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arXiv:2109.02588 · Entropy

Common Coherence Witnesses and Common Coherent States

Bang-Hai Wang, Ziheng Ding, Zhihao Ma, Shao-Ming Fei

Published Aug 31, 2021• 6 citations
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Abstract

We show the properties and characterization of coherence witnesses. We show methods for constructing coherence witnesses for an arbitrary coherent state. We investigate the problem of finding common coherence witnesses for certain class of states. We show that finitely many different witnesses W1,W2,⋯,Wn can detect some common coherent states if and only if ∑i=1ntiWi is still a witnesses for any nonnegative numbers ti(i=1,2,⋯,n). We show coherent states play the role of high-level witnesses. Thus, the common state problem is changed into the question of when different high-level witnesses (coherent states) can detect the same coherence witnesses. Moreover, we show a coherent state and its robust state have no common coherence witness and give a general way to construct optimal coherence witnesses for any comparable states.

DOI
10.3390/e23091136

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We show the properties and characterization of coherence witnesses. We show methods for constructing coherence witnesses for an arbitrary coherent state. We investigate the problem of finding common coherence witnesses for certain class of states. We show that finitely many different witnesses W1,W2,⋯,Wn can detect some common coherent states if and only if ∑i=1ntiWi is still a witnesses for any nonnegative numbers ti(i=1,2,⋯,n). We show coherent states play the role of high-level witnesses. Thus, the common state problem is changed into the question of when different high-level witnesses (coherent states) can detect the same coherence witnesses. Moreover, we show a coherent state and its robust state have no common coherence witness and give a general way to construct optimal coherence witnesses for any comparable states.
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, the l1{l}_{1}l1​ norm, and the l2{l}_{2}l2​ norm of coherence measures. Furthermore, we compare one of our coherence witnesses 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.
The question of under what conditions different witnesses (e.g., W1,W2{W}_{1},{W}_{2}W1​,W2​) may detect some common entangled states [i.e., there exists some state $\ensuremath{\rho}$ so that $\mathrm{Tr}({W}_{1}\ensuremath{\rho})<0$ and $\mathrm{Tr}({W}_{2}\ensuremath{\rho})<0$] is answered for both finite-dimensional and infinite-dimensional bipartite systems. Finitely many different witnesses W1,W2,…,Wn{W}_{1},{W}_{2},\dots{},{W}_{n}W1​,W2​,…,Wn​ can detect some common entangled states if and only if ${\ensuremath{\sum}}_{i=1}^{n}{d}_{i}{W}_{i}$ is still a witness for any nonnegative numbers d1,d2,…,dn{d}_{1},{d}_{2},\dots{},{d}_{n}d1​,d2​,…,dn​ with ${\ensuremath{\sum}}_{i=1}^{n}{d}_{i}=1$; they cannot detect any common entangled state if and only if ${\ensuremath{\sum}}_{i=1}^{n}{c}_{i}{W}_{i}$ is a positive operator for some nonnegative numbers c1,c2,…,cn{c}_{1},{c}_{2},\dots{},{c}_{n}c1​,c2​,…,cn​ with ${\ensuremath{\sum}}_{i=1}^{n}{c}_{i}=1$. For two witnesses W1{W}_{1}W1​ and W2{W}_{2}W2​ more can be said. First, W1{W}_{1}W1​ and W2{W}_{2}W2​ can detect the same set of entangled states if and only if W1=aW2{W}_{1}=a{W}_{2}W1​=aW2​ for some number a>0a>0a>0. Second, W2{W}_{2}W2​ can detect more entangled states than W1{W}_{1}W1​ can if and only if W1=aW2+D{W}_{1}=a{W}_{2}+DW1​=aW2​+D for some number a>0a>0a>0 and a positive operator DDD. As an application, some characterizations of the optimal witnesses are given and some structural properties of the decomposable optimal witnesses are presented.
Abstract Quantum coherence is one of the most basic characteristics of quantum mechanics. Here we give some methods to detect and measure quantum coherence. Firstly, we propose a coherence criterion without full quantum state tomography based on partial transposition. Moreover, we present a coherent nonlinear detection strategy from witnesses, in which we find that for some coherent states, normal witness detection fails but our nonlinear detection succeeds. In addition, we prove that when the nonlinear detection on the two copies of the coherent state fails, the nonlinear detection on the three copies may be successful. Finally, due to the difficulty in calculating robustness of coherence for general states, we introduce a lower bound for coherent robustness based on the witness operator, and after comparing our lower bound with the currently known lower bound, one show that our lower bound is better. Coherence is believed to play a crucial role in quantum information tasks, making the detection and quantization of coherence particularly significant. Therefore, these results help to open up new avenues for advancement in quantum theory.
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.
Abstract 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 [ m , M ] d . We present necessary and sufficient conditions of detecting quantum coherence of d − dimensional quantum states based on C [ m , M ] d . 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.