Enhanced entanglement criterion via symmetric informationally complete measurements

Entanglement criteria based on quantum Fisher information

The correlation matrix (CM) criterion is a recently derived powerful sufficient condition for the presence of entanglement in bipartite quantum states of arbitrary dimensions. It has been shown that it can be stronger than the positive partial transpose (PPT) criterion, as well as the computable cross norm or realignment (CCNR) criterion in different situations. However, it remained as an open question whether there existed sets of states for which the CM criterion could be stronger than both criteria simultaneously. Here, we give an affirmative answer to this question by providing examples of entangled states that scape detection by both the PPT and CCNR criteria whose entanglement is revealed by the CM condition. We also show that the CM can be used to measure the entanglement of pure states and obtain lower bounds for the entanglement measure known as tangle for general (mixed) states.

The general individual (non-coherent) attack on the ping-pong protocol with completely entangled pairs of three-dimensional quantum systems (qutrits) is analyzed. The expression for amount of the eavesdropper's information as functions from probability of attack detection is derived. It is shown, that the security of the ping-pong protocol with pairs of qutrits is higher the security of the protocol with pairs of qubits. It is also shown, that with the use by legitimate users in a control mode two mutually unbiased measuring bases the ping-pong protocol with pairs of qutrits, similar to the protocol with groups of qubits, possesses only asymptotic security and requires additional methods for its security amplification.

From an operational point of view, several new entanglement detection criteria are proposed using quantum designs. These criteria are constructed by considering the correlations defined with quantum designs. Counter‐intuitively, the criteria with more settings are exactly equivalent to the corresponding ones with the minimal number of settings, namely the symmetric informationally complete positive operator‐valued measures (SIC POVMs). Fundamentally, this observation highlights the potentially unique role played by SIC POVMs in quantum information processing. Experimentally, this provides the minimal number of settings that one should choose for detecting entanglement. Furthermore, it is found that nonlinear criteria are not always better than linear ones for the task of entanglement detection.

Realignment Operation and CCNR Criterion of Separability for States in Infinite-Dimensional Quantum Systems

We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the 'realignment' or 'computable cross norm' criterion; in particular, we highlight the fact that this criterion relies only on the Schmidt equivalence class of a state. Then, the relevance — with regard to the characterization of entanglement — of the 'symmetric polynomials' in the Schmidt coefficients and a new family of separability criteria that generalize the realignment criterion are discussed. Various interesting open problems are proposed.

Inspired by the ‘computable cross norm’ or ‘realignment’ criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states are derived. We conjecture that these conditions, which are weaker than the parent criterion, can be strengthened in such a way to obtain a new family of criteria that are independent of the original one. This conjecture is supported by numerical examples for the low dimensional cases. These ideas can be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement.

On the basis of methods of quantum information theory the eavesdropping attack with the use of auxiliary quantum systems on ping – pong protocol with completely entangled pair of three – dimensional quantum systems – qutrits is analyzed. The formula for quantity of the eavesdropper’s information as functions from probability of attack’s detection is deduced. It is shown, that the resistance against the attack of ping – pong protocol with pairs of qutrits is above the resistance of the protocol with pair of qubits

We propose a family of entanglement witnesses and corresponding positive maps that are not completely positive based on local orthogonal observables. As applications the entanglement witness of a 3x3 bound entangled state [P. Horodecki, Phys. Lett. A 232, 333 (1997)] is explicitly constructed and a family of dxd bound entangled states is introduced, whose entanglement can be detected by permuting local orthogonal observables. The proposed criterion of separability can be physically realized by measuring a Hermitian correlation matrix of local orthogonal observables.

We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows us to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria that have been proposed to complement the positive partial transpose criterion---the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others---are in fact a corollary of the covariance matrix criterion.

Entanglement detection and estimation are fundamental problems in quantum information science. Compared with discrete-variable states, for which lots of efficient entanglement detection criteria and lower bounds of entanglement measures have been proposed, the continuous-variable entanglement is much less understood. Here we shall present a family of entanglement witnesses based on continuous-variable local orthogonal observables (CVLOOs) to detect and estimate entanglement of Gaussian and non-Gaussian states, especially for bound entangled states. By choosing an optimal set of CVLOOs, our entanglement witness is equivalent to the realignment criterion and can be used to detect bound entanglement of a class of 2+2 mode Gaussian states. Via our entanglement witness, lower bounds of two typical entanglement measures for arbitrary two-mode continuous-variable states are provided.

Improved bounds on some entanglement criteria in bipartite quantum systems

We present an explicit construction of entanglement witnesses for depolarized states in arbitrary finite dimension. For infinite dimension we generalize the construction to twin-beams perturbed by Gaussian noises in the phase and in the amplitude of the field. We show that entanglement detection for all these families of states requires only three local measurements. The explicit form of the corresponding set of local observables (quorom) needed for entanglement witness is derived.

Entanglement witnesses are invaluable for efficient quantum entanglement certification without the need for expensive quantum state tomography. Yet, standard entanglement witnessing requires multiple measurements and its bounds can be elusive as a result of experimental imperfections. Here, we introduce and demonstrate a novel procedure for entanglement detection which simply and seamlessly improves any standard witnessing procedure by using additional available information to tighten the witnessing bounds. Moreover, by relaxing the requirements on the witness operators, our method removes the general need for the difficult task of witness decomposition into local observables. We experimentally demonstrate entanglement detection with our approach using a separable test operator and a simple fixed measurement device for each agent. Finally, we show that the method can be generalized to higher-dimensional and multipartite cases with a complexity that scales linearly with the number of parties.

This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (SIC POVMs) in this paper are called regular quantum 2-designs with degree 1. The assigned vectors span b2 equiangular lines. These objects had been investigated since the 1960s, but only a few solutions were known in complex vector spaces. In this thesis further maximal analytic and numerical solutions were given (today a lot more solutions are known) and it was (probably for the first time) conjectured that solutions exist in any dimension b (generated by the Weyl–Heisenberg group and with a certain additional symmetry of order 3).