Universal freezing of asymmetry
aa r X i v : . [ qu a n t - ph ] F e b Universal freezing of asymmetry
Da-Jian Zhang,
Xiao-Dong Yu, Hua-Lin Huang, and D. M. Tong ∗ Department of Physics, Shandong University, Jinan 250100, China School of Mathematics, Shandong University, Jinan 250100, China School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China (Dated: September 16, 2018)Asymmetry of quantum states is a useful resource in applications such as quantum metrology, quantum com-munication, and reference frame alignment. However, asymmetry of a state tends to be degraded in physicalscenarios where environment-induced noise is described by covariant operations, e.g., open systems constrainedby superselection rules, and such degradations weaken the abilities of the state to implement quantum informa-tion processing tasks. In this paper, we investigate under which dynamical conditions asymmetry of a stateis totally una ff ected by the noise described by covariant operations. We find that all asymmetry measures arefrozen for a state under a covariant operation if and only if the relative entropy of asymmetry is frozen forthe state. Our finding reveals the existence of universal freezing of asymmetry, and provides a necessary andsu ffi cient condition under which asymmetry is totally una ff ected by the noise. Symmetry is a central concept in quantum mechanics, de-scribing invariant features of a quantum system with respectto the action of a group of transformations [1]. For a specificsymmetry, two relevant notions are asymmetric states and co-variant operations, which are the states that break the sym-metry and the quantum operations that respect the symmetry,respectively. In the physical world, all elementary interac-tions are expected to have specific symmetries [2]. For exam-ple, the interactions that do not have preferred direction arerotationally invariant and hence have SO(3) symmetry. Thepresence of a symmetry in a system generally imposes re-strictions on the manipulation of the system, which results innontrivial limitations on the implementation of quantum in-formation processing tasks. Interestingly, asymmetric statescan be exploited to overcome the restrictions and allow one toimplement quantum information processing tasks that wouldotherwise be forbidden [3]. For example, in the presence of aconservation law, it is forbidden to measure exactly an observ-able that does not commute with a conserved quantity, but it isstill possible to measure approximatively the observable withthe aid of asymmetric states [4, 5]. Asymmetry of states isa useful resource for implementing quantum information pro-cessing tasks [3], and the exploitation of asymmetric states hasbeen carried out in applications, such as quantum metrology[6–9], quantum communication [10, 11], and reference framealignment [12–14].By taking asymmetry as a physical resource, a resourcetheory of asymmetry, just like the resource theory of entan-glement [15], has been recently developed. The abilities ofan asymmetric state to overcome the restriction imposed by asymmetry are analogous to the abilities of an entangled stateto overcome the restriction of local operations and classicalcommunication (LOCC). Asymmetric states and covariant op-erations in the asymmetry theory correspond respectively toentangled states and LOCC in the entanglement theory, or re-source states and free operations in a general resource the-ory [16]. Over the recent years, a lot of e ff ort has been de-voted to the formulation of a unified and quantitative theory of ∗ [email protected] asymmetry, which aims to quantify the abilities of asymmet-ric states for implementing quantum information processingtasks [17–29]. The asymmetry theory was first used to mea-sure the quality of a quantum reference frame [17–19], sub-sequently exploited to find the consequences of symmetriesfor open quantum dynamics [22–25], and recently linked tothe resource theory of coherence [26–29]. It turns out that theasymmetry theory is applicable to a wide spectrum of phys-ical contexts, and based on them, a number of asymmetrymeasures, such as the unique asymptotic measure of frame-ness [18], the relative entropy of frameness [19], the Holevoasymmetry measure [23], the Wigner-Yanase skew informa-tion [26], and the quantum Fisher information [27], have beenproposed.Asymmetry of a state is a useful resource for quantum infor-mation processing, but it may su ff er from degradation arisingfrom the interaction between the system and its environment.Indeed, in many physical scenarios, e.g., all open systemsconstrained by superselection rules, the environment-inducednoise is described by covariant operations [30], whose ac-tions tend to destroy asymmetry of the state and hence makethe state less useful for implementing quantum informationprocessing tasks. A challenge in exploiting the resource istherefore to preserve asymmetry of states from the degrada-tion caused by the covariant noise, i.e., the noise describedby covariant operations. When the asymmetry of a state isfrozen, i.e., remains constant, the ability of the state to im-plement some quantum information processing task will notbe weakened if the ability exploited in the task is based onthe frozen asymmetry measure. However, there are many dif-ferent asymmetry measures, each of which is used to quan-tify one ability of a state to implement a di ff erent quantuminformation processing task, and one asymmetry measure be-ing frozen does not imply other asymmetry measures beingfrozen, too. The covariant noise may not weaken some abili-ties of a state if these abilities are based on the frozen asym-metry measures, but it can still weaken the other abilities thatare based on unfrozen asymmetry measures. Only the statewith universal freezing of asymmetry can keep all the abili-ties of asymmetry resource totally una ff ected by the covariantnoise. Here, by the phrase, universal freezing of asymmetry,we mean that the asymmetry of a state is frozen regardless ofasymmetry measures adopted, i.e., all asymmetry measures ofthe state are frozen under a certain covariant operation. Thequestion then is as follows: under which dynamical conditionsdoes the universal freezing of asymmetry occur for a state un-der a covariant operation? This is an important issue, sinceonly in this case asymmetry of a state is totally una ff ected bythe covariant noise. In this paper, we address this issue. Wewill show that all asymmetry measures are frozen for a stateunder a covariant operation if and only if the relative entropyof asymmetry is frozen for the state.Note that similar issues on how to preserve other resourcesof a state from the degradation caused by noise have beenwidely addressed. For instance, the preservation of entangle-ment, the freezing of quantum correlations, and the freezing ofquantum coherence were investigated in Refs. [31–35], Refs.[36–42], and Refs. [43–45], respectively. The present investi-gation aims to fill the gap in the resource theory of asymmetry.To present our finding clearly, it is instructive to specifysome notions, such as symmetric states, asymmetric states,covariant operations, and asymmetry measures.Consider a quantum system equipped with a Hilbert space H . Let G be a group of physical transformations acting on H through a unitary representation U g . The group G to-gether with its unitary representation specifies the symmetryunder consideration. We represent the transformation associ-ated with the group element g by the map U g , i.e., U g ( ρ ) = U g ρ U † g .A state δ is said to be a symmetric state with respect to G if U g ( δ ) = δ, for all g ∈ G . The set of all symmetric states is denoted by S . All other states are called asymmetric states with respectto G . Hereafter, we use ρ to represent a general state, and δ specially to denote a symmetric state.A quantum operation Λ is said to be a covariant operationwith respect to G if Λ ◦ U g = U g ◦ Λ , (1)for all g ∈ G . That is, the transformation realized by applyingfirst U g and then Λ is equivalent to that realized by applyingfirst Λ and then U g .A functional A mapping states to real numbers can be takenas an asymmetry measure if it satisfies the following two con-ditions:(i) A ( ρ ) ≥ ρ , and A ( ρ ) = ρ ∈ S ;(ii) A ( ρ ) ≥ A ( Λ ( ρ )) for all covariant operations Λ , that is, A isnon-increasing under covariant operations.One of the asymmetry measures is the relative entropy ofasymmetry A r [46]. It is defined as A r ( ρ ) = min δ ∈S S ( ρ k δ ) , (2)where S ( ρ k δ ) = Tr ρ (log ρ − log δ ) is the quantum relativeentropy. In the case that G is a finite or compact Lie group,this measure admits a closed-form expression [19], A r ( ρ ) = S ( ρ k Λ G ( ρ )) = S ( Λ G ( ρ )) − S ( ρ ) , (3) where S ( ρ ) = − Tr( ρ log ρ ) is the von Neumann entropy,and Λ G is the G -twirling operation, defined as Λ G ( ρ ) = R G dgU g ρ U † g with the integral being performed over the Haarmeasure. Note that for a finite group, there is Λ G ( ρ ) = | G | P g ∈ G U g ρ U † g with | G | being the order of the group.With these notions, we can now state our main finding as atheorem. Theorem. A ( ρ t ) = A ( ρ ) for all asymmetry measures A ifand only if A r ( ρ t ) = A r ( ρ ), where ρ t = Λ t ( ρ ) with Λ t beinga covariant operation and ρ being an initial state.We only need to prove that A ( ρ t ) = A ( ρ ) if A r ( ρ t ) = A r ( ρ ),since A r is certainly frozen if all asymmetry measures arefrozen.First, we show that S ( Λ t ( ρ ) k Λ t ( δ )) = S ( ρ k δ ), where δ denotes the symmetric state achieving the minimum in theexpression A r ( ρ ) = min δ ∈S S ( ρ k δ ). By definition, A r ( ρ ) = S ( ρ k δ ) . (4)Since the quantum relative entropy is contracting under com-pletely positive and trace-preserving (CPTP) maps [47, 48],we have S ( Λ t ( ρ ) k Λ t ( δ )) ≤ S ( ρ k δ ) . (5)On the other hand, as Λ t is a covariant operation mappingsymmetric states to symmetric states, we have Λ t ( δ ) ∈ S andhence A r ( ρ t ) = min δ ∈S S ( Λ t ( ρ ) k δ ) ≤ S ( Λ t ( ρ ) k Λ t ( δ )) . (6)Combining Eqs. (4), (5), and (6), we obtain A r ( ρ t ) ≤ S ( Λ t ( ρ ) k Λ t ( δ )) ≤ S ( ρ k δ ) = A r ( ρ ) . (7)In the condition of A r ( ρ t ) = A r ( ρ ), Eq. (7) leads to A r ( ρ t ) = S ( Λ t ( ρ ) k Λ t ( δ )) , (8)and S ( Λ t ( ρ ) k Λ t ( δ )) = S ( ρ k δ ) . (9)Equation (8) implies that Λ t ( δ ) is the symmetric state achiev-ing the minimum in the expression A r ( ρ t ) = min δ ∈S S ( ρ t k δ ),while Eq. (9) shows that the equality for the contractivity ofquantum relative entropy in Eq. (5) is attained.In passing, we would like to point out that if the symmetrygroup G is restricted to finite or compact Lie groups, the aboveproof for Eq. (9) can be simplified by resorting to Eq. (3). In-deed, from Eq. (3), it follows that A r ( ρ ) = S ( ρ k Λ G ( ρ ))and A r ( ρ t ) = S ( ρ t k Λ G ( ρ t )). Noting that δ = Λ G ( ρ ) and Λ G ◦ Λ t = Λ t ◦ Λ G , we have Λ G ( ρ t ) = Λ G ◦ Λ t ( ρ ) =Λ t ◦ Λ G ( ρ ) = Λ t ( δ ). Hence, there is A r ( ρ ) = S ( ρ k δ )and A r ( ρ t ) = S ( Λ t ( ρ ) k Λ t ( δ )). Equation (9) then follows im-mediately from the condition A r ( ρ t ) = A r ( ρ ).Second, we demonstrate that there exists a covariant op-eration R t such that R t ( ρ t ) = ρ and R t ( δ t ) = δ , where δ t = Λ t ( δ ). Hereafter, we use δ t to represent Λ t ( δ ) for sim-plicity. Let Λ t ( ρ ) = X n K n ( t ) ρ K † n ( t ) (10)be the Kraus representation of Λ t , where K n ( t ) are the Krausoperators satisfying P n K † n ( t ) K n ( t ) = I . From the celebratedresult about the contractivity of quantum relative entropy [49,50], it follows that Eq. (9) is valid if and only if there exists aCPTP map R t such that R t ( ρ t ) = ρ and R t ( δ t ) = δ . (11)In the case that δ t is invertible, R t can be given explicitly bythe formula [50] R t ( ρ ) = X n δ K † n ( t ) δ − t ρδ − t K n ( t ) δ . (12)We therefore only need to prove that the CPTP map expressedby Eq. (12) is covariant. For convenience, we rewrite Eq. (12)as follows, R t = R ◦ R ◦ R , (13)where R ( ρ ) = δ ρδ , R ( ρ ) = P n K † n ( t ) ρ K n ( t ), and R ( ρ ) = δ − t ρδ − t . In order to prove that Eq. (12) defines a covariantoperation, it su ffi ces to show that each R i , i = , ,
3, fulfillsEq. (1), i.e., R i ◦U g = U g ◦ R i , for all g ∈ G . Since δ is a sym-metric state, there is U g ( δ ) = δ , i.e., [ δ , U g ] =
0. This im-plies that δ and U g are simultaneously diagonalizable, whichfurther implies that δ and U g are simultaneously diagonaliz-able, because δ shares common eigenvectors with δ . Hence,[ δ , U g ] =
0. It follows that R ◦ U g ( ρ ) = δ U g ρ U † g δ = U g δ ρδ U † g = U g ◦ R ( ρ ). That is, R fulfills Eq. (1). Sim-ilarly, we can prove that R fulfills Eq. (1). Now, it remainsto show that R fulfills Eq. (1), too. By definition, there is U g ◦ Λ t = Λ t ◦ U g . As an immediate consequence, the equal-ity Tr[ X U g ◦ Λ t ( Y )] = Tr[ X Λ t ◦ U g ( Y )] holds for any oper-ators X and Y . Inserting the explicit expressions of U g and Λ t into this equality, we have Tr { XU g [ P n K n ( t ) YK † n ( t )] U † g } = Tr { X [ P n K n ( t ) U g YU † g K † n ( t )] } . Since the trace of a matrixis cyclic, i.e., Tr( AB ) = Tr( BA ), for two arbitrary ma-trices A and B , we have Tr { [ P n K † n ( t ) U † g XU g K n ( t )] Y } = Tr { U † g [ P n K † n ( t ) XK n ( t )] U g Y } . Noting that U † g = U g − and U g = U † g − , we further have Tr[ R ◦ U g − ( X ) Y ] = Tr[ U g − ◦ R ( X ) Y ]. Since this equality holds for any operators X and Y ,there is R ◦ U g − = U g − ◦ R . Letting g in this equation runover all elements of G , we then have R ◦ U g = U g ◦ R , forall g ∈ G , which means that R fulfills Eq. (1). Therefore, Eq.(12) defines a covariant operation satisfying Eq. (11). In thecase that δ t is not invertible, instead of Eq. (12), R t should beexpressed as R t ( ρ ) = X n δ K † n ( t ) δ − t ρδ − t K n ( t ) δ + P ρ P , (14)where P is the orthogonal projector onto the kernel of δ t , and δ − t is defined to be the square root of the Moore-Penrosepseudoinverse of δ t , i.e., δ − t = P λ i , λ − i | φ i ih φ i | , providedthat the spectral decomposition of δ t reads δ t = P i λ i | φ i ih φ i | . Similarly, one can show that Eq. (14) defines a covariant op-eration satisfying Eq. (11).Third, with the foregoing arguments, it is ready to show theconclusion A ( ρ t ) = A ( ρ ). By combining the two covariantoperations Λ t and R t , there is ρ Λ t −−→ ρ t R t −→ ρ . (15)Since all the asymmetry measures A are non-increasing undercovariant operations, Eq. (15) results in A ( ρ ) ≥ A ( ρ t ) ≥ A ( ρ ) , (16)which implies that A ( ρ t ) = A ( ρ ). This completes the proof ofthe theorem.Our theorem provides a necessary and su ffi cient conditionunder which asymmetry of a state is totally una ff ected bythe covariant noise. It is applicable to all kinds of symme-try groups. When the symmetry group is a finite or compactLie group, the relative entropy of asymmetry admits a closed-form expression in Eq. (3), and consequently our theoremprovides a computable criterion for identifying the states withuniversal freezing of asymmetry. All the states with universalfreezing of asymmetry can be obtained by solving the equa-tion A r ( ρ t ) = A r ( ρ ), although it may be di ffi cult to solve ana-lytically this equation to obtain all the solutions since the cal-culation of entropy is complicated. However, it is generallyunnecessary to obtain all the solutions. In practical applica-tions, researchers are usually interested only in some specialstates. In this case, one only needs to examine the desiredstates, to which our theorem is quite useful.In the following, we demonstrate the usefulness of our the-orem by presenting two examples, of which one is about timeevolution of an open system and another is about measure-ment on a system. Example 1: time evolution. –Consider the time evolution ofan open system subject to a superselection rule [30]. For sim-plicity, we suppose that the system is composed of two qubitsand the superselection rule is associated with the group U (1)with the unitary representation U θ = exp[ i θ ( σ z ⊗ I + I ⊗ σ z )],where σ z = | ih | − | ih | . The model can be generalized tothe multiqubit case. As discussed in Ref. [30], the superselec-tion rule restricts the allowed dynamics of the system to thosethat are covariant with respect to the associated group. Giventhis situation, we consider the time evolution of the systemdescribed by the following covariant operation, Λ t ( ρ ) = (1 − p ) ρ + p σ z ⊗ σ + ρσ z ⊗ σ − + p σ z ⊗ σ − ρσ z ⊗ σ + , (17)where σ + = | ih | , σ − = | ih | , and 0 ≤ p ≤ t .Let us examine a family of pure states, expressed as | ϕ i = λ | i + λ | i , (18)where λ m are complex numbers satisfying | λ | + | λ | =
1. Byusing our theorem, we now show that all asymmetry measuresare frozen forever for any initial state | ϕ i under the covariantoperation Λ t . To this end, we only need to show that A r ( ρ t )is a constant, where ρ t = Λ t ( ρ ) with Λ t being defined in Eq.(17) and ρ = | ϕ ih ϕ | being an initial state.Direct calculations show that ρ t = (1 − p )( λ | i + λ | i )( λ ∗ h | + λ ∗ h | ) + p ( λ | i − λ | i )( λ ∗ h | − λ ∗ h | ) . Noting that the G -twirling operation is Λ G ( ρ ) = P i = P i ρ P i ,where P = | ih | , P = | ih | + | ih | , and P = | ih | , we further have Λ G ( ρ t ) = (1 − p ) (cid:16) | λ | | ih | + | λ | | ih | (cid:17) + p (cid:16) | λ | | ih | + | λ | | ih | (cid:17) . We can then obtain the relative entropy of asymmetry, A r ( ρ t ) = S ( ρ t k Λ G ( ρ t )) = S ( Λ G ( ρ t )) − S ( ρ t ) = X m = (cid:2) − (1 − p ) | λ m | log (cid:16) (1 − p ) | λ m | (cid:17) − p | λ m | × log (cid:16) p | λ m | (cid:17) (cid:3) + (1 − p ) log(1 − p ) + p log p = X m = −| λ m | log | λ m | = A r ( ρ ) . (19)Equation (19) shows that A r ( ρ t ) is a constant. This impliesthat all asymmetry measures manifest freezing forever for thetwo-qubit system initially in the state expressed by Eq. (18)undergoing the time evolution defined by Eq. (17). That is,universal freezing of asymmetry occurs in this case. Example 2: measurement process. – Consider the measure-ment process discussed first in Ref. [51]. Although we havethus far focused on the preservation of asymmetry from thedegradation caused by environment-induced noise, our theo-rem is also applicable to the case where the degradation ofasymmetry arises from a measurement process as long as theprocess can be described by covariant operations.As shown in Refs. [51–53], performing a covariant mea-surement has some back reaction on a quantum referenceframe and degrades asymmetry of the reference frame. In or-der to demonstrate the degradation of a phase reference frame,the authors of Ref. [51] considered the estimation task of mea-suring the phase of a large number of qubits relative to a singlephase reference frame. Their estimation task consists of a se-quence of covariant measurements, which are performed onthe combined systems consisting of the reference frame andeach qubit, one after another. As a result of these repeatedmeasurements, the reference frame becomes more and moreuseless for implementing their estimation task. It implies thatthe asymmetry of the reference frame is degraded [51]. Inthe following, we show that if the reference frame is initiallyprepared in some special states, the asymmetry of the refer-ence frame manifests universal freezing for a certain numberof measurements and is therefore totally una ff ected by thesemeasurements.The authors of Ref. [51] used an oscillator mode to actas the phase reference frame. The Hilbert space of the phasereference frame is then the Fock space spanned by {| n i , n = , , , . . . } . The symmetry under consideration is the group U (1) with the unitary representation U θ = exp( i θ N ), where N is the number operator. Accordingly, the G -twirling operationis Λ G ( ρ ) = P ∞ n = | n ih n | ρ | n ih n | , whose e ff ect is to project ontoeigenvectors of the group generator N . As a result of a singlemeasurement, the state of the phase reference frame is updatedby the covariant operation [51], Λ ( ρ ) = ρ + | ih | ρ | ih | + A † ρ A + A ρ A † , (20)where A = P ∞ n = | n ih n + | . In this situation, the time index t issimply an integer specifying the number of measurements thathave taken place. The state of the phase reference frame fol-lowing the t -th measurement is ρ t = Λ ( ρ t − ), with ρ denotingthe initial state prior to any measurement.We examine a family of pure states, expressed as λ | N i + λ | N i + · · · + λ M | (2 M + N i , (21)where λ m are complex numbers satisfying P Mm = | λ m | =
1, and N and M are positive integers. Hereafter, we use | ϕ n i to de-note the state λ | N + n i + λ | N + n i + · · · + λ M | (2 M + N + n i for simplicity. Then, the state in Eq. (21) can be simply writ-ten as | ϕ i . By using our theorem, we show that if the numberof measurements is less than N , the asymmetry of the phasereference frame initially in the state expressed by Eq. (21)manifests universal freezing. To this end, we only need toshow that the relative entropy of asymmetry A r ( ρ t ) are con-stants, where ρ = | ϕ ih ϕ | and t < N .By detail calculations, we obtain ρ t = t X n = − t p n ( t ) | ϕ n ih ϕ n | , with p n ( t ) = X n ≤ k ≤ t + n tk ! t − kk − n ! ! t − n + k , where k represents an nonnegative integer and (cid:16) ·· (cid:17) denotes thebinomial coe ffi cient, i.e., (cid:16) tk (cid:17) = t ! k !( t − k )! . Further calculationsshow that Λ G ( ρ t ) = t X n = − t M X m = p n ( t ) | λ m | | (2 m + N + n ih (2 m + N + n | . With the aid of the above expressions, we can obtain therelative entropy of asymmetry, A r ( ρ t ) = S ( ρ t k Λ G ( ρ t )) = S ( Λ G ( ρ t )) − S ( ρ t ) = − t X n = − t M X m = p n ( t ) | λ m | log h p n ( t ) | λ m | i + t X n = − t p n ( t ) log p n ( t ) = − M X m = | λ m | log | λ m | = A r ( ρ ) . (22)Equation (22) shows that the relative entropy of asymmetryfor the state ρ t is constant. Therefore, all asymmetry measuresmanifest freezing for the phase reference frame initially in thestate expressed by Eq. (21). Universal freezing of asymmetryoccurs in this case, too.In conclusion, we have investigated the freezing phe-nomenon of asymmetry and put forward a theorem on thisissue. It shows that all measures of asymmetry are frozen fora state under a covariant operation if and only if the relativeentropy of asymmetry is frozen for the state. This theoremis applicable to all kinds of covariant operations defined byall groups, including but not limited to finite and compactLie groups. Our finding reveals the existence of universalfreezing of asymmetry, and more importantly provides a nec-essary and su ffi cient condition under which asymmetry of astate is totally una ff ected by the covariant noise. Note that similar issues about other resources such as quantum corre-lations and quantum coherence have been widely addressed,and the freezing phenomenon of correlations and the freez-ing phenomenon of coherence have already been found. Ourinvestigation fills a gap in the resource theory of asymmetry.This work was supported by the China Postdoctoral ScienceFoundation (Grant No. 2016M592173). X.D.Y. acknowl-edges support from the National Natural Science Foundationof China (Grant No. 11575101). H.L.H. acknowledges sup-port from the National Natural Science Foundation of China(Grant No. 11571199). D.M.T. acknowledges support fromthe National Basic Research Program of China (Grant No.2015CB921004). [1] A. Messiah, Quantum Mechanics (North-Holland, Amsterdam,1962).[2] I. Marvian and R. B. Mann, Phys. Rev. A , 022304 (2008).[3] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev. Mod.Phys. , 555 (2007).[4] E. P. Wigner, Z. Phys. , 101 (1952).[5] H. Araki and M. M. Yanase, Phys. Rev. , 622 (1960).[6] V. Giovannetti, S. Lloyd, and L. Maccone, Science , 1330(2004).[7] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96,010401 (2006).[8] V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photon. , 222(2011).[9] M. J. W. Hall and H. M. Wiseman, Phys. Rev. X , 041006(2012).[10] L.-M. Duan, M. Lukin, I. Cirac, and P. Zoller, Nature (London) , 413 (2001).[11] N. Gisin and R. Thew, Nat. Photon. , 165 (2007).[12] A. Peres and P. F. Scudo, Phys. Rev. Lett. , 167901 (2001).[13] E. Bagan, M. Baig, and R. Mu˜noz-Tapia, Phys. Rev. Lett. ,257903 (2001).[14] G. Chiribella, G. M. D’Ariano, P. Perinotti, and M. F. Sacchi,Phys. Rev. Lett. , 180503 (2004).[15] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki,Phys. Rev. Lett. Rev. Mod. Phys. , 865 (2009).[16] F. G. S. L. Brand˜ao and G. Gour, Phys. Rev. Lett. , 070503(2015).[17] J. A. Vaccaro, F. Anselmi, H. M. Wiseman, and K. Jacobs, Phys.Rev. A , 032114 (2008).[18] G. Gour and R. W. Spekkens, New J. Phys. , 033023 (2008).[19] G. Gour, I. Marvian, and R. W. Spekkens, Phys. Rev. A ,012307 (2009).[20] B. Toloui, G. Gour, and B. C. Sanders, Phys. Rev. A , 022322(2011).[21] M. Skotiniotis and G. Gour, New J. Phys. , 073022 (2012).[22] I. Marvian and R. W. Spekkens, New. J. Phys. ,033001(2013).[23] I. Marvian and R. W. Spekkens, Nat. Commun. , 3821 (2014).[24] I. Marvian and R. W. Spekkens, Phys. Rev. A , 014102(2014).[25] I. Marvian and R. W. Spekkens, Phys. Rev. A , 062110(2014).[26] I. Marvian, R. W. Spekkens, and P. Zanardi, Phys. Rev. A ,052331 (2016).[27] B. Yadin and V. Vedral, Phys. Rev. A , 022122 (2016). [28] M. Piani, M. Cianciaruso, T. R. Bromley, C. Napoli, N. John-ston, and G. Adesso, Phys. Rev. A , 042107 (2016).[29] I. Marvian and R. W. Spekkens, arXiv:1602.08049v2.[30] S. D. Bartlett and H. M. Wiseman, Phys. Rev. Lett. , 097903(2003).[31] Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. A , 052330 (2010).[32] S. C. Hou, X. L. Huang, and X. X. Yi, Phys. Rev. A , 012336(2010).[33] Z.-X. Man, Y.-J. Xia, and R. Lo Franco, Sci. Rep. , 13843(2015).[34] R. Lo Franco, Quantum Inf. Proc. , 2393 (2016).[35] X. Q. Shao, Z. H. Wang, H. D. Liu, and X. X. Yi, Phys. Rev. A , 032307 (2016).[36] L. Mazzola, J. Piilo, and S. Maniscalco, Phys. Rev. Lett. ,200401 (2010).[37] L. Mazzola, J. Piilo, and S. Maniscalco, Int. J. Quantum Inf. ,981 (2011).[38] B. You and L.-X. Cen, Phys. Rev. A , 012102 (2012).[39] Y.-Q. L¨u, J.-H. An, X.-M. Chen, H.-G. Luo, and C. H. Oh, Phys.Rev. A , 012129 (2013).[40] B. Aaronson, R. Lo Franco, and G. Adesso, Phys. Rev. A ,012120 (2013).[41] T. Chanda, A. K. Pal, A. Biswas, A. Sen(De), and U. Sen, Phys.Rev. A , 062119 (2015).[42] M. Cianciaruso, T. R. Bromley, W. Roga, R. Lo Franco, and G.Adesso, Sci. Rep. , 10177 (2015).[43] T. R. Bromley, M. Cianciaruso, and G. Adesso, Phys. Rev. Lett. , 210401 (2015).[44] X.-D. Yu, D.-J. Zhang, C. L. Liu, and D. M. Tong, Phys. Rev.A , 060303(R) (2016).[45] I. A. Silva et al. , Phys. Rev. Lett. , 160402 (2016).[46] This measure has been studied in the context of quantum ref-erence frames, in which it was termed the relative entropy offrameness [19]. In order to indicate its general applicability forvarious physical contexts, we call this measure the relative en-tropy of asymmetry in this paper.[47] G. Lindblad, Commun. Math. Phys. , 147 (1975).[48] A. Uhlmann, Commun. Math. Phys. , 21 (1977).[49] D. Petz, Rev. Math. Phys. , 123 (1986).[50] P. Hayden, R. Jozsa, D. Petz, and A. Winter, Commun. Math.Phys. , 359 (2004).[51] S. D. Bartlett, T. Rudolph, R. W. Spekkens, and P. S. Turner,New J. Phys. , 58 (2006).[52] D. Poulin and J. Yard, New J. Phys. , 156 (2007). [53] J. C. Boileau, L. Sheridan, M. Laforest, and S. D. Bartlett, J. Math. Phys.49