Imaginarity has shown to be an important resource in quantum information. The witness theory of quantum resource, such as entanglement witness, coherence witness, and imaginarity witness, has been established, in particular entanglement witness and coherence witness have been extensively explored. We present here another class of imaginarity witnesses beyond the one in [{Phys. Lett. A} \textbf{530}, 130135 (2025)]. Within our framework, any nonreal Hermitian operator under the reference basis is an imaginarity witness and only finite of them can detect all the imaginarity states. As the common approaches that were explored in the witness theories of entanglement and coherence, we then explore the relations between these imaginarity witnesses in different cases: (i) when they can detect common imaginary states, (ii) when they can detect the same sets of imaginary states, and (ii) when they obey the finer relation. Finally, we define an imaginarity measure in terms of witnesses, termed witnessed imaginarity, and prove that it coincides with both the trace norm of imaginarity and the robustness of imaginarity.
We study the Witness Set problem, a natural dual to the classical Art Gallery problem. In the Witness Set problem, we are given a polygon $P$ and an integer $k$ as input, and the objective is to determine whether $P$ has a witness set of size at least $k$. A point set $X$ in $P$ is called a witness set if every point in $P$ is visible from at most one point in $X$. For simple polygons, we show that Witness Set lies in both $NP$ and $XP$. This stands in sharp contrast to its dual, the Art Gallery problem, which was recently shown to be $\exists \mathbb{R}$-complete by Abrahamsen et al. and is therefore neither in $NP$ nor admits a polynomial-size discretization unless $NP=\exists \mathbb{R}$. In contrast, we prove that Witness Set for simple polygons admits a finite discretization of size $n^{f(k)}$ for some function $f$. For comparison, even for simple polygons, Efrat and Har-Peled gave an algorithm for Art Gallery running in time $n^{O(k)}$ using tools from real algebraic geometry, and it appears difficult to obtain such algorithms without this machinery. On the other hand, our approach for Witness Set is purely combinatorial and relies on discretization, leading to an $n^{f(k)}$-
Given a set $Γ$ of $k$ unlabelled posets, each of size $n$, we say that a poset $Q$ is a \emph{witness} to $Γ$ if $Γ$ is the set of downsets of size $n$ of $Q$. We say that $Q$ is a \emph{minimal witness} if it does not contain a proper downset that is itself a witness to $Γ$. Motivated by the causal set approach to quantum gravity, we study the upper bound on the size of minimal witnesses as a function of $n$ and $k$. We show that there is no linear upper bound of the form $n+k+c$ for any constant $c$. We introduce the \emph{exchange graph of downsets} as a new tool to study this scenario, and use it to show that all minimal witnesses $Q$ satisfy the bound $|Q|\leq nk-n$, and that when $k=3$ there is at least one minimal witness $Q$ that satisfies the bound $|Q|\leq \frac{3}{2}(n+1)$.
Statistical witness indistinguishability is a relaxation of statistical zero-knowledge which guarantees that the transcript of an interactive proof reveals no information about which valid witness the prover used to generate it. In this paper we define and initiate the study of QSWI, the class of problems with quantum statistically witness indistinguishable proofs. Using inherently quantum techniques from Kobayashi (TCC 2008), we prove that any problem with an honest-verifier quantum statistically witness indistinguishable proof has a 3-message public-coin malicious-verifier quantum statistically witness indistinguishable proof. There is no known analogue of this result for classical statistical witness indistinguishability. As a corollary, our result implies SWI is contained in QSWI. Additionally, we extend the work of Bitansky et al. (STOC 2023) to show that quantum batch proofs imply quantum statistically witness indistinguishable proofs with inverse-polynomial witness indistinguishability error.
Coherence is a fundamental resource in quantum information processing, which can be certified by a coherence witness. In order to detect all the coherent states, we introduce a useful concept of coherence witness and structure the set of coherence witnesses $C^{d}_{[m,M]}$. We present necessary and sufficient conditions of detecting quantum coherence of $d-$dimensional quantum states based on $C^{d}_{[m,M]}$. Moreover, we show that each coherent state can be detected by one of the coherence witnesses in a finite set. The corresponding finite set of coherence witnesses is presented explicitly, which detects all the coherent states.
We define the stringent coherence witness as an observable which has zero mean value for all of incoherent states and hence a nonzero mean value indicates the coherence. The existence of such witnesses are proved for any finite-dimension states. Not only is the witness efficient in testing whether the state is coherent, the mean value is also quantitatively related to the amount of coherence contained in the state. For an unknown state, the modulus of the mean value of a normalized witness provides a tight lower bound of $l_1$-norm of coherence in the state. When we have some previous knowledge of the state, the optimal witness is derived such that its measured mean value, called the witnessed coherence, equals to the $l_1$-norm of coherence. One can also fix the witness and implement some incoherent operations on the state before the witness is measured. In this case, the measured mean value cannot reach the witnessed coherence if the initial state and the fixed witness does not match well. Based this result, we design a quantum coherence game. Our results provides a way to directly measure the coherence in arbitrary finite dimension states and an operational interpretation of the
We study extremal entanglement witnesses on a bipartite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus $\textrm{Tr}\,Ωρ\geq 0$ when $Ω$ is a witness and $ρ$ a pure product state, $ρ=ψψ^{\dagger}$ with $ψ=φ\otimesχ$. The set of witnesses of unit trace is a compact convex set, defined by its extremal points. The expectation value $f(φ,χ)=\mathrm{Tr}\,Ωρ$ as a function of $φ$ and $χ$ is a nonnegative biquadratic form. Every zero of $f(φ,χ)$ imposes real-linear constraints on $f$ and $Ω$. The Hessian matrix at the zero must be nonnegative. Its eigenvectors with zero eigenvalue, if any, we call Hessian zeros. A zero of $f(φ,χ)$ is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic otherwise. We prove that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, the constraints defined by its zeros and Hessian zeros determine all directions in which to search for witness
An entanglement witness approach to quantum coherent state key distribution and a system for its practical implementation are described. In this approach, eavesdropping can be detected by a change in sign of either of two witness functions, an entanglement witness S or an eavesdropping witness W. The effects of loss and eavesdropping on system operation are evaluated as a function of distance. Although the eavesdropping witness W does not directly witness entanglement for the system, its behavior remains related to that of the true entanglement witness S. Furthermore, W is easier to implement experimentally than S. W crosses the axis at a finite distance, in a manner reminiscent of entanglement sudden death. The distance at which this occurs changes measurably when an eavesdropper is present. The distance dependance of the two witnesses due to amplitude reduction and due to increased variance resulting from both ordinary propagation losses and possible eavesdropping activity is provided. Finally, the information content and secure key rate of a continuous variable protocol using this witness approach are given.
A pair $\langle G_0, G_1 \rangle$ of graphs admits a mutual witness proximity drawing $\langle Γ_0, Γ_1 \rangle$ when: (i) $Γ_i$ represents $G_i$, and (ii) there is an edge $(u,v)$ in $Γ_i$ if and only if there is no vertex $w$ in $Γ_{1-i}$ that is ``too close'' to both $u$ and $v$ ($i=0,1$). In this paper, we consider infinitely many definitions of closeness by adopting the $β$-proximity rule for any $β\in [1,\infty]$ and study pairs of isomorphic trees that admit a mutual witness $β$-proximity drawing. Specifically, we show that every two isomorphic trees admit a mutual witness $β$-proximity drawing for any $β\in [1,\infty]$. The constructive technique can be made ``robust'': For some tree pairs we can suitably prune linearly many leaves from one of the two trees and still retain their mutual witness $β$-proximity drawability. Notably, in the special case of isomorphic caterpillars and $β=1$, we construct linearly separable mutual witness Gabriel drawings.
If a laser- or particle beam-driven plasma wakefield accelerator operates in the linear or moderately nonlinear regime, injecting an externally produced particle bunch (witness) to be accelerated may encounter an alignment problem. Witness alignment tolerances can be relaxed by using a damper, an additional particle bunch produced by the same injector and propagating at a submillimeter distance ahead of the witness. If misaligned, the damper perturbs the wakefield in such a way that the witness shifts on-axis with no quality loss.
Unitarity and the optical theorem are used to derive the reduced density matrices of Compton scattering in the presence of a witness particle. Two photons are initially entangled wherein one photon participates in Compton scattering while the other is a witness, i.e. does not interact with the electron. Unitarity is shown to require that the entanglement entropy of the witness photon does not change after its entangled partner undergoes scattering. The final mutual information of the electronic and witness particle's polarization is nonzero for low energy Compton scattering. This indicates that the two particles become correlated in spite of no direct interaction. Assuming an initial maximally entangled state, the change in entanglement entropy of the scattered photon's polarization is calculated in terms of Stokes parameters. A common ratio of areas occurs in the final reduced density matrix elements, von Neumann entropies, Stokes parameter, and mutual information. This common ratio consists of the Thomson scattering cross-section and an accessible regularized scattering area.
Quantum coherence has wide-ranging applications from quantum thermodynamics to quantum metrology, quantum channel discrimination and even quantum biology. Thus, detecting and quantifying coherence are two fundamental problems in quantum resource theory. Here, we introduce feasible methods to detect and estimate the coherence by constructing coherence witnesses for any finite-dimensional states. Our coherence witnesses detect coherent states by testing whether the expectation value of the witness is negative or not. Two typical coherence witnesses are proposed and discussed based on our witness-constructing method, which are also used to estimate the robustness of coherence, $l_1$-norm and $l_2$-norm of coherence measures. Furthermore, we compare one of our coherence witness with a previously introduced witness, by proving that our witness is strictly stronger than that previous witness. We also present an application of coherence in a quantum metrology task, in which we estimate an unknown parameter by measuring our coherence witness.
There always exists an entanglement witness for every entangled quantum state. Negativity of the expectation value of an entanglement witness operator guarantees entanglement of the corresponding state, given that the measurement devices involved are perfect, i.e., the performed measurements actually constitute the witness operator for the state under consideration. In a realistic situation, there are two possible ways of measurements to drive the process away from the ideal one. Firstly, wrong measurements may be performed, and secondly, while the measurement operators are implemented correctly, the detection process is noisy. Entanglement witnesses are prone to both of these imperfections. The concept of measurement-device-independent entanglement witnesses was introduced to remove the first problem. We analyze the "detection loophole" in the context of measurement-device-independent entanglement witnesses, which deal with the second problem of imprecise measurements. We obtain an upper bound on the entanglement witness function in the measurement-device-independent entanglement witness scenario, below which entanglement is guaranteed for given non-ideal detector efficiencies, th
We propose a generalized form of optimal teleportation witness to demonstrate their importance in experimental detection of the larger set of entangled states useful for teleportation in higher dimensional systems. The interesting properties of our witness reveal that teleportation witness can be used to characterize mixed state entanglement using Schmidt numbers. Our results show that while every teleportation witness is also a entanglement witness, the converse is not true. Also, we show that a hermitian operator is a teleportation witness iff it is a decomposable entanglement witness. In addition, we analyze the practical significance of our study by decomposing our teleportation witness in terms of Pauli and Gell-Mann matrices, which are experimentally measurable quantities.
We present a null witness of the dimension of a quantum system, discriminating real, complex and classical spaces, based on equality due to linear independence. The witness involves only a single measurement with sufficiently many outcomes and prepared input states. In addition, for intermediate dimensions, the witness bounds saturate for a family of equiangular tight frames including symmetric informationally complete positive operator valued measures. Such a witness requires a minimum of resources, being robust against many practical imperfections. We also discuss errors due to finite statistics.
Bell inequalities, considered within quantum mechanics, can be regarded as non-optimal witness operators. We discuss the relationship between such Bell witnesses and general entanglement witnesses in detail for the Bell inequality derived by Clauser, Horne, Shimony, and Holt (CHSH). We derive bounds on how much an optimal witness has to be shifted by adding the identity operator to make it positive on all states admitting a local hidden variable model. In the opposite direction, we obtain tight bounds for the maximal proportion of the identity operator that can be subtracted from such a CHSH witness, while preserving the witness properties. Finally, we investigate the structure of CHSH witnesses directly by relating their diagonalized form to optimal witnesses of two different classes.
In a celebrated paper, Valiant and Vazirani raised the question of whether the difficulty of NP-complete problems was due to the wide variation of the number of witnesses of their instances. They gave a strong negative answer by showing that distinguishing between instances having zero or one witnesses is as hard as recognizing NP, under randomized reductions. We consider the same question in the quantum setting and investigate the possibility of reducing quantum witnesses in the context of the complexity class QMA, the quantum analogue of NP. The natural way to quantify the number of quantum witnesses is the dimension of the witness subspace W in some appropriate Hilbert space H. We present an efficient deterministic procedure that reduces any problem where the dimension d of W is bounded by a polynomial to a problem with a unique quantum witness. The main idea of our reduction is to consider the Alternating subspace of the d-th tensor power of H. Indeed, the intersection of this subspace with the d-th tensor power of W is one-dimensional, and therefore can play the role of the unique quantum witness.
The primary goal of entanglement theory is to determine convertibility conditions for two quantum states. Up until now, this has always been done with the use of entanglement monotones. With the exception of the negativity, such quantities tend to be rather uncomputable. We instead promote the idea of conversion witnesses in this paper. A conversion witness is a function on pairs of states and whose value determines whether a state can be converted into another. We construct a conversion witness that can be efficiently computed for arbitrary states in systems of any size. This conversion witness is always better than the negativity at detecting when two entangled states are not interconvertible. Furthermore, when considering states of two-qubit systems, this new conversion witness is sometimes better than the entanglement of formation. This shows that the study of conversion witness is in fact useful, and may have applications in resource theories beyond that of entanglement.
Teleportation witnesses are hermitian operators which can identify useful entanglement for quantum teleportation. Here we provide a systematic method to construct teleportation witnesses from entanglement witnesses corresponding to general qudit systems. The witnesses so constructed are shown to be optimal for qubit and qutrit systems, and therefore detect the largest set of states useful for teleportation within a given class. We demonstrate the action of the witness pertaining to different classes of states in qubits and qutrits. Decomposition of the witness in terms of spin operators facilitiates experimental identification of useful resources for teleportation.
The ultrafine entanglement witness, introduced in [F. Shahandeh, M. Ringbauer, J.C. Loredo, and T.C. Ralph, Phys. Rev. Lett. \textbf{118}, 110502 (2017)], can seamlessly and easily improve any standard entanglement witness. In this paper, by combining the constraint and the test operators, we rotate the hyperplane determined by the test operator and improve further the original ultrafine entanglement witness. In particular, we present a series of new ultrafine entanglement witnesses, which not only can detect entangled states that the original ultrafine entanglement witnesses cannot detect, but also have the merits that the original ultrafine entanglement witnesses have.