arXiv subjects
Publications and source records attributed to Samuel R. Hedemann.
We present new generalizations of Cauchy-Schwarz (CS) inequalities to multiple vectors and use them to derive multi-operator quantum uncertainty relations and propose multi-operator squeezing.
We prove that all states (mixed or pure) of qubit-qutrit ($2\times 3$) systems have entanglement-preserving unitary (EPU) equivalence to a compact subset of true-generalized X (TGX) states called EPU-minimal TGX states which we give explicitly. Thus, for any spectrum-entanglement combination achievable by general states, there exists an EPU-minimal TGX state of the same spectrum and entanglement. We use I-concurrence to measure entanglement and give an explicit formula for it for all $2\times 3$ minimal TGX states (a more general set than EPU-minimal TGX states) whether mixed or pure, yielding its minimum average value over all decompositions. We also give a computable I-concurrence formula for a more general family called minimal super-generalized X (SGX) states, and give optimal decompositions for minimal SGX states and all of their subsets.
We present a full definition of mixed maximally entangled (MME) states for multipartite systems, generalizing their existing definition for bipartite systems by using multipartite Schmidt decomposition. MME states are a special kind of maximally entangled mixed state (MEMS) for which every pure decomposition state in all decompositions is maximally entangled. Thus, MME states have entanglement 1 by all valid unit-normalized entanglement measures, whereas general MEMS can have entanglement less than 1. Multipartite MME states likely have important applications such as remote state preparation, and also set critical performance goals for entanglement measures.
We present six new measures of nonlocal correlation for discrete multipartite quantum systems; correlance, statance, probablance, strong discordance, discordance, and diagonal discordance. The correlance measures all nonlocal correlation (even bound entanglement), and is exactly computable for all pure and mixed states. Statance and probablance are not yet computable, but motivate the strong discordance (for nonlocal correlation beyond that achievable by a strictly classical state), discordance (a measure of all nonlocal correlation in distinguishably quantum states), and diagonal discordance (for nonlocal correlation in diagonal states), all of which are exactly computable for all states. We discuss types of correlation and notions of classicality, and compare correlance, strong discordance, and discordance to quantum discord. We also define diagonal correlance to handle strictly classical probability distributions, providing a powerful tool with wide-ranging applications.
We examine the entanglement properties of a system that represents two driven microwave cavities each optomechanically coupled to two separate driven optical cavities which are connected by a single-mode optical fiber. The results suggest that it may be possible to achieve near-maximal entanglement of the microwave cavities, thus allowing a teleportation scheme to enable interactions for hybrid quantum computing of superconducting qubits with optical interconnects.
A multipartite entanglement measure called the ent is presented and shown to be an entanglement monotone, with the special property of automatic normalization. Necessary and sufficient conditions are developed for constructing maximally entangled states in every multipartite system such that they are true-generalized X states (TGX) states, a generalization of the Bell states, and are extended to general nonTGX states as well. These results are then used to prove the existence of maximally entangled basis (MEB) sets in all systems. A parameterization of general pure states of all ent values is given, and proposed as a multipartite Schmidt decomposition. Finally, we develop an ent vector and ent array to handle more general definitions of multipartite entanglement, and the ent is extended to general mixed states, providing a general multipartite entanglement measure.
We propose and examine several candidates for universal multipartite entanglement measures. The most promising candidate for applications needing entanglement in the full Hilbert space is the ent-concurrence, which detects all entanglement correlations while distinguishing between different types of distinctly multipartite entanglement, and simplifies to the concurrence for two-qubit mixed states. For applications where subsystems need internal entanglement, we develop the absolute ent-concurrence which detects the entanglement in the reduced states as well as the full state.
We present an explicit family of two-qubit X states with entanglement-preserving unitary (EPU) equivalence to the set of general states; that is, for any spectrum-entanglement combination achievable by general states, this family contains an X state of the same spectrum and entanglement. This idea was originally conjectured by the author and supported with strong numerical evidence in arXiv:1310.7038. Then, in Ann. Phys. 351 (2014) 79, the authors proved the existence of such two-qubit unitary transformations, but found the parameters to be transcendental, eluding explicit solution. Here, by a different method, we prove the existence of such transformations, obtain a compact implicit solution for them, and provide an exact, explicit form of the desired X-state family.
The Cayley-Hamilton problem of expressing functions of matrices in terms of only their eigenvalues is well-known to simplify to finding the inverse of the confluent Vandermonde matrix. Here, we give a highly compact formula for the inverse of any matrix, and apply it to the confluent Vandermonde matrix, achieving in a single equation what has only been achieved by long iterative algorithms until now. As a prime application, we use this result to get a simple formula for explicit exponential operators in terms of only their eigenvalues, with an emphasis on application to finite discrete quantum systems with time dependence. This powerful result permits explicit solutions to all Schrödinger and von Neumann equations for time-commuting Hamiltonians, and explicit solutions to any degree of approximation in the non-time-commuting case. The same methods can be extended to general finite discrete open systems to get explicit quantum operations for time evolution using effective joint systems, and the exact solution of all finite discrete Baker-Campbell-Hausdorff formulas.
We consider the problems of maximizing the entanglement negativity of X-form qubit-qutrit density matrices with (i) a fixed spectrum and (ii) a fixed purity. In the first case, the problem is solved in full generality whereas, in the latter, partial solutions are obtained by imposing extra spectral constraints such as rank-deficiency and degeneracy, which enable a semidefinite programming treatment for the optimization problem at hand. Despite the technically-motivated assumptions, we provide strong numerical evidence that three-fold degenerate X states of purity $P$ reach the highest entanglement negativity accessible to arbitrary qubit-qutrit density matrices of the same purity, hence characterizing a sparse family of likely qubit-qutrit maximally entangled mixed states.
A method is presented for achieving entanglement-free teleportation of a quantum state subject to any quantum noise. We apply this as a light-speed noise-resistant communicator, but also treat the possibility of a quantum ansible, a device for effectively superluminal communication and quantum broadcasting. The results suggest a "no-projector theorem" analogous to the no-cloning theorem. We then show how to build a pseudo-ansible for connection-free light-speed communication.
It is generally assumed that on-off detectors with single-photon sensitivity cannot distinguish coherent states from phase-mixed coherent states without some form of quadrature-based tomography. Here, we show that it is theoretically possible to distinguish these states without quadrature-based methods by comparing the vacuum probability of one output port of a variable beam splitter (VBS) that has two different phase-mixed coherent states as its inputs to the well-known case of dual coherent-state inputs. As an application, a method is proposed to test whether a laser field is in a coherent state over a given time by simply measuring the VBS output populations with an on-off single-photon detector. Unfortunately, timing limitations of present technology prohibit such simple tests, but the technique may become practical in the future.
We prove that if any error channel has a Kraus decomposition that is simultaneously correctable and Hilbert-Schmidt (HS) complete, then the existence of Kraus sets with these properties guarantees the correctability of all quantum channels. As a proof of the existence of such Kraus sets, the $n$-level depolarization channel is shown to have a random-unitary (RU) decomposition that is both HS complete and correctable due its RU nature, thereby proving that all quantum channels are correctable. As an application, conditions for universal error-correction operations are presented.
Non-Markovian evolution of an open quantum system can be induced by the memory effects of a reservoir. Although a reservoir with stronger memory effects may seem like it should cause stronger non-Markovian effects on the system of interest, this seemingly intuitive thinking may not always be correct. We illustrate this by investigating a qubit (a two-level atom) that is coupled to a hierarchical environment, which contains a single-mode cavity and a reservoir consisting of infinite numbers of modes. We show how the non-Markovian character of the system is influenced by the coupling strength between the qubit and cavity and the correlation time of the reservoir. In particular, we found a new phenomenon whereby the qubit Markovian and non-Markovian transition exhibits a anomalous pattern in a parameter space depicted by the coupling strength and the correlation time of the reservoir.
Strong numerical evidence is presented suggesting that all two-qubit mixed states are equivalent to X states by a single entanglement-preserving unitary (EPU) transformation, so that the concurrence of such an X state equals that of the original general state. An X-state parameterization of a general two-qubit state is given, allowing all states to have their concurrence parametrically specified. A new kind of entanglement measure is proposed, relating a general state's entanglement to that of a pure state in the same system. New states called "H States" are presented, having fully parametric concurrence and purity, with the intention of using them to construct entanglement-preserving depolarization channels, which may aid development of the new entanglement measure. A theory of "true-generalized" X states (TGX states) is proposed for the general case of $N$-partite systems. While such states do not generally have the literal "X" shape, evidence is shown that they are the true generalizations of X states in larger systems, since they appear to always be EPU-equivalent to general states of all ranks, whereas literal X states generally are not. An example of this is given for $2\times 3$, including the proposition of the $2\times 3$ maximally entangled mixed states (MEMS). If the claim that TGX states are universal is valid, then any entanglement measure may be computable in a simpler form by using the EPU-equivalence between general states and TGX states.
In this paper, we propose an explicit scheme to fully recover a multiple-qubit state subject to a phase damping noise. We establish the theoretical framework and the operational procedure to restore an unknown initial quantum state for an N-qubit model interacting with either individual baths or a common bath. We give an explicit construction of the random unitary (RU) Kraus decomposition for an N-qubit model interacting with a common bath. We also demonstrate how to use only one unitary reversal operation to restore an arbitrary state with phase damping noise. In principle, the initial state can always be recovered with a success probability of 1. Interestingly, we found that non-RU decomposition can also be used to restore some particular entangled states. This may open a new path to restore a quantum state beyond the standard RU scheme.