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Register a new userDetecting qubit entanglement : an alternative to the PPT test
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We propose a Partial Lorentz Transformation (PLT) test for detecting entanglement in a two qubit system. One can expand the density matrix of a two qubit system in terms of a tensor product of $(mathbb{I}, vec{sigma})$. The matrix $A$ of the coefficients that appears in such an expansion can be squared to form a $4times4$ matrix $B$. It can be shown that the eigenvalues $lambda_0, lambda_1, lambda_2, lambda_3$ of $B$ are positive. With the choice of $lambda_0$ as the dominant eigenvalue, the separable states satisfy $sqrt{lambda_1}+sqrt{lambda_2}+sqrt{lambda_3}leq sqrt{lambda_0}$. Violation of this inequality is a test of entanglement. Thus, this condition is both necessary and sufficient and serves as an alternative to the celebrated Positive Partial Transpose (PPT) test for entanglement detection. We illustrate this test by considering some explicit examples.
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We present the first experimental test that distinguishes between an event-based corpuscular model (EBCM) [H. De Raedt et al.: J. Comput. Theor. Nanosci. 8 (2011) 1052] of the interaction of photons with matter and quantum mechanics. The test looks at the interference that results as a single photon passes through a Mach-Zehnder interferometer [H. De Raedt et al.: J. Phys. Soc. Jpn. 74 (2005) 16]. The experimental results, obtained with a low-noise single-photon source [G. Brida et al.: Opt. Expr. 19 (2011) 1484], agree with the predictions of standard quantum mechanics with a reduced $chi^2$ of 0.98 and falsify the EBCM with a reduced $chi^2$ of greater than 20.
We realize a suite of logical operations on a distance-two logical qubit stabilized using repeated error detection cycles. Logical operations include initialization into arbitrary states, measurement in the cardinal bases of the Bloch sphere, and a universal set of single-qubit gates. For each type of operation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants, and quantify the difference through detailed characterization. In particular, we demonstrate process tomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration of high-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone on the road to quantum error correction with higher-distance superconducting surface codes.
This paper establishes single-letter formulas for the exact entanglement cost of generating bipartite quantum states and simulating quantum channels under free quantum operations that completely preserve positivity of the partial transpose (PPT). First, we establish that the exact entanglement cost of any bipartite quantum state under PPT-preserving operations is given by a single-letter formula, here called the $kappa$-entanglement of a quantum state. This formula is calculable by a semidefinite program, thus allowing for an efficiently computable solution for general quantum states. Notably, this is the first time that an entanglement measure for general bipartite states has been proven not only to possess a direct operational meaning but also to be efficiently computable, thus solving a question that has remained open since the inception of entanglement theory over two decades ago. Next, we introduce and solve the exact entanglement cost for simulating quantum channels in both the parallel and sequential settings, along with the assistance of free PPT-preserving operations. The entanglement cost in both cases is given by the same single-letter formula and is equal to the largest $kappa$-entanglement that can be shared by the sender and receiver of the channel. It is also efficiently computable by a semidefinite program.
Testing processes and workflows in information and Internet of Things systems is a major part of the typical software testing effort. Consistent and efficient path-based test cases are desired to support these tests. Because certain parts of software system workflows have a higher business priority than others, this fact has to be involved in the generation of test cases. In this paper, we propose a Prioritized Process Test (PPT), which is a model-based test case generation algorithm that represents an alternative to currently established algorithms that use directed graphs and test requirements to model the system under test. The PPT accepts a directed multigraph as a model to express priorities, and edge weights are used instead of test requirements. To determine the test-coverage level of test cases, a test-depth-level concept is used. We compared the presented PPT with five alternatives (i.e., the Process Cycle Test, a naive reduction of test set created by the Process Cycle Test, Brute Force algorithm, Set-covering Based Solution and Matching-based Prefix Graph Solution) for edge coverage and edge-pair coverage. To assess the optimality of the path-based test cases produced by these strategies, we used fourteen metrics based on the properties of these test cases and 59 models that were created for three real-world systems. For all edge coverage, the PPT produced more optimal test cases than the alternatives in terms of the majority of the metrics. For edge-pair coverage, the PPT strategy yielded similar results to those of the alternatives. Thus, the PPT strategy is an applicable alternative, as it reflects both the required test coverage level and the business priority in parallel.
We study the relation between qubit entanglement and Lorentzian geometry. In an earlier paper, we had given a recipe for detecting two qubit entanglement. The entanglement criterion is based on Partial Lorentz Transformations (PLT) on individual qubits. The present paper gives the theoretical framework underlying the PLT test. The treatment is based physically, on the causal structure of Minkowski spacetime, and mathematically, on a Lorentzian Singular Value Decomposition. A surprising feature is the natural emergence of Energy conditions used in Relativity. All states satisfy a Dominant Energy Condition (DEC) and separable states satisfy the Strong Energy Condition(SEC), while entangled states violate the SEC. Apart from testing for entanglement, our approach also enables us to construct a separable form for the density matrix in those cases where it exists. Our approach leads to a simple graphical three dimensional representation of the state space which shows the entangled states within the set of all states.
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