Shun-Long Luo^,?, Yuan Sun^?Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing 100049, China

Corresponding authors: ?E-mail:luosl@amt.ac.cn

Received:2019-06-29Online:2019-12-1

Fund supported:

*Supported by the National Natural Science Foundation of China under Grant .No. 11875317 the National Center for Mathematics and Interdisciplinary Sciences, and Chinese Academy of Sciences under Grant.No. Y029152K51

Abstract
Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables. In terms of the coherence measure based on the Wigner-Yanase skew information, we establish several uncertainty relations for coherence with respect to von Neumann measurements, mutually unbiased bases (MUBs), and general symmetric informationally complete positive operator valued measurements (SIC-POVMs), respectively. Since coherence is intimately connected with quantum uncertainties, the obtained uncertainty relations are of intrinsically quantum nature, in contrast to the conventional uncertainty relations expressed in terms of variance, which are of hybrid nature (mixing both classical and quantum uncertainties). From a dual viewpoint, we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement. In particular, it is shown that if the density operators representing the quantum states do not commute, then there is no measurement (reference basis) such that the coherence of these states can be simultaneously small.
Keywords： uncertainty relations;coherence;Wigner-Yanase skew information;quantum uncertainty;von Neumann measurements


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Shun-Long Luo, Yuan Sun. Uncertainty Relations for Coherence*. [J], 2019, 71(12): 1443-1447 doi:10.1088/0253-6102/71/12/1443

1 Introduction

The Heisenberg uncertainty principle arising from incompatible (noncommuting) observables asserts a fundamental limit to quantum measurements, and is one of the characteristic consequences of quantum mechanics with deep connections to the Bohr complementarity principle. Uncertainty relations, as manifestations of the Heisenberg uncertainty principle, have been extensively and intensively studied with a wide range of applications in many fields. For example, it is closely related to quantum measurement and signal processing,^[1-2] preparation of state,^[3] complementarity,^[4-5] entanglement detections,^[6-9] quantum coherence,^[10-13] quantum non-locality.^[14-16] There are various quantitative characterizations of uncertainty relations such as entropic uncertainty relations,^[17-20] uncertainty relations based on variance and the Wigner-Yanase skew information,^[21-25] and so on.

Coherence is intrinsically related to superposition which differentiates quantum mechanics from classical mechanics. In recent years, there are increasing interests in quantitative studies of coherence.^[26-40] As a kind of quantum resource, coherence plays an important role in a variety of operational applications in asymmetry,^[41-45] metrology,^[46] quantum key distributions,^[47] thermodynamics,^[48-50] quantum computation and communication.^[51-53]

Since coherence of a quantum state depends on the choice of measurements (reference bases), it is natural to study the relations of coherence between two or more different measurements, or coherence of different states with respect to a fixed measurement. In fact, there are several investigations on uncertainty relations for quantum coherence,^[10-13] as well as the complementarity of coherence in different bases.^[54-55] Based on the skew information introduced by Wigner and Yanase,^[56] an information-theoretic measure of coherence has been introduced in Refs. ^[35--37] (see Sec. 2), which has an operational interpretation as quantum uncertainty, in sharp contrast to the conventional notion of variance (which usually involves both classical and quantum uncertainties). The aim of this paper is to employ this coherence measure to characterize uncertainty relations for coherence in an intrinsically quantum fashion.

The paper is organized as follows. In Sec. 2, we establish several uncertainty relations for coherence with respect to arbitrary von Neumann measurements (orthonormal bases), as well as with respect to mutually unbiased bases (MUBs). In Sec. 3, we study coherence with respect to general symmetric informationally complete positive operator valued measurements (SIC-POVMs), and derive the corresponding uncertainty relations. We obtain some uncertainty relations for several quantum states with respect to a common measurement in Sec. 4. Finally, we summarize in Sec. 5.

2 Uncertainty Relations for Coherence with Respect to von Neumann Measurements

Let $\rho$ be a quantum state (density operator) and $\Pi=$ $\{|i\rangle\langle i|\}$ be a von Neumann measurement (i.e., $\{|i\rangle \}$ constitutes an orthonormal basis for the system Hilbert space), then one can consider coherence of $\rho$ with respect to $\Pi$. In this paper, we employ the coherence measure

(1)$$ C(\rho,\Pi)=\sum_{i}I(\rho,|i\rangle\langle i|)\,,$$
introduced in Refs. ^[35--37] to characterize uncertainty relations. Here

$ I(\rho,H)=-\frac{1}{2}{\rm tr}[\sqrt{\rho},H]^2 $

is the skew information introduced by Wigner and Yanase in 1963,^[56] $H$ is an arbitrary observable (Hermitian operator). It is remarkable that the skew information enjoys many nice properties and has several interpretations as non-commutativity (between $\rho$ and $H$), quantum Fisher information (of $\rho$ with respect to $H$), asymmetry (of $\rho$ with respect to $H$), quantum uncertainty (of $H$ in $\rho$),^{[21-24,37-38]} etc. More generally, for any POVM $M=\{M_i\}$ (i.e., $M_i\geq 0, \sum _i M_i={\bf 1}),$ we may define a bona fide measure for coherence of $\rho $ with respect to $M$ as^[37]

(2) $$ C(\rho, M)=\sum _i I(\rho, M_i)\,.$$
The above measure reduces to that defined by Eq. ({1}) when $M$ is a von Neumann measurement. In general, the measurement operators in $M$ may not be mutually orthogonal projections (e.g., coherent states in quantum optics and spin systems), and in this case $C(\rho, M)$ generalizes $C(\rho, \Pi)$ considerably.

Let $\Pi_1=\{|{u_i}\rangle \langle u_i|\}$, $\Pi_2=\{|{v_i}\rangle\langle v_i|\}$ be two arbitrary von Neumann measurements, then our first result is the following inequality

(3)$$ C(\rho,\Pi_1)+C(\rho,\Pi_2)\geq \frac{1}{2}\|[\sqrt{\rho},U^{\dagger}\sqrt{\rho}U]\|_{F}^{2}\,,$$
which may be regarded as a kind of uncertainty relations for coherence of the quantum state $\rho$ with respect to the measurements $\Pi_1$ and $\Pi _2$. Here $||A||_{F}=({\rm tr}A^{\dagger}A)^{1/2}$ is the Frobenius norm, $U$ is the unitary operator defined by $U|{u_i}\rangle=|{v_i}\rangle$, i.e., $U=\sum _i |v_i\rangle \langle u_i|$.

To prove inequality ({3}), we first recall a mathematical result in Ref. ^[57], Remark 5.1: If one of the matrices $A$ and $B$ is non-negative, then

(4) $$ \|[A,B]\|_{F}\leq\|A\|_{F}\|B\|_{F}\,.$$
Similar to the method in Ref. ^[58], consider the decompositions of the matrices

(5) $$ \sqrt{\rho}=X+D,\qquad U^{\dagger}\sqrt{\rho}U=Y+D_U\,,$$
where $D$ and $D_U$ are the diagonal parts of $\rho$ and $U^{\dagger}\rho U$ respectively. It is obvious that $X$, $Y$ have zero diagonal elements.Then by inequality ({4}), the triangle inequality of the norm $\|\cdot\|_{F}$, and the elementary inequality $x^2+y^2\geq(x+y)^2/2$ for any real numbers $x$, $y$, we have

$ \|[\sqrt{\rho},U^{\dagger}\sqrt{\rho}U]\|_{F}^2=\|[X+D,Y+D_U]\|_{F}^2\\ \quad =\|[X+D,Y]+[X+D,D_U]\|_{F}^2\\ \quad \leq(\|[X+D,Y]\|_{F}+\|[X+D,D_U]\|_{F})^2\\ \quad =(\|[X+D,Y]\|_{F}+\|[X,D_U]\|_{F})^2\\ \quad \leq(\|\sqrt{\rho}\|_{F}\|Y\|_{F}+\|X\|_{F}\|D_U\|_{F})^2\\ \quad \leq(\|Y\|_{F}+\|X\|_{F})^2 \leq2\|Y\|_{F}^2+2\|X\|_{F}^2\,.$

Therefore

$ 1/2\|[\sqrt{\rho},U^{\dagger}\sqrt{\rho}U]\|_{F}^2 \leq\|Y\|_{F}^2+\|X\|_{F}^2\\ \quad =\sum_{i\neq j}(|\langle u_i|\sqrt{\rho}|u_j\rangle|^2+|\langle u_i|U^{\dagger}\sqrt{\rho}U|u_j\rangle|^2)\\ \quad =\sum_{i\neq j}|\langle u_i|\sqrt{\rho}|u_j\rangle|^2+\sum_{i\neq j}|\langle v_i|\sqrt{\rho}|v_j\rangle|^2\\ \quad =C(\rho,\Pi_1)+C(\rho,\Pi_2)\,,$

which completes the proof.

From an alternative perspective, we have the following inequality

(6) $$ C(\rho,\Pi_1)+C(\rho,\Pi_2)\geq \frac 12 ({1-||G||_{F}^{2}})\,,$$
which is also a kind of uncertainty relations for coherence. Here $G=D_1TD_2$ with

$ D_1={\rm diag}(\langle u_1|\rho|u_1\rangle^{1/4},\langle u_2|\rho|u_2\rangle^{1/4},\ldots,\langle u_d|\rho|u_d\rangle^{1/4})\,,\\\ D_2={\rm diag}(\langle v_1|\rho|v_1\rangle^{1/4},\langle v_2|\rho|v_2\rangle^{1/4},\ldots,\langle v_d|\rho|v_d\rangle^{1/4})\,,\\\ T=(|\langle u_i|v_j\rangle|)\,.$

The derivation of inequality ({6}) is as follows.

$ C(\rho,\Pi_1)+C(\rho,\Pi_2)\\\ \quad =1-{\rm tr}\sqrt{\rho}\Pi_1(\sqrt{\rho})+1-{\rm tr}\sqrt{\rho}\Pi_2(\sqrt{\rho})\\\ \quad \geq1-{\rm tr}\sqrt{\rho}\sqrt{\Pi_1(\rho)}+1-{\rm tr}\sqrt{\rho}\sqrt{\Pi_2(\rho)}\\\ \quad =(D_{H}^2(\rho,\Pi_1(\rho))+D_{H}^2(\rho,\Pi_2(\rho)))/2\\\ \quad \geq D_{H}^2(\Pi_1(\rho),\Pi_2(\rho))/4\\\ \quad =(1-{\rm tr}\sqrt{\Pi_1(\rho)}\sqrt{\Pi_2(\rho)})/2\\\ \quad =\Big(1-\sum_{ij}\sqrt{\langle u_i|\rho|u_i\rangle}\sqrt{\langle v_j|\rho|v_j\rangle}|\langle u_i|v_j\rangle|^2\Big)\Big/2\\\ \quad =\frac 12({1-\|G\|_{F}^{2}})\,,$

where

$ D_{H}^2(\rho,\sigma)=||\sqrt \rho -\sqrt \sigma ||_{F}^2=2(1-{\rm tr}\sqrt{\rho}\sqrt{\sigma})$

is the square of quantum Hellinger distance between $\rho$ and $\sigma$.

The first inequality in the above derivation follows from the Kadison inequality, which states that^[59] $\Phi(A)^2\leq\Phi(A^2)$ for any unital and positive quantum operation $\Phi$ and Hermitian operator $A$,while the second inequality is obtained by the triangle inequality of quantum Hellinger distance.

Combining the above two uncertainty relations, we have

(7) $$ C(\rho,\Pi_1)+C(\rho,\Pi_2)\geq \frac 12 \max\{C_1,C_2\}\,,$$
where

$ C_1=\|[\sqrt{\rho},U^{\dagger}\sqrt{\rho}U]\|_{F}^{2}\,,\quad C_2=1-\|G\|_{F}^{2}\,.$

Next, we consider coherence with respect to mutually unbiased bases (MUBs).

Recall that two orthonormal bases $B_1=\{|b_{1j}\rangle : j=1,2,\ldots, d \}$ and $B_2=\{|b_{2j}\rangle : j=1,2,\ldots, d \}$ of a $d$-dimensional system Hilbert space are mutually unbiased if^[60-61]

$|\langle b_{1j}|b_{2k}\rangle |^2=1/d\,, \quad \text{for all} \ \ j,k\,.$ When the dimension $d$ is a prime power (i.e., $d=p^k$ for a prime number $p$ and a positive integer $k$), there exists a complete set of $d+1$ MUB $B_\nu =\{|b_{\nu j}\rangle : j =1,2,\ldots, d\}$, $\nu =1,2,\ldots,d+1$.^[60-61]

In Ref. ^[62], we have obtained the following exact uncertainty relation

(8) $$ \sum_{\nu =1}^{d+1}C(\rho, B_\nu)=d-({\rm tr}\sqrt{\rho})^2\,,$$
for a complete set of MUBs $B_\nu$, $\nu=1, 2, \ldots, {d+1}$.In particular, for any pure state $\rho$, we have

(9) $$ \sum_{\nu =1}^{d+1}C(\rho, B_\nu)=d-1\,.$$
Here we extend the above exact uncertainty relation to any $m$ MUBs $B_{\nu}$, $\nu =1,2, \ldots, m,$ as follows

(10) $$\sum_{\nu=1}^m C(\rho,B_{\nu})\geq(m-1)\Big(1-\frac{({\rm tr}\sqrt{\rho})^2}{d}\Big)\,.$$
To establish the above result, noting that^[63]

$\sum_{\nu=1}^mP(B_\nu|\rho)\leq \frac{m-1}{d}+{\rm tr} \rho^2\,,$

where $P(B_\nu |\rho)=\sum_j\langle b_{\nu j}|\rho|b_{\nu j}\rangle^2$.

Replacing $\rho$ with $\sqrt{\rho}/{\rm tr}\sqrt{\rho}$ and using the definition of $C(\rho,B_{\nu})$, we obtain the desired inequality (10).In particular, for any pure state $\rho$, we have

(11)$$ \sum_{\nu=1}^{m}C(\rho,B_{\nu})\geq\frac{(m-1)(d-1)}{d}\,.$$

3 Uncertainty Relations for Coherence with Respect to SIC-POVMs

In this section, we study coherence of a state with respect to a general SIC-POVM, and derive some ncertainty relations for coherence of a state with respect to a family of SIC-POVMs.

Consider a $d$-dimensional system, let $H_d$ be the set of all $d\times d$ Hermitian operators and $T_d$ be the set of all $d\times d$ traceless Hermitian operators. A set of $d^2$ non-negative operators $P=\{P_i: i=1,2,\ldots, d^2\}$ (not necessarily of rank 1) is called a general symmetric informationally complete positive operator valued measurement (SIC-POVMs),^[64] if

(i) It is a POVM: $P_i\geq0$, $\sum_{i=1}^{d^2}P_i={\bf 1}$, where ${\bf 1}$ is the identity matrix.

(ii) It is symmetric: ${\rm tr}P_i^2={\rm tr}P_j^2\neq{1}/{d^3}$ for all $i,j=1,2,\ldots,d^2$,and ${\rm tr}(P_iP_j)={\rm tr}(P_lP_m)$ for all $i\neq j$ and $l\neq m$.

Gour and Kalev have shown that there is a one-to-one correspondence between SIC-POVMs and orthonormal bases of $T_d$ in the following sense:^[64]

Let $\{F_i:i=1, 2, \ldots, d^2-1\}$ be an orthonormal base

of $T_d$, that is, ${\rm tr}F_i=0$ and ${\rm tr}F_iF_j=\delta_{ij}$, $i,j =1, 2,\ldots,d^2-1$. Put $F=\sum_{i=1}^{d^2-1}F_i$ and

(12)$$ t_0=-\frac{1}{d^2}\min \Big\{ \frac{1}{\lambda_i}: i=1, 2,\ldots,d^2-1 \Big\}\,,$$
(13)$$ t_1=-\frac{1}{d^2}\max \Big\{ \frac{1}{\mu_i}: i=1, 2, \ldots,d^2-1 \Big\}\,,$$
with $\lambda_i$ and $\mu_i$ the maximum and minimum eigenvalues of $F-d(d+1)F_i$, respectively.For any $ 0\neq t\in[t_0,t_1],$ take $ P_i(t)=\frac{1}{d^2}{\bf 1}+t(F-d(d+1)F_i),\quad i=1,\ldots,d^2-1\,,\\\ P_{d^2}(t)=\frac{1}{d^2}{\bf 1}+t(d+1)F\,,$

then $P(t)=\{P_i(t): i=1, 2, \ldots,d^2\}$ constitutes a general SIC POVM. Conversely, any SIC POVM is of the above form for some orthonormal basis $\{F_i:i=1, 2,\ldots, d^2-1\}$ of $T_d$.Take $F_0={\bf 1}$, then it is obvious that $\{F_i:i=0,1, 2,\ldots, d^2-1\}$ is an orthonormal basis for the space $H_d$ of all Hermitian operators.

With the above preparation, we can state our result for coherence $C(\rho, P(t))=\sum_{i=1}^{d^2}I(\rho,P_i(t))$ of the state $\rho$ with respect to the SIC-POVM $P(t)=\{P_i(t): i=1, 2, \ldots,d^2\}$ as

(14)$$ C(\rho, P(t))=t^2d^2(d+1)^2(d-({\rm tr}\sqrt{\rho})^2)\,,$$
where $t\in [t_0,t_1]$. This may be regarded as an exact uncertainty relation for the family of operators $P_i(t)$ in the state $\rho$.

To prove Eq. ({14}), noting that from Ref. ^[24], we know that

$ \sum_{i=0}^{d^2-1}I(\rho,F_i)=d-({\rm tr}\sqrt{\rho})^2\,,$

from which we have

$ C(\rho, P(t))=\sum_{i=1}^{d^2}I(\rho,P_i(t))\\\ \quad =-\frac{1}{2}\sum_{i=1}^{d^2}{\rm tr}[\sqrt{\rho},P_i(t)]^2\\\ \quad =-\frac{t^2}{2}\Big(\sum_{i=1}^{d^2-1}{\rm tr}[\sqrt{\rho},F-d(d+1)F_i]^2\\\ \qquad +{\rm tr}[\sqrt{\rho},(d+1)F]^2\Big)\\\ \quad =t^2d^2(d+1)^2\sum_{i=1}^{d^2-1}({\rm tr}\rho F_i^2-{\rm tr}\sqrt{\rho}F_i\sqrt{\rho}F_i)\\\ \quad =t^2d^2(d+1)^2\sum_{i=1}^{d^2-1}I(\rho,F_i)\\\ \quad =t^2d^2(d+1)^2\sum_{i=0}^{d^2-1}I(\rho,F_i)\\\ \quad =t^2d^2(d+1)^2(d-({\rm tr}\sqrt{\rho})^2)\,.$

From the above result, we readily obtain that for arbitrary $m$ SIC-POVMs $P^{(\nu )}(t_\nu ), \nu =1,2,\ldots, m$, where $t_\nu $ is the corresponding $t$ constants in their representations in terms of orthonormal bases of $T_d$, we have

$ \sum_{\nu =1}^{m}C(\rho, P^{(\nu)}(t_\nu ) )=\Big(\sum_{\nu=1}^{m}t_\nu ^2\Big)(d^2+d)^2(d-({\rm tr}\sqrt{\rho})^2\,.$

In particular, for any pure state $\rho$, we have

(15) $$\sum_{\nu =1}^{m}C(\rho, P^{(\nu)}(t_\nu ) )=\Big(\sum_{\nu =1}^{m}t_\nu ^2\Big)(d^3-d)^2\,.$$
These are uncertainty relations for coherence with respect to several SIC-POVMs.

4 Uncertainty Relations for Coherence of Quantum States with Respect to a Common Measurement

Coherence is a relative concept involving both quantum states and measurements. In the previous sections, we have discussed uncertainty relations with respect to different measurements by fixing a quantum state. From a dual viewpoint, we study uncertainty relations of different quantum states by fixing a measurement in this section, and obtain the following result

(16)$$ C(\rho,\Pi)+C(\sigma,\Pi)\geq \frac{1}{2}\|[\sqrt{\rho},\sqrt{\sigma}]\|_{F}^2\,,$$
which may be regarded as an uncertainty relation for coherence of two quantum states with respect to a common measurement (reference basis).Here $\rho$ and $\sigma$ are arbitrary quantum states, and $\Pi=\{|i\rangle\langle i|\}$ is any von Neumann measurement.

Equation ({16}) can be derived by decomposing $\sqrt{\rho}$ and $\sqrt{\sigma}$ as Eq. ({5}).This shows that, if two states are not commutative, then there is no measurement (reference basis)such that the coherence of these two states can be simultaneously small.

We further reveal a link between uncertainty relations for coherence and quantumness of ensembles. Consider a quantum ensemble $\mathcal{E}=\{(p_i,\rho_i): i=1,2,\ldots, n\}$ with $ Q(\mathcal{E})=-\sum_{i,j=1}^n\sqrt{p_ip_j}{\rm tr}[\sqrt{\rho_i},\sqrt{\rho_j}]^2\,,$ as a measure of quantumness in Ref. ^[65]. Here $\rho_i$ are density operators and $p_i\geq 0, \sum _{i=1}^np_i=1.$

From the above discussion, we get that

$ Q(\mathcal{E}) \leq \;2\sum_{i,j=1}^n\sqrt{p_ip_j}(C(\rho_i,\Pi)+C(\rho_j,\Pi)) \\=4\Big(\sum_{i=1}^n \sqrt{p_i}\Big)\Big(\sum_{i=1}^n \sqrt{p_i}C(\rho_i,\Pi)\Big)\\\ \leq;4\sqrt{n}\sum_{i=1}^n \sqrt{p_i}C(\rho_i,\Pi)\,.$

Consequently, the weighted average of coherence

$ \sum_{i=1}^n\sqrt{p_i}C(\rho_i,\Pi)$

of the quantum ensemble satisfies

$ \sum_{i=1}^n\sqrt{p_i}C(\rho_i,\Pi)\geq \frac{Q(\mathcal{E})}{4\sqrt{n}}\,.$

This relation sets a lower bound to coherence in terms of quantumness of the quantum ensemble.

5 Summary

By employing the coherence measure based on skew information, we have derived some uncertainty relations with respect to von Neumann measurements as well as with respect to SIC POVMs. In the special cases of a family of von Neumann measurements, we have considered uncertainty relations for coherence with respect to any MUBs (not necessary two MUBs), and have proved that the lower bound is a positive constant for pure states.

From a dual perspective, we have obtained some trade-off relations for coherence of different quantum states and quantum ensembles with respect to a common measurement. These results imply that if two density operators are not commutative, then there is no reference basis such that their coherence are simultaneously small.

We emphasize that the coherence measure based on the skew information has a natural interpretation as quantum uncertainty,^[36-37] consequently the uncertainty relations for coherence obtained here can be regarded as genuinely quantum uncertainty relations.

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We study the distribution of dynamical quantities in various one-dimensional disordered models, the critical behavior of which is described by an infinite randomness fixed point. In the disordered contact process, the survival probability P(t) is found to show multiscaling in the critical point, meaning that P(t)=t-δ, where the (environment and time-dependent) exponent δ has a universal limit distribution when t→∞. The limit distribution is determined by the strong disorder renormalization group method analytically in the end point of a semi-infinite lattice, where it is found to be exponential, while, in the infinite system, conjectures on its limiting behaviors for small and large δ, which are based on numerical results, are formulated. By the same method, the survival probability in the problem of random walks in random environments is also shown to exhibit multiscaling with an exponential limit distribution. In addition to this, the (imaginary-time) spin-spin autocorrelation function of the random transverse-field Ising chain is found to have a form similar to that of survival probability of the contact process at the level of the renormalization approach. Consequently, a relationship between the corresponding limit distributions in the two problems can be established. Finally, the distribution of the spontaneous magnetization in this model is also discussed.

[20] J. Zhang, Y. Zhang, C. Yu , Quantum Inf. Process. 14(2015) 2239.
DOI:10.1007/s11128-015-0950-zURL [Cited within: 1]

[21] S. Luo , Phys. Rev. Lett. 91(2003) 180403.
DOI:10.1103/PhysRevLett.91.180403URLPMID:14611271 [Cited within: 2]
The Wigner-Araki-Yanase theorem puts a limitation on the measurement of observables in the presence of a conserved quantity, and the notion of Wigner-Yanase skew information quantifies the amount of information on the values of observables not commuting with the conserved quantity. We demonstrate that the statistical idea underlying the skew information is the Fisher information in the theory of statistical estimation. A quantum Cramér-Rao inequality and a new uncertainty relation in terms of the skew information are established, which shed considerable new light on the relationships between quantum measurement and statistical inference. The result is applied to estimating the evolution speed of quantum states.

[22] S. Luo , Phys. Rev. A 72 ( 2005) 042110.


[23] S. Luo , Theor. Math. Phys. 143(2005) 681.
DOI:10.1007/s11232-005-0098-6URL

[24] S. Luo , Phys. Rev. A 73 ( 2006) 022324.
[Cited within: 2]

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[26] T. Baumgratz, M. Cramer, M. B. Plenio , Phys. Rev. Lett. 113(2014) 140401.
DOI:10.1103/PhysRevLett.113.140401URLPMID:25325620 [Cited within: 1]
We introduce a rigorous framework for the quantification of coherence and identify intuitive and easily computable measures of coherence. We achieve this by adopting the viewpoint of coherence as a physical resource. By determining defining conditions for measures of coherence we identify classes of functionals that satisfy these conditions and other, at first glance natural quantities, that do not qualify as coherence measures. We conclude with an outline of the questions that remain to be answered to complete the theory of coherence as a resource.

[27] D. Girolami , Phys. Rev. Lett. 113(2014) 170401.
DOI:10.1103/PhysRevLett.113.170401URLPMID:25379903
Quantum coherence is the key resource for quantum technology, with applications in quantum optics, information processing, metrology, and cryptography. Yet, there is no universally efficient method for quantifying coherence either in theoretical or in experimental practice. I introduce a framework for measuring quantum coherence in finite dimensional systems. I define a theoretical measure which satisfies the reliability criteria established in the context of quantum resource theories. Then, I present an experimental scheme implementable with current technology which evaluates the quantum coherence of an unknown state of a d-dimensional system by performing two programmable measurements on an ancillary qubit, in place of the O(d2) direct measurements required by full state reconstruction. The result yields a benchmark for monitoring quantum effects in complex systems, e.g., certifying nonclassicality in quantum protocols and probing the quantum behavior of biological complexes.

[28] L. H. Shao, Z. Xi, H. Fan, Y. Li , Phys. Rev. A 91 ( 2015) 042120.
DOI:10.1103/PhysRevE.91.042120URLPMID:25974451
The nonbacktracking operator for a graph is the adjacency matrix defined on directed edges of the graph. The operator was recently shown to perform optimally in spectral clustering in sparse synthetic graphs and have a deep connection to belief propagation algorithm. In this paper we consider nonbacktracking operator for Ising model on a general graph with a general coupling distribution and study the spectrum of this operator analytically. We show that spectral algorithms based on this operator is equivalent to belief propagation algorithm linearized at the paramagnetic fixed point and recovers replica-symmetry results on phase boundaries obtained by replica methods. This operator can be applied directly to systems with multiple states like Hopfield model. We show that spectrum of the operator can be used to determine number of patterns that stored successfully in the network, and the associated eigenvectors can be used to retrieve all the patterns simultaneously. We also give an example on how to control the Hopfield model, i.e., making network more sparse while keeping patterns stable, using the nonbacktracking operator and matrix perturbation theory.

[29] A. Streltsov, U. Singh, H. S. Dhar , Phys. Rev. Lett. 115(2015) 020403.
DOI:10.1103/PhysRevLett.115.020403URLPMID:26207452
Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative characterization of coherence as an operational resource are still very limited. Here we show that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. This finding allows us to define a novel general class of measures of coherence for a quantum system of arbitrary dimension, in terms of the maximum bipartite entanglement that can be generated via incoherent operations applied to the system and an incoherent ancilla. The resulting measures are proven to be valid coherence monotones satisfying all the requirements dictated by the resource theory of quantum coherence. We demonstrate the usefulness of our approach by proving that the fidelity-based geometric measure of coherence is a full convex coherence monotone, and deriving a closed formula for it on arbitrary single-qubit states. Our work provides a clear quantitative and operational connection between coherence and entanglement, two landmark manifestations of quantum theory and both key enablers for quantum technologies.

[30] X. Yuan, H. Zhou, Z. Cao, X. Ma , Phys. Rev. A 92 ( 2015) 022124.
DOI:10.1103/PhysRevE.92.022124URLPMID:26382361
Behavior of two-time autocorrelation during the phase separation in solid binary mixtures is studied via numerical solutions of the Cahn-Hilliard equation as well as Monte Carlo simulations of the Ising model. Results are analyzed via state-of-the-art methods, including the finite-size scaling technique. Full forms of the autocorrelation in space dimensions 2 and 3 are obtained empirically. The long-time behavior is found to be power law, with exponents unexpectedly higher than the ones for the ferromagnetic ordering. Both Cahn-Hilliard and Ising models provide consistent results.

[31] C. Napoli, T. R. Bromley, M. Cianciaruso , et al., Phys. Rev.Lett. 116(2016) 150502.
DOI:10.1103/PhysRevLett.116.150502URLPMID:27127946
Quantifying coherence is an essential endeavor for both quantum foundations and quantum technologies. Here, the robustness of coherence is defined and proven to be a full monotone in the context of the recently introduced resource theories of quantum coherence. The measure is shown to be observable, as it can be recast as the expectation value of a coherence witness operator for any quantum state. The robustness of coherence is evaluated analytically on relevant classes of states, and an efficient semidefinite program that computes it on general states is given. An operational interpretation is finally provided: the robustness of coherence quantifies the advantage enabled by a quantum state in a phase discrimination task.

[32] A. Winter and D. Yang, Phys. Rev. Lett. 116(2016) 120404.
DOI:10.1103/PhysRevLett.116.120404URLPMID:27058063
We establish an operational theory of coherence (or of superposition) in quantum systems, by focusing on the optimal rate of performance of certain tasks. Namely, we introduce the two basic concepts-"coherence distillation" and "coherence cost"-in the processing quantum states under so-called incoherent operations [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. We, then, show that, in the asymptotic limit of many copies of a state, both are given by simple single-letter formulas: the distillable coherence is given by the relative entropy of coherence (in other words, we give the relative entropy of coherence its operational interpretation), and the coherence cost by the coherence of formation, which is an optimization over convex decompositions of the state. An immediate corollary is that there exists no bound coherent state in the sense that one would need to consume coherence to create the state, but no coherence could be distilled from it. Further, we demonstrate that the coherence theory is generically an irreversible theory by a simple criterion that completely characterizes all reversible states.

[33] K. Bu, U. Singh, S. M. Fei , et al., Phys. Rev.Lett. 119(2017) 150405.
DOI:10.1103/PhysRevLett.119.150405URLPMID:29077456
The operational characterization of quantum coherence is the cornerstone in the development of the resource theory of coherence. We introduce a new coherence quantifier based on maximum relative entropy. We prove that the maximum relative entropy of coherence is directly related to the maximum overlap with maximally coherent states under a particular class of operations, which provides an operational interpretation of the maximum relative entropy of coherence. Moreover, we show that, for any coherent state, there are examples of subchannel discrimination problems such that this coherent state allows for a higher probability of successfully discriminating subchannels than that of all incoherent states. This advantage of coherent states in subchannel discrimination can be exactly characterized by the maximum relative entropy of coherence. By introducing a suitable smooth maximum relative entropy of coherence, we prove that the smooth maximum relative entropy of coherence provides a lower bound of one-shot coherence cost, and the maximum relative entropy of coherence is equivalent to the relative entropy of coherence in the asymptotic limit. Similar to the maximum relative entropy of coherence, the minimum relative entropy of coherence has also been investigated. We show that the minimum relative entropy of coherence provides an upper bound of one-shot coherence distillation, and in the asymptotic limit the minimum relative entropy of coherence is equivalent to the relative entropy of coherence.

[34] A. Streltsov, G. Adesso, M. B. Plenio , Rev. Mod. Phys. 89(2017) 041003.
DOI:10.1103/PhysRevE.89.041003URLPMID:24827179
We derive an exact equation governing two-particle backwards mean-squared dispersion for both deterministic and stochastic tracer particles in turbulent flows. For the deterministic trajectories, we probe the consequences of our formula for short times and arrive at approximate expressions for the mean-squared dispersion which involve second order structure functions of the velocity and acceleration fields. For the stochastic trajectories, we analytically compute an exact t3 contribution to the squared separation of stochastic paths. We argue that this contribution appears also for deterministic paths at long times and present direct numerical simulation results for incompressible Navier-Stokes flows to support this claim. We also numerically compute the probability distribution of particle separations for the deterministic paths and the stochastic paths and show their strong self-similar nature.

[35] C. Yu , Phys. Rev. A 95 ( 2017) 042337.
[Cited within: 2]

[36] S. Luo and Y. Sun , Phys. Rev. A 96 ( 2017) 022136.
DOI:10.1103/PhysRevE.96.022136URLPMID:28950615 [Cited within: 1]
We investigate the dynamics of a one-dimensional p-wave superconductor with next-nearest-neighbor hopping and superconducting interaction derived from a three-spin interacting Ising model in transverse field by mapping to Majorana fermions. The next-nearest-neighbor hopping term leads to a new topological phase containing two zero-energy Majorana modes at each end of an open chain, compared to a nearest-neighbor p-wave superconducting chain. We study the Majorana survival probability (MSP) of a particular Majorana edge state when the initial Hamiltonian (H_{i}) is changed to the quantum critical as well as off-critical final Hamiltonian (H_{f}), which additionally contains an impurity term (H_{imp}) that breaks the time-reversal invariance. For the off-critical quenching inside the new topological phase with H_{f}=H_{i}+H_{imp}, and small impurity strength (λ_{d}), we observe a perfect oscillation of the MSP as a function of time with a single frequency (determined by the impurity strength λ_{d}) that can be analyzed from an equivalent two-level problem. On the other hand, the MSP shows a beating like structure with time for quenching to the phase boundary separating the topological phase (with two edge Majoranas at each edge) and the nontopological phase where the additional frequency is given by inverse of the system size. We attribute this behavior of the MSP to the modification of the energy levels of the final Hamiltonian due to the application of the impurity term.

[37] S. Luo and Y. Sun , Phys. Rev. A 96 ( 2017) 022130.
DOI:10.1103/PhysRevE.96.022130URLPMID:28950574 [Cited within: 5]
We study the dynamics of overdamped Brownian particles diffusing in conservative force fields and undergoing stochastic resetting to a given location at a generic space-dependent rate of resetting. We present a systematic approach involving path integrals and elements of renewal theory that allows us to derive analytical expressions for a variety of statistics of the dynamics such as (i) the propagator prior to first reset, (ii) the distribution of the first-reset time, and (iii) the spatial distribution of the particle at long times. We apply our approach to several representative and hitherto unexplored examples of resetting dynamics. A particularly interesting example for which we find analytical expressions for the statistics of resetting is that of a Brownian particle trapped in a harmonic potential with a rate of resetting that depends on the instantaneous energy of the particle. We find that using energy-dependent resetting processes is more effective in achieving spatial confinement of Brownian particles on a faster time scale than performing quenches of parameters of the harmonic potential.

[38] Y. Sun, Y. Mao, S. Luo , Europhys. Lett. 118(2017) 60007.
DOI:10.1209/0295-5075/118/60007URL [Cited within: 1]

[39] M. N. Bera, T. Qureshi, M. A. Siddiqui, A. K. Pati , Phys. Rev. A 92 ( 2015) 012118.
DOI:10.1103/PhysRevE.92.012118URLPMID:26274135
The quantum analog of Carnot cycles in few-particle systems consists of two quantum adiabatic steps and two isothermal steps. This construction is formally justified by use of a minimum work principle. It is then shown, using minimal assumptions of work or heat in nanoscale systems, that the heat-to-work efficiency of such quantum heat engine cycles can be further optimized via two conditions regarding the expectation value of some generalized force operators evaluated at equilibrium states. In general the optimized efficiency is system specific, lower than the Carnot efficiency, and dependent upon both temperatures of the cold and hot reservoirs. Simple computational examples are used to illustrate our theory. The results should be an important guide towards the design of favorable working conditions of a realistic quantum heat engine.

[40] E. Bagan, J. A. Bergou, S. S. Cottrell, M. Hillery , Phys. Rev. Lett. 116(2016) 160406.
DOI:10.1103/PhysRevLett.116.160406URLPMID:27152780 [Cited within: 1]
We find two relations between coherence and path information in a multipath interferometer. The first builds on earlier results for the two-path interferometer, which used minimum-error state discrimination between detector states to provide the path information. For visibility, which was used in the two-path case, we substitute a recently defined l_{1} measure of quantum coherence. The second is an entropic relation in which the path information is characterized by the mutual information between the detector states and the outcome of the measurement performed on them, and the coherence measure is one based on relative entropy.

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DOI:10.1038/ncomms4821URLPMID:24819237
Noether's theorem is a fundamental result in physics stating that every symmetry of the dynamics implies a conservation law. It is, however, deficient in several respects: for one, it is not applicable to dynamics wherein the system interacts with an environment; furthermore, even in the case where the system is isolated, if the quantum state is mixed then the Noether conservation laws do not capture all of the consequences of the symmetries. Here we address these deficiencies by introducing measures of the extent to which a quantum state breaks a symmetry. Such measures yield novel constraints on state transitions: for nonisolated systems they cannot increase, whereas for isolated systems they are conserved. We demonstrate that the problem of finding non-trivial asymmetry measures can be solved using the tools of quantum information theory. Applications include deriving model-independent bounds on the quantum noise in amplifiers and assessing quantum schemes for achieving high-precision metrology.

[43] I. Marvian and R. W. Spekkens , Phys. Rev. A 94 ( 2016) 052324.


[44] M. Piani, M. Cianciaruso, T. R. Bromley , et al., Phys. Rev. A 93 ( 2016) 042107.
DOI:10.1103/PhysRevE.93.042107URLPMID:27176254
We study the directed Abelian sandpile model on a square lattice, with K downward neighbors per site, K>2. The K=3 case is solved exactly, which extends the earlier known solution for the K=2 case. For K>2, the avalanche clusters can have holes and side branches and are thus qualitatively different from the K=2 case where avalanche clusters are compact. However, we find that the critical exponents for K>2 are identical with those for the K=2 case, and the large-scale structure of the avalanches for K>2 tends to the K=2 case.

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Recent studies have developed fundamental limitations on nanoscale thermodynamics, in terms of a set of independent free energy relations. Here we show that free energy relations cannot properly describe quantum coherence in thermodynamic processes. By casting time-asymmetry as a quantifiable, fundamental resource of a quantum state, we arrive at an additional, independent set of thermodynamic constraints that naturally extend the existing ones. These asymmetry relations reveal that the traditional Szilárd engine argument does not extend automatically to quantum coherences, but instead only relational coherences in a multipartite scenario can contribute to thermodynamic work. We find that coherence transformations are always irreversible. Our results also reveal additional structural parallels between thermodynamics and the theory of entanglement.

[50] P. ′Cwikliński, M. Studziński, M. Horodecki, J. Oppenheim , Phys. Rev. Lett. 115(2015) 210403.
DOI:10.1103/PhysRevLett.115.210403URLPMID:26636834 [Cited within: 1]
The second law of thermodynamics places a limitation into which states a system can evolve into. For systems in contact with a heat bath, it can be combined with the law of energy conservation, and it says that a system can only evolve into another if the free energy goes down. Recently, it's been shown that there are actually many second laws, and that it is only for large macroscopic systems that they all become equivalent to the ordinary one. These additional second laws also hold for quantum systems, and are, in fact, often more relevant in this regime. They place a restriction on how the probabilities of energy levels can evolve. Here, we consider additional restrictions on how the coherences between energy levels can evolve. Coherences can only go down, and we provide a set of restrictions which limit the extent to which they can be maintained. We find that coherences over energy levels must decay at rates that are suitably adapted to the transition rates between energy levels. We show that the limitations are matched in the case of a single qubit, in which case we obtain the full characterization of state-to-state transformations. For higher dimensions, we conjecture that more severe constraints exist. We also introduce a new class of thermodynamical operations which allow for greater manipulation of coherences and study its power with respect to a class of operations known as thermal operations.

[51] H. L. Shi, S. Y. Liu, X. H. Wang , et al., Phys. Rev. A 95 ( 2017) 032307.
DOI:10.1103/PhysRevE.95.032307URLPMID:28415219 [Cited within: 1]
''Three is a crowd" is an old proverb that applies as much to social interactions as it does to frustrated configurations in statistical physics models. Accordingly, social relations within a triangle deserve special attention. With this motivation, we explore the impact of topological frustration on the evolutionary dynamics of the snowdrift game on a triangular lattice. This topology provides an irreconcilable frustration, which prevents anticoordination of competing strategies that would be needed for an optimal outcome of the game. By using different strategy updating protocols, we observe complex spatial patterns in dependence on payoff values that are reminiscent to a honeycomb-like organization, which helps to minimize the negative consequence of the topological frustration. We relate the emergence of these patterns to the microscopic dynamics of the evolutionary process, both by means of mean-field approximations and Monte Carlo simulations. For comparison, we also consider the same evolutionary dynamics on the square lattice, where of course the topological frustration is absent. However, with the deletion of diagonal links of the triangular lattice, we can gradually bridge the gap to the square lattice. Interestingly, in this case the level of cooperation in the system is a direct indicator of the level of topological frustration, thus providing a method to determine frustration levels in an arbitrary interaction network.

[52] Y. L. Su, S. Y. Liu, X. H. Wang , et al., Sci. Rep. 8(2018) 11081.
DOI:10.1038/s41598-018-29342-5URLPMID:30038348
We investigate the behavior of coherence in scattering quantum walk search on complete graph under the condition that the total number of vertices of the graph is significantly larger than the marked number of vertices we are searching, N???v. We find that the consumption of coherence represents the increase of the success probability for the searching, also it is related to the efficiency of the algorithm in oracle queries. If no coherence is consumed or an incoherent state is utilized, the algorithm will behave as the classical blind search, implying that coherence is responsible for the speed-up in this quantum algorithm over its classical counterpart. The effect of noises, in particular of photon loss and random phase shifts, on the performance of algorithm is studied. Two types of noise are considered because they arise in the optical network used for experimental realization of scattering quantum walk. It is found that photon loss will reduce the coherence and random phase shifts will hinder the interference between the edge states, both leading to lower success probability compared with the noise-free case. We then conclude that coherence plays an essential role and is responsible for the speed-up in this quantum algorithm.

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We establish two complementarity relations for the relative entropy of coherence in quantum information processing, i.e., quantum dense coding and teleportation. We first give an uncertainty-like expression relating local quantum coherence to the capacity of optimal dense coding for bipartite system. The relation can also be applied to the case of dense coding by using unital memoryless noisy quantum channels. Further, the relation between local quantum coherence and teleportation fidelity for two-qubit system is given.

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The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.