 The Authors. Published by Elsevier on behalf of the American College of Cardiology Foundation

Quantum Iterated Function Systems

Iterated functions system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum iterated functions system (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting a QIFS consists of completely positive maps acting in the space of density operators. We present exemplary classical IFSs, the invariant measure of which exhibits fractal structure, and study properties of the corresponding QIFSs and their invariant states.Comment: 12 pages, 1 figure include

Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities

Employing Hilbert-Schmidt measure, we explicitly compute and analyze a number of determinantal product (bivariate) moments |rho|^k |rho^{PT}|^n, k,n=0,1,2,3,..., PT denoting partial transpose, for both generic (9-dimensional) two-rebit (alpha = 1/2) and generic (15-dimensional) two-qubit (alpha=1) density matrices rho. The results are, then, incorporated by Dunkl into a general formula (Appendix D6), parameterized by k, n and alpha, with the case alpha=2, presumptively corresponding to generic (27-dimensional) quaternionic systems. Holding the Dyson-index-like parameter alpha fixed, the induced univariate moments (|rho| |rho^{PT}|)^n and |rho^{PT}|^n are inputted into a Legendre-polynomial-based (least-squares) probability-distribution reconstruction algorithm of Provost (Mathematica J., 9, 727 (2005)), yielding alpha-specific separability probability estimates. Since, as the number of inputted moments grows, estimates based on |rho| |rho^{PT}| strongly decrease, while ones employing |rho^{PT}| strongly increase (and converge faster), the gaps between upper and lower estimates diminish, yielding sharper and sharper bounds. Remarkably, for alpha = 2, with the use of 2,325 moments, a separability-probability lower-bound 0.999999987 as large as 26/323 = 0.0804954 is found. For alpha=1, based on 2,415 moments, a lower bound results that is 0.999997066 times as large as 8/33 = 0.242424, a (simpler still) fractional value that had previously been conjectured (J. Phys. A, 40, 14279 (2007)). Furthermore, for alpha = 1/2, employing 3,310 moments, the lower bound is 0.999955 times as large as 29/64 = 0.453125, a rational value previously considered (J. Phys. A, 43, 195302 (2010)).Comment: 47 pages, 12 figures; slightly expanded and modified for journal publication; this paper incorporates and greatly extends arXiv:1104.021