aa r X i v : . [ qu a n t - ph ] F e b Measure of genuine coherence based of quasi-relative entropy
Anna Vershynina
Department of Mathematics, Philip Guthrie Hoffman Hall, University of Houston, 3551 Cullen Blvd., Houston, TX77204-3008, USA
March 1, 2021
Abstract
We present a genuine coherence measure based on a quasi-relative entropy as a difference between quasi-entropies of the dephased and the original states. The measure satisfies non-negativity and monotonicityunder genuine incoherent operations (GIO). It is strongly monotone under GIO in two- and three-dimensions,or for pure states in any dimension, making it a genuine coherence monotone. We provide a bound on theerror term in the monotonicity relation in terms of the trace distance between the original and the dephasedstates. Moreover, the lower bound on the coherence measure can also be calculated in terms of this tracedistance.
Quantum coherence is a fundamental property of quan-tum systems, describing the existence of quantuminterference. It is widely used in thermodynamics[1, 9, 19], transport theory [28, 38], and quantum optics[13, 30], among few applications. Recently, problemsinvolving coherence included quantification of coher-ence [2, 22, 26, 27, 31, 40], distribution [25], entangle-ment [7, 33], operational resource theory [5, 7, 12, 37],correlations [17, 20, 34], with only a few referencesmentioned in each. See [32] for a more detailed review.As a golden standard it is taken that any “good” co-herence measure should satisfy four criteria presentedin [2]: vanishing on incoherent states; monotonicity un-der incoherent operations; strong monotonicity underincoherent operations, and convexity. Alternatively,the last two properties can be substituted by an ad-ditivity for subspace independent states, which wasshown in [40].A number of ways has been proposed as a coher-ence measure, but only a few satisfy all necessary cri- teria [2, 41, 42]. A broad class of coherence measureare defined as the minimal distance D to the set ofincoherent states I , as C D ( ρ ) = min δ ∈I D ( ρ, δ ) . In [2], it was shown that coherence vanishes on incoher-ent states when the distance vanishes only on identicalstates; the measure is monotone when the distance iscontractive under quantum channels; and it is convexwhen the distance is jointly convex. Strong monotonic-ity property is more challenging to pinpoint. Measuresthat satisfy the strong monotonicity that have been in-troduced up to date, are based on l , relative entropy,Tsallis entropy, and real symmetric concave functionson a probability simplex.Another approach to generate physically relevantcoherence measures is to consider different incoherentoperations. The largest class of incoherent operationsis called maximally incoherent (MIO), and it consists ofall completely-positive trace-preserving (CPTP) mapsthat preserve the set of incoherent states. The smallerset, called incoherent operations (IO) [2], has Krauss1operators that each preserve the set of incoherentstates (see Definition 2.4). A smaller set consists ofstrictly incoherent operations (SIO) [37, 39], which arethe result of action on a primary and ancillary systemsthat do not generate coherence on a primary system,see Definition 2.7. And the last class of operations thatis discussed in this paper, is called genuine incoherentoperations (GIO) [10], which act trivially on incoher-ent states, see Definition 2.5. See [6] for a larger list ofincoherent operations, and their comparison. For thesetypes of incoherent operations one may look at simi-lar properties as the ones presented in [2]. Restrictedto GIO, one would obtain a measure of genuine co-herence when it is non-negative and monotone, or acoherence monotone when it is also strongly monotoneunder GIO.In [10], the following genuine coherence measurewas proposed: C D ( ρ ) = D ( ρ k ∆( ρ )) , for a distance D , and ∆( ρ ) being the dephased statein a pre-fixed basis, see Notation 2.3. It was shownthat this is a genuine coherence measure if the dis-tance is contractive under unital operations. If fact,the monotonicity holds not only for GIO maps butfor dephasing-covariant incoherent operations (DIO)as well (the ones that commute with the dephasingoperator).Here we propose another genuine coherence mea-sure based on a quasi-relative entropy: C f ( ρ ) = S f (∆( ρ )) − S f ( ρ ) , here S f ( ρ ) is a quasi entropy, which could be definedin two ways, one of which is S f ( ρ ) = − S f ( ρ k| I ). Themotivation for this definition comes from the relativeentropy coherence. It was shown [2] that for a rela-tive entropy S ( ·k· ), there is a closed expression of adistance-based coherence measure:min δ ∈I S ( ρ k δ ) = S ( ρ k ∆( ρ )) = S (∆( ρ )) − S ( ρ ) . In general, for quasi-relative entropies neither ofthese equalities will hold. This can be seen for Tsallis relative entropy, which is a particular case of a quasi-relative entropy. The closest incoherent state is givenin [27], and it is not a dephased state ∆( ρ ). The secondequality does not hold either in general.We show that quasi-relative entropy coherence,which we call f -coherence, is unique for pure states,non-negative, zero if and only if a state is incoherent,and monotone under GIO maps. Moreover, we givea lower bound on this coherence in terms of a tracedistance between a state and its dephased state, weprovide an if and only if condition on a GIO map thatsaturates the monotonicity relation, and bound the er-ror term in the monotonicity relation. Additionally, weinvestigate when the f -coherence would be monotoneunder a larger class of SIO maps.We show that f -coherence saturates strong mono-tonicity under GIO maps in two- and three-dimensions,and it satisfies the strong monotonicity under GIOmaps in any dimensions for pure states. Let H be a d -dimensional Hilbert space. Let us fix abasis E = {| j i} dj =1 of vectors in H . A state δ is called incoherent if it canbe represented as follows δ = X j δ j | j i h j | . For a state on multiple systems, the incoherentstates is defined as follows [3, 33].
A state δ on H = N k H k is calledincoherent if it can be represented as follows δ = X j δ j | j i h j | , where j = ( j , . . . , j N ) with j k = 1 , . . . , d k is a vectorof indices, and | j i = N k | j k i = | j i ⊗ · · · ⊗ | j N i . Denote the set of incoherent states for a fixed basis E = {| j i} j as I = { ρ = X j p j | j i h j |} . A dephasing operation in E basis is the followingmap: ∆( ρ ) = X j h j | ρ h j | | j i h j | . A CPTP map Φ with the followingKraus operators Φ( ρ ) = X n K n ρK ∗ n , is called the incoherent operation (IO) or incoher-ent CPTP (ICPTP), when the Kraus operators satisfy K n I K ∗ n ⊂ I , for all n , besides the regular completeness relation P n K ∗ n K n =1l . Any reasonable measure of coherence C ( ρ ) shouldsatisfy the following conditions • (C1) C ( ρ ) = 0 if and only if ρ ∈ I ; • (C2) Non-selective monotonicity under IO maps(monotonicity) C ( ρ ) ≥ C (Φ( ρ )) ; • (C3) Selective monotonicity under IO maps(strong monotonicity) C ( ρ ) ≥ X n p n C ( ρ n ) , where p n and ρ n are the outcomes and post-measurement states ρ n = K n ρK ∗ n p n , p n = Tr K n ρK ∗ n . • (C4) Convexity, X n p n C ( ρ n ) ≥ C X n p n ρ n ! , for any sets of states { ρ n } and any probabilitydistribution { p n } . These properties are parallel with the entanglementmeasure theory, where the average entanglement is notincreased under the local operations and classical com-munication (LOCC). Notice that coherence measuresthat satisfy conditions (C3) and (C4) also satisfies con-dition (C2).In [10] a class of incoherence operations was de-fined, called genuinely incoherent operations (GIO) asquantum operations that preserve all incoherent states. An IO map Λ is called a genuinelyincoherent operation (GIO) is for any incoherentstate δ ∈ I , Λ( δ ) = δ . An operation Λ is GIO if and only if all Kraus rep-resentations of Λ has all Kraus operators diagonal in apre-fixed basis [10].Conditions (C2), (C3) and (C4) can be restrictedto GIO maps to obtain different classes of coherencemeasures.
In this case, a measure of genuinecoherence satisfies at least (G1) and (G2). And if acoherence measure fulfills conditions (G1), (G2), (G3)it is called genuine coherence monotone . A larger class of IO maps was defined in [37, 39].
An IO map Λ is called strictly in-coherent operations (SIO) if its Kraus representa-tion operator commute with dephasing, i.e. for Λ( ρ ) = P j K j ρK ∗ j , we have for any j , K j ∆( ρ ) K ∗ j = ∆( K j ρK ∗ j ) . Since Kraus operators of GIO maps are diagonal in E basis, any GIO map is SIO as well, i.e. GIO ⊂ SIO,[10].One may consider an additional property, closelyrelated to the entanglement theory: • (C5) Uniqueness for pure states: for any purestate | ψ i coherence takes the form: C ( ψ ) = S (∆( ψ )) , where S is the von Neumann entropy and ∆ isthe dephasing operation defined as∆( ρ ) = X j h j | ρ | j i | j i h j | . Quantum quasi-relative entropy was introduced byPetz [23, 24] as a quantum generalization of a classicalCsisz´ar’s f -divergence [8]. It is defined in the con-text of von Neumann algebras, but we consider onlythe Hilbert space setup. Let H be a finite-dimensionalHilbert space, and ρ and σ be two states (given bydensity operators). For strictly positive bounded opera-tors A and B acting on a finite-dimensional Hilbertspace H , and for any function f : (0 , ∞ ) → R , thequasi-relative entropy (or sometimes referred to as the f -divergence) is defined as S f ( A || B ) = Tr( f ( L B R − A ) A ) , where left and right multiplication operators are definedas L B ( X ) = BX and R A ( X ) = XA . There is a straightforward way to calculate thequasi-relative entropy from the spectral decompositionof operators [16, 36]. Let A and B have the followingspectral decomposition A = X j λ j | φ j i h φ j | , B = X k µ k | ψ k i h ψ k | . (2.1)the set {| φ k i h ψ j |} j,k forms an orthonormal basis of B ( H ), the space of bounded linear operators, with re-spect to the Hilbert-Schmidt inner product defined as h A, B i = Tr( A ∗ B ). By [36], the product of left andright multiplication operators can be written as L B R − A = X j,k µ k λ j P j,k , (2.2)where P j,k : B ( H ) → B ( H ) is defined by P j,k ( X ) = | ψ k i h φ j | h ψ k | X | φ j i . The quasi-relative entropy is calculated as follows S f ( A || B ) = X j,k λ j f (cid:18) µ k λ j (cid:19) | h ψ k | | φ j i | . (2.3) ([23]) For states, i.e. trace one positivedensity matrices ρ and σ , the quasi-relative entropy isbounded below by S f ( ρ k σ ) ≥ f (1) . The equality happens for a non-linear function f if andonly if ρ = σ . It is natural to require the quasi-relative entropyto be zero for equal state, and therefore we assumethroughout the paper that f (1) = 0.For an operator convex function, f , the quasi-relative entropy is jointly convex and monotone underCPTP maps [16]. The equality in monotonicity holdsif and only if the map is reversible on these two states,i.e. for two states ρ and σ with supp ρ ⊂ supp σ , anda CPTP map Λ, the equality S f ( ρ k σ ) = S f (Λ( ρ ) k Λ( σ ))is satisfied if and only if R σ (Λ( ρ )) = ρ , where R σ is the Petz’s recovery map defined as R σ ( ω ) = σ / Λ ∗ (cid:16) Λ( σ ) − / ω Λ( σ ) − / (cid:17) σ / . (2.4) Throughout the paper we will as-sume that the function f is operator convex and f (1) =0 . For any function f , its transpose ˜ f is defined as˜ f ( x ) = xf (cid:18) x (cid:19) , x ∈ (0 . ∞ ) . The transpose ˜ f of an operator convex function f on(0 , ∞ ) is operator convex again, [16]. From (2.3) itfollows that S ˜ f ( ρ k σ ) = S f ( σ k ρ ) . For f ( x ) = − log x , the quasi-relativeentropy becomes the Umegaki relative entropy S − log ( ρ k σ ) = S ( ρ k σ ) = Tr( ρ log ρ − ρ log σ ) . For p ∈ ( − , and p = 0 , let ustake the function f p ( x ) := 1 p (1 − p ) (1 − x p ) , which is operator convex. The quasi-relative entropyfor this function is calculated to be S f p ( ρ || σ ) = 1 p (1 − p ) (cid:0) − Tr( σ p ρ − p ) (cid:1) . For p ∈ ( − , take q = 1 − p ∈ (0 , ,the function f q ( x ) = 11 − q (1 − x − q ) is operator convex. The quasi-relative entropy for thisfunction is known as Tsallis q -entropy S q ( ρ k σ ) = 11 − q (cid:0) − Tr( ρ q σ − q ) (cid:1) . f -entropy For a convex, operator monotone decreasing function f , such that f (1) = 0, define entropy two ways. The f -entropy is defied as S f ( ρ ) := f (1 /d ) − S f ( ρ k I/d ) . (3.1)ˆ S f ( ρ ) := − S f ( ρ k I ) . (3.2) Let us use a notation ˜ S f for either S f or ˆ S f . f -entropy is non-negative, and is zeroon pure states.Proof. Let { λ j } be the eigenvalues of ρ . Then from(2.3) we have S f ( ρ ) = f (1 /d ) − X j λ j f (cid:18) dλ j (cid:19) , (3.3)and ˆ S f ( ρ ) = − X j λ j f (cid:18) λ j (cid:19) . (3.4)A sequence of eigenvalues { λ j } is majorized by a se-quence { , , . . . , } . Since a perspective function (or a transpose function) xf (1 /x ) is convex for a convexfunction f [16], this implies that by results on Schur-concavity [14, 21, 29] we have X j λ j f (cid:18) dλ j (cid:19) ≤ f (1 /d ) . Here, if needed, we adopt a convention 0 · ±∞ := 0[15].Since f is monotonically decreasing and f (1) = 0,for any 0 ≤ λ j ≤ f (cid:16) λ j (cid:17) ≤
0. Thus, ˆ S f ≥ . When ρ = | Ψ i h Ψ | is a pure state, there is only oneeigenvalue λ = 1. Then S f ( | Ψ i h Ψ | ) = f (1 /d ) − f (1 /d ) = 0 , and ˆ S f ( | Ψ i h Ψ | ) = − f (1) = 0 . The maximum value of f -entropy isreached on the maximally mixed state I/d and it is S f ( ρ ) ≤ f (1 /d ) , and ˆ S f ( ρ ) ≤ − f ( d ) . Proof.
From Theorem 2.9, S f ( ρ k I/d ) ≥
0, or since f is convex, we have X j λ j f (cid:18) dλ j (cid:19) ≥ f X j λ j dλ j = f (1) = 0 . Similarly, X j λ j f (cid:18) λ j (cid:19) ≥ f X j λ j λ j = f ( d ) . From (3.3) and (3.4), the result follows. Clearly, when ρ = I/d , we have S f ( I/d ) = f (1 /d ) − f (1 /d ), andfrom (3.4) we have ˆ S f ( I/d ) = − f ( d ) . The f -entropies are concave in ρ . Let { p k } be a probability distribution and ρ k be some states,then for ρ = P k p k ρ k , we have ˜ S f ( ρ ) ≥ X k p k ˜ S f ( ρ k ) . Proof.
This immediately follows from the joint convex-ity of f -divergence [15, 16]. The f -entropies are invariant underunitaries.Proof. Since a unitary operation
U ρU ∗ does notchange the eigenvalues of ρ , and the f -entropies arethe functions of eigenvalues of ρ , this implies that f -entropies are invariant under any operations that pre-serve eigenvalues. The f -entropies are non-decreasingunder untial CPTP maps, i.e. for any linear CPTPmap Λ , such that Λ( I ) = I , we have ˜ S f (Λ( ρ )) ≥ ˜ S f ( ρ ) . Proof.
Let us denote σ = I or σ = I/d , which corre-sponds to the appropriate f -entropy. Then˜ S f (Λ( ρ )) − ˜ S f ( ρ ) = S f ( ρ k σ ) − S f (Λ( ρ ) k σ ) (3.5)= S f ( ρ k σ ) − S f (Λ( ρ ) k Λ( σ )) ≥ . (3.6)The last equality holds since Λ is unital, and the in-equality holds due to the monotonicity of f -divergenceunder CPTP maps [18, 23, 35]. In a d -dimensional Hilbert space H , fix a basis E = {| j i} d − j =0 . For any entropy function S , which isnon-decreasing under CPTP maps, define coherence asfollows: C S ( ρ ) := S (∆( ρ )) − S ( ρ ) . (4.1)In particular, for any operator convex and oper-ator monotone decreasing function f , define two f -coherence measures. For entropy defined in (3.1), C f ( ρ ) := S f (∆( ρ )) − S f ( ρ ) . (4.2) For entropy defined in (3.2), ˆ C f ( ρ ) := ˆ S f (∆( ρ )) − ˆ S f ( ρ ) . (4.3) Let us denote ˜ C f as either one C f or ˆ C f for shortness. If { λ j } are the eigenvalues of ρ , and the diagonalelements of ρ in E basis are χ j = h j | ρ | j i , then from(3.3) and (3.4), we have C f ( ρ ) = X j λ j f (cid:18) dλ j (cid:19) − X j χ j f (cid:18) dχ j (cid:19) , (4.4)and ˆ C f ( ρ ) = X j λ j f (cid:18) λ j (cid:19) − X j χ j f (cid:18) χ j (cid:19) . (4.5) Since f ( x ) = − log( x ) is operator convex, coherencemeasure defined above coincides with [2]: C ( ρ ) = ˆ C f ( ρ ) = S log (∆( ρ )) − S log ( ρ ) (4.6)= X j λ j log λ j − X j χ j log χ j (4.7)= S (∆( ρ )) − S ( ρ ) (4.8)= S ( ρ k ∆( ρ )) (4.9)= min δ ∈I S ( ρ k δ ) . (4.10) The function f ( x ) = − α (1 − x − α ) is operator convexfor α ∈ (0 , C α ( ρ ) = d α − − α X j χ αj − X j λ αj = d α − ˆ C α ( ρ ) . (4.11) For any pure state coherence becomes an entropy of adephased state: C S ( ψ ) = S (∆( ψ )) . This holds since entropies are zero on pure states. C S and, in particular, ˜ C f are non-negative.Proof. By assumption S is non-decreasing underCPTP maps, it follows that C is non-negative.This holds for f -entropies as well due to Theorem3.6, since the dephasing operation is unital.Clearly, for any incoherent state ρ , coherence C S ( ρ ) = 0. Having no information on the saturationcondition for a general entropy S , it is impossible tosay what happens in the other direction. Consider f -coherences (4.2) and (4.3). ˜ C f ( ρ ) = 0 if and only if ρ ∈ I is inco-herent state.Proof. In Theorem 3.6, the equality in the only in-equality (3.6) holds if and only if there is a recoverymap R such that R (∆( ρ )) = ρ and R ( I ) = I , [15, 16].By (2.4), this map admits the following explicit form:denoting σ = I R σ ( ω ) = σ / ∆ ∗ (cid:16) ∆( σ ) − / ω ∆( σ ) − / (cid:17) σ / , where ∆ ∗ is a dual map of ∆ . Since ∆ is a linear unitalGIO map, we have R σ ( ω ) = ∆ ∗ ( ω ) . (5.1)Therefore, condition R σ (∆( ρ )) = ρ implies that ρ = ∆ ∗ (∆( ρ )) . (5.2)Since ∆ ∗ = ∆, we have that ρ = ∆( ρ ), which happensif and only if ρ ∈ I . Thus, C f ( ρ ) = 0 = ˆ C f ( ρ ) if andonly if ρ ∈ I .A strengthening of the monotonicity inequality for f -divergence was presented in [4]. Using this result, weobtain the following lower bound on f -coherence. Let f be an operator monotone de-creasing function, and T > . Suppose for someconstant c > , there is a constant C > so that d t ≤ CT c d µ f ( t ) for t ∈ [ T − , T ] . Then there is an ex-plicitly computable constant K f ( ρ ) depending only on the smallest non-zero eigenvalue of ρ , C and c , suchthat, C f ( ρ ) ≥ K f ( ρ ) k ρ − ∆( ρ ) k c )1 . (5.3) Here, k A k = Tr | A | = Tr √ A ∗ A is the trace-norm ofan operator. From this inequality, the above condition of a zerocoherence becomes apparent, i.e. C f ( ρ ) = 0 if and onlyif ρ ∈ I .The upper bound given below extends the upperbound for a relative entropy of coherence [2] to any f -coherence. The coherence is upper bounded by C f ( ρ ) ≤ f (1 /d ) , and ˆ C f ( ρ ) ≤ − f ( d ) . The maximum value is reached for a maximally coher-ent pure state ρ = | ψ i h ψ | , with | ψ i = √ d P j | j i .Proof. This follows from the upper bound on the f -entropy Theorem 3.3, and the definition of coherence˜ C f ( ρ ) = ˜ S f (∆( ρ )) − ˜ S f ( ρ ) . For a pure state the entropy is zero, ˜ S f ( | ψ i h ψ | ) = 0.The dephasing operation applied to the state | ψ i = √ d P j | j i gives a maximally mixed state I/d . The the-orem follows from the fact that the entropy is maximalon maximally mixed state. C S and, in particular, ˜ C f is monotoneunder GIO.Proof. Any GIO map Λ is also SIO, and, in particular,Λ commutes with the dephasing operation. Therefore,∆(Λ( ρ )) = Λ(∆( ρ )) = ∆( ρ ), the last equality is dueto the fact that ∆( ρ ) ∈ I and Λ as GIO preservesincoherent states. Therefore, C S ( ρ ) − C S (Λ( ρ )) (5.4)= S (∆( ρ )) − S ( ρ ) − S (∆(Λ( ρ ))) + S (Λ( ρ )) (5.5)= S (Λ( ρ )) − S ( ρ ) (5.6) ≥ , (5.7)since Λ is a CPTP map and S is non-increasing un-der CPTP maps. For f -coherences, the last inequal-ity holds to the Theorem 3.6 since a GIO map is uni-tal. For GIO map Λ , the equality ˜ C f ( ρ ) = ˜ C f (Λ( ρ )) happens if and only if any Kraus representation of Λ( ρ ) = P j K j ρK ∗ j mush have operators K j = P n k jn | n i h n | that satisfy: for any n, m such that h n | ρ | m i 6 = 0 , it must be that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j k jn k jm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 . Proof.
Similarly, to the positivity section, equality in(5.7) happens if and only if there is a recovery map R such that R (Λ( ρ )) = ρ and R ( I ) = I , [15, 16]. By(2.4), this map admits the following explicit form: de-noting σ = I R σ ( ω ) = σ / Λ ∗ (cid:16) Λ( σ ) − / ω Λ( σ ) − / (cid:17) σ / , where Λ ∗ is a dual map of Λ. Since Λ is a linear unitalGIO map, we have R σ ( ω ) = Λ ∗ ( ω ) . (5.8)Therefore, condition R σ (Λ( ρ )) = ρ implies that ρ = Λ ∗ (Λ( ρ )) . (5.9)Denote a Kraus representation of Λ as Λ( ρ ) = P j K i ρK ∗ j . From [10], since Λ is GIO, any Krausrepresentation of Λ has diagonal operators, i.e. each K j = P n k jn | n i h n | is diagonal in basis E . Since P j K ∗ j K j = I , we have P j | k jn | = 1 for every n .The dual map is Λ ∗ ( ρ ) = P j K ∗ j ρK j . Therefore, (5.9)becomes ρ = X ji K ∗ j K i ρ (cid:0) K ∗ j K i (cid:1) ∗ . Writing both sides in basis E gives X nm h n | ρ | m i | n i h m | (5.10)= X nm X ij k jn k in k jm k im h n | ρ | m i | n i h m | (5.11)= X nm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j k jn k jm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h n | ρ | m i | n i h m | . (5.12)This implies that for every n, m such that h n | ρ | m i 6 = 0we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j k jn k jm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 . (5.13)This clearly confirms that any incoherent state satu-rates monotonicity for GIO maps.If ρ is a coherent state, i.e. there exist n, m suchthat h n | ρ | m i 6 = 0, to saturate monotonicity the mapΛ should satisfy (5.13). Note that by Cauchy-Schwarzinequality we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j k jn k jm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X j | k jn | X j | k jm | = 1 . The equality above happens if and only if there exists ascalar α nm ∈ C such that for any j : k jn = α nm k jm .Applying the strengthening of monotonicity in-equality for f -divergences [4], we obtain a strength-ening on the monotonicity inequality for f -coherence. Let Λ be any GIO map. Let f be anoperator monotone decreasing function, and T > .Suppose for some constant c > , there is a constant C > so that d t ≤ CT c d µ f ( t ) for t ∈ [ T − , T ] . Thenthere is an explicitly computable constant K f ( ρ ) de-pending only on the smallest non-zero eigenvalue of ρ , C and c , such that, C f ( ρ ) − C f (Λ( ρ )) ≥ K f ( ρ ) k ρ − Λ ∗ (Λ( ρ )) k c )1 . (5.14)The next theorem shows that ˜ C f is not in generalmonotone under SIO operations. If ˜ C f is monotone under all SIO, thenfor all states ρ and | i ∈ E , we have ˜ C f ( ρ ⊗ I/d ) = ˜ C f ( ρ ⊗ | i h | ) . In other words, if C f is monotone under SIO, then forall states with eigenvalues { λ j } and diagonal elements { χ j } in the basis E , the following holds X j λ j f (cid:18) dλ j (cid:19) − X j χ j f (cid:18) dχ j (cid:19) = X j λ j f (cid:18) d λ j (cid:19) − X j χ j f (cid:18) d χ j (cid:19) . (5.15) And, if ˆ C f is monotone under SIO, then for all stateswith eigenvalues { λ j } and diagonal elements { χ j } inthe basis E , the following holds X j λ j f (cid:18) λ j (cid:19) − X j χ j f (cid:18) χ j (cid:19) (5.16)= X j λ j f (cid:18) dλ j (cid:19) − X j χ j f (cid:18) dχ j (cid:19) . (5.17) Proof.
First, note that from (4.4) we have: for | i ∈ E , C f ( ρ ⊗ | i h | ) = X j λ j f (cid:18) d λ j (cid:19) − X j χ j f (cid:18) d χ j (cid:19) , (5.18)and C f ( ρ ⊗ I/d ) = C f ( ρ ) = X j λ j f (cid:18) dλ j (cid:19) − X j χ j f (cid:18) dχ j (cid:19) . (5.19)Moreover,ˆ C f ( ρ ⊗| i h | ) = ˆ C f ( ρ ) = X j λ j f (cid:18) λ j (cid:19) − X j χ j f (cid:18) χ j (cid:19) , (5.20)andˆ C f ( ρ ⊗ I/d ) = X j λ j f (cid:18) dλ j (cid:19) − X j χ j f (cid:18) dχ j (cid:19) . (5.21)Let us consider two examples of SIO \ GIO maps.1. Let Φ( ρ ) = I/d be the depolarizing quantum chan-nel, which in Kraus form can be written asΦ( ρ ) = I/d = d − X ij =0 K ij ρK ∗ ij , where K ij = √ d | i i h j | . Define an operation on a tensor product Hilbertspace as followsΛ( ω ) = X ij ( I ⊗ K ij ) ω ( I ⊗ K ij ) ∗ . (5.22)Clearly, Λ is not a GIO, since its Kraus operators arenot diagonal in E ⊗ E basis, or sinceΛ( ρ ⊗ | i h | ) = ρ ⊗ Φ( | i h | ) = ρ ⊗ I/d (5.23) = ρ ⊗ | i h | ∈ E ⊗ E . (5.24)But Λ is SIO, since for any n, m ( I ⊗ K nm )(∆( ω )( I ⊗ K ∗ nm ) (5.25)= 1 d ( I ⊗ | n i h m | ) X ij h ij | ω | ij i | ij i h ij | ( I ⊗ | m i h n | )(5.26)= 1 d X ij h ij | ω | ij i | i i h i | ⊗ | n i h m | | j i h j | | m i h n | (5.27)= 1 d X i h im | ω | im i | in i h in | , (5.28)and∆(( I ⊗ K nm ) ω ( I ⊗ K ∗ nm )) (5.29)= 1 d X ij h ij | ( I ⊗ | n i h m | ) ω ( I ⊗ | m i h n | ) | ij i | ij i h ij | (5.30)= 1 d X i h im | ω | im i | in i h in | (5.31)(5.32)Therefore, Λ is a SIO map.For either C f or ˆ C f , consider˜ C f (Λ( ρ ⊗ | i h | )) = ˜ C f ( ρ ⊗ Φ( | i h | ))) (5.33)= ˜ C f ( ρ ⊗ I/d ) . (5.34)2. Consider another example, let Ψ( ρ ) = | i h | be the erasure channel, which in Kraus form can bewritten asΨ( ρ ) = | i h | = d − X j =0 K j ρK ∗ j , where K j = | i h j | . M ( ω ) = X j ( I ⊗ K j ) ω ( I ⊗ K j ) ∗ . (5.35)Clearly, M is not a GIO, since its Kraus operators arenot diagonal in E ⊗ E basis, or since M ( ρ ⊗ I/d ) = ρ ⊗ Ψ( I/d ) = ρ ⊗ | i h | (5.36) = ρ ⊗ I/d ∈ E ⊗ E . (5.37)But M is SIO, since for any n ,( I ⊗ K n )∆( ω )( I ⊗ K ∗ n ) (5.38)= X ij h ij | ω | ij i ( I ⊗ | i h n | ) | ij i h ij | ( I ⊗ | n i h | )(5.39)= X i h in | ω | in i | i i h i | ⊗ | i h | . (5.40)and∆ (( I ⊗ K n ) ω ( I ⊗ K ∗ n )) (5.41)= X ij h ij | ( I ⊗ | i h n | ) ω ( I ⊗ | n i h | ) | ij i | ij i h ij | (5.42)= X i h in | ω | in i | i i h i | . (5.43)Therefore, M is an SIO map.For either C f or ˆ C f , consider˜ C f ( M ( ρ ⊗ I/d )) = ˜ C f ( ρ ⊗ Ψ( I/d ))) (5.44)= ˜ C f ( ρ ⊗ | i h | ) . (5.45)Now, compare (5.34) and (5.45). In order for mono-tonicity of f -coherence to hold under all SIO, theremust be an equality˜ C f ( ρ ⊗ I/d ) = ˜ C f ( ρ ⊗ | i h | ) . Invoking (5.18-5.21) we have the result stated in thetheorem.Note that both (5.15) and (5.17) hold for the loga-rithmic function f ( x ) = − log( x ), but fail for the powerfunction f ( x ) = − α (1 − x − α ). This is in line with thefact that the relative entropy of coherence is monotoneunder SIO, and it shows that Tsallis coherence failsmonotonicity for SIO. f -coherences ˜ C f saturate strong mono-tonicity for convex mixtures of diagonal unitaries.Therefore, ˜ C f saturates strong monotonicity underGIO in two- and three-dimensions.Proof. Consider an example of GIO, which is a proba-bilistic mixture of diagonal unitaries: for some α j > P j α j = 1, defineΛ( ρ ) = X j α j U j ρU ∗ j , where for some ρ jn the unitaries U j are diagonal in E ,i.e. U j = X n e iφ jn | n i h n | . In [10] it has been shown that all GIO are of suchform for dimensions two and three, but it is no longerthe case for higher dimensions.Note that for σ = I or σ = I/d and for all unitaries U , we have S f ( U ρU ∗ k σ ) = S f ( ρ k σ ) . (5.46)Taking U j diagonal in E above, it follows that∆( U j ρU ∗ j ) = ∆( ρ ) . Therefore, ˜ C f saturates the strong monotonicity underconvex mixtures of diagonal unitaries: X j α j ˜ C f ( U j ρU ∗ j ) (5.47)= X j α j (cid:2) S f ( U j ρU ∗ j k σ ) − S f (∆( U ρU ∗ ) k σ ) (cid:3) (5.48)= X j α j [ S f ( ρ k σ ) − S f (∆( ρ ) k σ )] (5.49)= ˜ C f ( ρ ) . (5.50)1 Expanding the set of operations to in-clude all unitaries (not necessarily diagonal in E ),forces ˜ C f to be invariant under all unitaries if it ismonotone under them. This results from the follow-ing observation: if ˜ C f is monotone under all unitaries U and all states ρ , then, since (5.46) holds, it must bethat S f (∆( U ρU ∗ ) k σ ) ≥ S f (∆( ρ ) k σ ) . But taking a unitary V = U ∗ and an initial state ω = U ρU ∗ above, results in the opposite inequality: S f (∆( V ωV ∗ ) k σ ) = S f (∆( ρ ) k σ ) (5.51) ≥ S f (∆( U ρU ∗ ) k σ ) = S f (∆( ω ) k σ ) . (5.52)Therefore, the above inequality must be equality,which makes ˜ C f invariant under unitaries. For any pure state ρ , the f -coherences are strongly monotone under GIO maps inany finite dimension.Proof. Let us denote σ = I or σ = I/d depending onthe f -coherence we are considering. For a GIO map Λwith Kraus operators K j , denote p j = Tr K j ρK ∗ j , ρ j = 1 p j K j ρK ∗ j . For a pure state ρ , states ρ j are also pure. Therefore,˜ C f ( ρ ) − X j p j ˜ C f ( ρ j ) (5.53)= X j p j S f (∆( ρ j ) k σ ) − S f (∆( ρ ) k σ ) . (5.54)Since any GIO map is an SIO map as well, it followsthat ∆( ρ j ) = 1 p j K j ∆( ρ ) K ∗ j . Dephased state ∆( ρ ) is diagonal in E basis witheigenvalues χ j , i.e. ∆( ρ ) = P j χ j | j i h j | . The f -divergence is S f (∆( ρ ) k I ) = X n χ n f (cid:18) χ n (cid:19) . Kraus operators of GIO map are diagonal is E ba-sis, K j = P n k jn | n i h n | , with P j | k jn | = 1 for all j .Then K j ∆( ρ ) K ∗ j = X n χ n | k jn | | n i h n | . And X j p j S f (∆( ρ j ) k I ) = X jn χ n | k jn | f (cid:18) p j χ n | k jn | (cid:19) . Since f is convex, we have for every n : X j | k jn | f (cid:18) p j χ n | k jn | (cid:19) ≥ f X j p j χ n (5.55)= f (cid:18) χ n (cid:19) . (5.56)Similarly, S f (∆( ρ ) k I/d ) = X n χ n f (cid:18) dχ n (cid:19) , and X j p j S f (∆( ρ j ) k I/d ) = X jn χ n | k jn | f (cid:18) p j dχ n | k jn | (cid:19) . Because f is convex, for any n : X j | k jn | f (cid:18) p j dχ n | k jn | (cid:19) ≥ f X j p j dχ n (5.57)= f (cid:18) dχ n (cid:19) . (5.58)And thus, P j p j S f (∆( ρ j ) k σ ) ≥ S f (∆( ρ ) k σ ) . Whichimplies that for any pure state ρ , the f -coherence isstrongly monotone under GIO:˜ C f ( ρ ) ≥ X j p j ˜ C f ( ρ j ) . Acknowledgments.
A. V. is supported by NSFgrant DMS-1812734.2
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