Max-Planck-Institut fur¨ Mathematik in den Naturwissenschaften Leipzig

Coherence measures with respect to general quantum measurements

by

Jianwei Xu, Lian-He Shao, and Shao-Ming Fei

Preprint no.: 70 2020

Coherence measures with respect to general quantum measurements

Jianwei Xu1, Lian-He Shao2, and Shao-Ming Fei3,4∗ 1College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China 2School of Computer Science, Xi’an Polytechnic University, Xi’an, 710048, China 3School of Mathematical Sciences, Capital Normal University, Beijing 100048, China and 4Max-Planck-Institute for Mathematics in the Sciences, Leipzig 04103, Germany

Quantum coherence with respect to orthonormal bases has been studied extensively in the past few years. Recently, Bischof, et al. [Phys. Rev. Lett. 123, 110402 (2019)] generalized it to the case of general positive operator-valued (POVM) measurements. Such POVM-based coherence, including the block coherence as special cases, have significant operational interpretations in quantifying the advantage of quantum states in processing. In this work we first establish an alternative framework for quantifying the block coherence and provide several block coherence measures. We then present several coherence measures with respect to POVM measurements, and prove a conjecture on the l1-norm related POVM coherence measure.

PACS numbers: 03.65.Ud, 03.67.Mn, 03.65.Aa

I. INTRODUCTION with maximal information about the state, which gen- eralizes the results from [2]. It has been shown that the Quantum coherence is a characteristic feature of quan- relative entropy of POVM-coherence is equal to the cryp- tum mechanics, with wide applications in superconduc- tographic randomness gain [20]. It provides an important tivity, quantum thermodynamics and biological process- operational meaning to the concept of coherence with re- es. From a resource-theoretic perspective the quantifi- spect to a general measurement. Generalizing the usual cation of quantum coherence has attracted much at- coherence theory from an orthonormal basis to a generic tention and various kinds of coherence measures have POVM had been also the efforts made in [21, 22]. been proposed [1–15]. Let ρ be a density operator in d-dimensional complex H. Under a fixed {| ⟩}d orthonormal basis i i=1 of H, the state ρ is called in- coherent if ⟨i|ρ|j⟩ = 0 for any i ≠ j [1]. Otherwise ρ is called coherent. The coherence theory has achieved After establishing a framework for quantifying the fruitful results in the past few years (for recent reviews POVM coherence [18, 20], Bischof, et al. developed see e.g. [16, 17]). [18, 20] a scheme by employing the Naimark extension From another perspective, the orthonormal basis to embed the POVM coherence into the block coherence {| ⟩}d proposed in [23] in a lager Hilbert space. The Naimark i i=1 corresponds to a rank-1 projective measurement {| ⟩⟨ |}d ⟨ | | ⟩ extension [24, 25] states that any POVM can be extend- (von Neumann measurement) i i i=1, and i ρ j = 0 is equivalent to |i⟩⟨i|ρ|j⟩⟨j| = 0. This observation leads ed to a projective measurement in a larger Hilbert space. one to view the coherence with respect to the orthonor- The block coherence was defined with respect to pro- {| ⟩}d jective measurements, not necessarily rank-1. With this mal basis i i=1 as the coherence with respect to the {| ⟩⟨ |}d scheme, the relative entropy of POVM coherence Crel, the rank-1 projective measurement i i i=1. Along this idea, the concept of coherence can be generalized to the robustness POVM coherence Crot were proposed. Re- cases of general measurements. Recently, Bischof, et al. cently, the structures of different incoherent operations [18] have generalized the concept of coherence to the case for POVM coherence were investigated [26]. For simplic- of general quantum measurements, i.e., positive operator- ity, we call the coherence theory with respect to fixed valued measures () and established the resource orthonormal bases the standard coherence theory. As theory of coherence based on POVMs. One motivation of the generalizations of the standard coherence, both the this generalization is due to the fact that POVMs may be block coherence and the POVM coherence reduce to the more advantageous compared to von Neumann measure- standard coherence in the case of the von Neumann mea- ment in many applications [19]. Moreover, the coherence surement. of a state with respect to a POVM can be interpreted as the cryptographic randomness generated by measur- ing the POVM on the state [20]. Namely, the amount of POVM coherence in a state is equal to the unpredictabil- ity of measurement outcomes relative to an eavesdropper In the present work, we establish an alternative frame- work for quantifying the block coherence and provide sev- eral block coherence measures. We then present several POVM coherence measures. Meanwhile, we also prove a ∗Electronic address: [email protected] conjecture raised recently. 2

II. ALTERNATIVE FRAMEWORK FOR is called block incoherent with respect to the projective QUANTIFYING BLOCK COHERENCE measurement P ⊗ P ′ if ⊗ ′ AA′ ⊗ ′ ∀ ′ ̸ ′ A. Block incoherent states and block incoherent (Pi Pi′ )ρ (Pj Pj′ ) = 0, (i, i ) = (j, j ), (5) channels ′ ′ ′ ′ where (i, i ) ≠ (j, j ) means that i ≠ j or i ≠ j . ′ ′ We denote the set of all states on HAA by S(HAA ) The block coherence theory was introduced in [23]. We ′ ′ and the set of all channels on S(HAA ) by C(HAA ). A adopt the framework proposed in [20] for quantifying the ′ ′ AA C AA block coherence. Consider a quantum system A associat- ϕ on (H ) is called a block in- coherent if it admits an expression of Kraus operators ed with an m-dimensional complex Hilbert space H. One ′ ′ AA { AA } ⊕n ϕ = Kl l such that has partition H = i=1πi into∑ orthogonal subspaces πi n ′ ′ ′ of dimension dimπi = mi, i=1 mi = m. Correspond- ⊗ ′ AA AA AA † ⊗ ′ { }n (Pi Pi′ )Kl ρ (Kl ) (Pj Pj′ ) = 0 (6) ingly, one gets a projective measurement P = Pi i=1, ′ ′ with each projector satisfying Pi(H) = πi. A state ρ on for all l and (i, i ) ≠ (j, j ). We denote the set of all block H is called block incoherent (BI) with respect to P if AA′ AA′ incoherent channels on C(H ) by CBI(H ) and call AA′ { AA′ } ∀ ̸ such an expression ϕ = Kl l a block incoherent PiρPj = 0, i = j, (1) ′ decomposition of ϕAA . or ∑n B. An alternative framework for quantifying the ρ = PiρPi. (2) block coherence i=1 We denote the set of all quantum states in H by S(H), A framework for quantifying the block coherence has and the set of all block incoherent quantum states by been established in [20]: any valid block coherence mea- IB(H). It is easy to check that sure C(ρ; P ) with respect to the projective measurement P should satisfy the conditions (B1-B4) below. ∑n (B1) Faithfulness: C(ρ; P ) ≥ 0 with equality if ρ ∈ IB(H) = { PiρPi|ρ ∈ S(H)}. (3) IB(H). i=1 (B2) Monotonicity: C(ϕBI(ρ); P ) ≤ C(ρ; P ) for any ∈ C A quantum channel is a completely positive and ϕBI BI(H). ∑ ≤ preserving (CPTP) linear map of quantum states [27]. (B3) Strong monotonicity: l plC(ρl; P ) C(ρ; P ) for any block incoherent decomposition ϕ = {K } ∈ A quantum channel ϕ is often∑ expressed by the Kraus BI l l † C † † operators {K } satisfying K K = I , where I is BI(H) of ϕBI, pl = tr(K∑lρKl ), ρl = K∑lρKl /pl. l l l l l m m ≤ the identity operator on H and † stands for the adjoint. A (B4) Convexity: C( j pjρj; P ) j pjC(ρj; P ) for quantum channel ϕ is called block incoherent if it admits any states {ρj}j and any probability distribution {pj}j. an expression of Kraus operators ϕ = {Kl}l such that This framework coincides with the one in the standard { }n coherence theory [1] if all Pi i=1 are rank-1. Note that † ∀ ∀ ̸ PiKlρKl Pj = 0, l, i = j (4) (B3) and (B4) together imply (B2). The framework of the standard coherence theory [1] for any ρ ∈ IB(H). Such an expression ϕ = {Kl}l is had been modified by adding an additivity condition in called a block incoherent decomposition of ϕ. We denote [28]. For the block coherence theory, we add the following the set of all quantum channels on H by C(H), and the condition: set of all block incoherent quantum channels by CBI(H). (B5) Block additivity: The concept of block coherence can be properly ex- tended to the multipartite systems via the tensor prod- C(p1ρ1 ⊕ p2ρ2; P ) = p1C(ρ1; P ) + p2C(ρ2; P ), (7) uct of the Hilbert spaces of the subsystems, similar to where p1 > 0, p2 > 0, p1 + p2 = 1, ρ1, ρ2 ∈ S(H), and for the case of standard coherence theory [16]. For bipartite { } ∪ { } { } ∪ ′ any partition P = Pk1 k1 Pk2 k2 such that k1 k1 systems, let A be another quantum system associating n ′ ′ {k } = {k} , {k } ∩ {k } = ∅ and ρ P = with the m -dimensional complex Hilbert space H . Par- 2 k2 k=1 1 k1 2 k2 1 k2 ′ ′ ′ ′ ⊕n ρ2Pk1 = 0 for any k1 and k2. titioning H = i=1πi into orthogonal subspaces πi of ∑ ′ ′ ′ ′ n ′ With condition (B5), we have the following theorem, dimension dimπi = mi, m = i=1 mi, one gets a pro- which establishes an alternative framework for quantify- ′ { ′}n′ jective measurement P = Pi i=1 with each projector ing the block coherence. ′ ′ ′ ′ Pi satisfying Pi (H ) = πi. Correspondingly one has con- Theorem 1. The framework given by conditions (B1) ′ ′ ′ ′ cepts such as S(H ), IB(H ), C(H ) and CBI(H ). For the to (B4) is equivalent to the one given by the conditions ′ ′ composite Hilbert space HAA = HA ⊗ HA associating (B1), (B2) and (B5). to the bipartite system AA′, we have the projective mea- [Proof] We first prove that conditions (B1) to (B4) im- ′ ′ ′ ′ ⊗ { ⊗ } ′ AA AA surement P P = Pi Pi′ ii . A state ρ on H ply (B1), (B2) and (B5). Suppose that (B1) to (B4) are 3 ∑ fulfilled. For the state p1ρ1 ⊕ p2ρ2 as given in (B5), we = pjρj ⊗ |1⟩⟨1|. (15) construct the BI channel ϕBI = {K1,K2} with K1 = j ∑ ∑ † P , K = P . We have K (p ρ ⊕ p ρ )K = k1 k1 2 k2 k2 1 1 1 2 2 1 ⊕ † Similarly, (B2), (B5), (14) and (15) together imply (B4). p1ρ1 and K2(p1ρ1 p2ρ2)K2 = p2ρ2. Then from (B3) we get We have provided an alternative framework for block coherence by proving that the conditions (B1) to (B4) C(p1ρ1 ⊕ p2ρ2; P ) ≥ p1C(ρ1; P ) + p2C(ρ2; P ). (8) are equivalent to the conditions (B1), (B2) and (B5). On the other hand, since p1ρ1 ⊕p2ρ2 = p1ρ1 +p2ρ2, from The similar condition (B5) in the standard coherence has (B4) we get particular advantages in calculating coherence of block diagonal states [29]. The condition (B5) in the block ⊕ ≤ C(p1ρ1 p2ρ2; P ) p1C(ρ1; P ) + p2C(ρ2; P ). (9) coherence may also simplify the calculations of the block Combining (8) and (9) we get the condition (B5). coherence for certain block diagonal states. Next we prove that (B1), (B2) and (B5) imply (B1) to (B4). Suppose conditions (B1), (B2) and (B5) are sat- ′ C. Several block coherence measures { }n ∈ C isfied. Let Kl l=1 BI(H) be a BI decomposition as- sociated to the system A. Consider the bipartite system AA′ with the aforementioned notation and ρ ∈ S(H). Under the framework of block coherence above, we ′ now provide several block coherence measures. Denote Let the state ρAA = ρ ⊗ |1⟩⟨1| undergo a BI channel P = {P }n a projective measurement on the Hilbert such that i i=1 ∑ space H. The following Propositions 1-5 provide block AA′ AA′ ⊗ ⊗ | ⟩⟨ | † ⊗ † coherence measures, see the detailed proofs in Appendix. ϕBI (ρ ) = (Kl Ul)(ρ 1 1 )(Kl Ul ) l Proposition 1. l1 norm of coherence ∑ ∑ † ⊗ | ⟩⟨ | = KlρKl l l , (10) || || Cl1 (ρ, P ) = PiρPj tr (16) l i≠ j where √ || || † ′ is a block coherence measure, where M tr =tr M M ∑n denotes the trace norm of the M. | − ⟩⟨ | Ul = k + l 1 k Proposition 2. For α ∈ (0, 1)∪(1, 2], coherence based k=1 on Tsallis relative entropy ′ are the unitary operators on A . From (B5), (10) gives 1 ∑ rise to C (ρ, P ) = { tr[(P ραP )1/α] − 1} (17) T,α α − 1 i i ∑ ∑ i † ⊗ | ⟩⟨ | ⊗ ′ C( KlρKl l l ; P P ) = plC(ρl; P ), (11) l l is a block coherence measure. ′ In particular, we have where P and P are rank-1 projective measurements, pl = † † Corollary 1. tr(KlρKl ), ρl = KlρKl /pl, and we have used lim CT,α(ρ, P ) = (ln 2)Crel(ρ, P ), (18) ′ α→1 C(ρl ⊗ |l⟩⟨l|; P ⊗ P ) = C(ρl; P ). (12) According to (B2), (10) and (11) together imply (B3). where ∑ Now consider the following state − Crel(ρ, P ) = tr(ρ log2 ρ) tr[(PiρPi) log2(PiρPi)], ∑n′ i AA′ ρ = plρl ⊗ |l⟩⟨l|, (13) (19) l=1 and ln is the natural logarithm. { }n′ { }n′ ⊂ with pl l=1 a probability distribution and ρl l=1 Proposition 3. Modified trace norm of coherence S {| ⟩}n′ ′ (H), l l=1 orthonormal basis of H . According to Ctr(ρ, P ) = min ||ρ − λσ||tr (20) (B5), we have λ>0,σ∈I (H) ∑ ∑ B ′ C( plρl ⊗ |l⟩⟨l|; P ⊗ P ) = plC(ρl; P ). (14) is a block coherence measure. l l Proposition 4. Coherence weight

AA′ Let ρ undergo a BI channel as Cw(ρ, P ) ′ { ≥ | − ∈ I ∈ S } ∑n = min s 0 ρ = (1 s)σ + sτ, σ B(H), τ (H) ′ ′ ′ σ,τ AA AA A ⊗ | ⟩⟨ | AA A ⊗ | ⟩⟨ | ϕBI (ρ ) = (I 1 k )ρ (I k 1 ) = min{s ≥ 0|ρ ≥ (1 − s)σ, σ ∈ IB(H)} (21) k=1 σ 4

† is a block coherence measure. Pi = V P iV, (28) Proposition 5. For α ∈ [ 1 , 1), coherence based on 2 { }n sandwiched R´enyi relative entropy with Aij ij=1 satisfying

1−α 1−α 1 α − ∑n CR,α(ρ, P ) = 1 − max ({tr[(ρ 2α σρ 2α ) ]} 1 α ) † ∈I σ B(H) AijAik = δjkId, (22) i=1 ∑n † is a block coherence measure. AikAjk = δijId, k=1 When P is a rank-1 projective measurement, Cl1 (ρ, P ) recovers the standard coherence measure Cl1 (ρ) proposed Ai1 = Ai. in Ref. [1], CT,α(ρ, P ) recovers the standard coherence ∈ C measure proposed in Ref. [9, 13, 30], Ctr(ρ, P ) recovers A channel ϕ (H) is called a POVM incoherent (PI) the standard coherence measure proposed in Ref. [28], channel if [20] ϕ allows∑ a Kraus operator decomposition { } † Cw(ρ, P ) recovers the standard coherence measure Cw(ρ) ϕ = Kl l with l Kl Kl = Id and there exists a BI ′ { ′} ∈ C proposed in Ref. [31], CR,α(ρ, P ) recovers the standard channel ϕ = Kl l BI(Hε) with respect to a canonical { }n coherence measure proposed in Ref. [14]. In particular, Naimark extension P = Pi i=1 such that when α = 1 , 2 † ′ ′† √ KlρK ⊗ |1⟩⟨1| = K (ρ ⊗ |1⟩⟨1|)K , ∀l, (29) √ √ l l l − 2 CR, 1 (ρ, P ) = 1 max (tr ρσ ρ) (23) ′ ′ 2 ∈I { } σ B(H) where Kl l is a BI decomposition of ϕ . For such case we call {Kl}l a PI decomposition of ϕ. recovers the standard coherence measure proposed in Ref. We denote the set of all PI states as IP(H), and the [32] when P is a rank-1 projective measurement. set of all PI channels as CPI(H). Note that IP(H) may be empty for some POVMs. Note also that such definition of PI operation does not depend on the choice of Naimark III. COHERENCE MEASURES WITH RESPECT extension [20], TO GENERAL QUANTUM MEASUREMENTS A coherence measure for states in Hilbert space H with { }n respect to a general quantum measurement E = Ei i=1 We study now the coherence measures with respec- should satisfy the following conditions (P1)-(P4) [20]: t to general quantum measurements [20]. A general (P1) Faithfulness: C(ρ, E) ≥ 0, with equality if ρ ∈ measurement or a POVM on d-dimensional Hilbert s- IP(H). pace H is given by a set of positive semidefinite op- ∑ (P2) Monotonicity: C(ϕPI(ρ),E) ≤ C(ρ, E), ∀ϕPI ∈ erators E = {E }n with n E = I the identity i i=1 i=1 i d CPI(H). ∑ on H. Projective measurement and rank-1 projective (P3) Strong monotonicity: plC(ρl,P ) ≤ C(ρ, P ), measurement are the special cases of POVM. Suppose l where {Kl}l is a PI decomposition of a PI channel, † { }n † † Ei = Ai Ai for any i. We also denote E = Ai i=1 with p =tr(K ρK ), ρ = K ρK /p . ∑ † l l l l l ∑l l ∑ n A A = I . Note that E = (U A )†(U A ) for any ≤ i=1 i i d i i i i i (P4) Convexity: C( j pjρj,E) j pjC(ρj,E), { }n unitary Ui i=1. {ρj}j ⊂ S(H), {pj}j a probability distribution. A state ρ is called an incoherent state with respect to Note that the definitions of PI states and PI channels E if [18] and the conditions (P1)-(P4) all include the projective measurements and the rank-1 projective measurements E ρE = 0, ∀i ≠ j. (24) i j as special cases [20]. We emphasize that the framework of { }n Note that this is equivalent to [18] POVM coherence measure is about POVM E = Ei i=1. Hence, any valid coherence measure in terms of {Ai}i † AiρA = 0, ∀i ≠ j. (25) should be invariant under the unitary transformation j { } → { } { }n Ai i UiAi i for any unitary Ui i=1 [20]. The POVM incoherent channel is defined via the An efficient scheme for constructing POVM coherence canonical Naimark extension [20]. For POVM E = {Ei = measures is as follows [18, 20] † }n Ai Ai i=1 on d-dimensional Hilbert space H, introduce {| ⟩}n C(ρ, E) = C(ε(ρ), P ), (30) an n-dimensional Hilbert space HR with i i=1 an or- thonormal basis of HR. A canonical Naimark extension { }n { }n where P = Pi i=1 of E = Ei i=1 is described by a unitary matrix V on H = H ⊗ H as [20] ∑n ε R † ε(ρ) = AiρA ⊗ |i⟩⟨j|, (31) ∑n j ij=1 V = Aij ⊗ |i⟩⟨j|, (26) ij=1 It can be checked that if C(ρε, P ) is a unitarily invari- { ⊗ | ⟩⟨ |}n P = P i = Id i i i=1, (27) ant block coherence measure satisfying conditions (B1) to 5

(B4), then C(ρ, E) defined above is a POVM coherence Cw(ρ, E) and CR,α(ρ, E) take the forms of Eqs. (33), measure satisfying conditions (P1) to (P4) [20]. Here ρε (34), (37), (38) and (39) under Eq. (30), respectively. ⊗ is any state on Hε = H HR. The unitary invariance (1). We prove that Cl1 (ρε, P ) is unitarily invariant. means that For any unitary U on Hε, we have

C(ρ , P ) = C(Uρ U †,UPU †) (32) C (Uρ U †,UPU †) ε ε ∑l1 ε † † † = ||UP iU UρεU UP jU ||tr for any unitary transformation U on Hε. Employing this ̸ scheme and using Propositions 1 to 5, we obtain the fol- ∑i=j lowing Theorem. || || = P iρεP j tr = Cl1 (ρε, P ), { † }n Theorem 2. Let E = Ei = Ai Ai i=1 be a POVM i≠ j on the Hilbert space H. The following quantities given in (1)-(5) are all POVM coherence measures with respect where we have used the fact that the trace norm is uni- to E. tarily invariant. It is easy to see that Cl1 (ρ, E) have the (1). l1 norm of coherence form of Eq. (33). ∑ (2). It is easy to see that CT,α(ρε, P ) is unitarily in- † || || variant. Now we prove that CT,α(ρ, E) has the form of Cl1 (ρ, E) = AiρAj tr. (33) i≠ j Eq. (34) under Eq. (30). For the unitary transformation V defined in Eq. (26), (2). For α ∈ (0, 1) ∪ (1, 2], coherence based on Tsallis ∑ ⊗ | ⟩⟨ | † † ⊗ | ⟩⟨ | relative entropy εV (ρ) = V (ρ 1 1 )V = AiρAj i j = ε(ρ). ∑ ij 1 † C (ρ, E) = { tr[(A ραA )1/α] − 1}, (34) T,α α − 1 i i As a result, i α 1/α and tr[(P i(εV (ρ)) P i) ] α ⊗ | ⟩⟨ | † 1/α = tr[(P iV (ρ 1 1 )Vi P ) ] lim CT,α(ρ, E) = (ln 2)Crel(ρ, E), (35) ∑ α→1 α † ⊗ | ⟩⟨ | 1/α = tr[(P i( Ajρ Ak j k )P i) ] where jk ∑ † 1/α † † = tr[(AiρA ⊗ |i⟩⟨i|) ] C (ρ, E) = tr(ρ log ρ) − tr[(A ρA ) log (A ρA )]. i rel 2 i i 2 i i † 1/α i = tr[(AiρAi ) ]. (36) Hence, CT,α(ρ, E) has the form of Eq. (34). Eq. (35) (3). Modified trace norm of coherence can be proved as Corollary 1. (3). It is easy to see that Ctr(ρ, E) has the form of Eq. Ctr(ρ, E) = min ||ε(ρ) − λσ||tr. (37) (37). Now we show that Ctr(ρε, P ) is unitarily invariant. ∈I λ>0,σ B(Hε) Note that (4). Coherence weight ∑n Ctr(ρε, P ) = min ||ρε − λ P iσP i||tr, λ>0,σ Cw(ρ, E) = min {s ≥ 0|ε(ρ) ≥ (1 − s)σ}. (38) i=1 σ∈IB(Hε) where σ is any density operator on Hε. ∈ 1 (5). For α [ 2 , 1), coherence based on sandwiched For any unitary U on Hε, we have R´enyi relative entropy † † Ctr(UρεU ,UPU ) CR,α(ρ, E) ∑n † † † 1−α 1−α 1 || − || α − = min UρεU λ UP iU σUP iU tr = 1 − max {tr[(ε(ρ 2α )σε(ρ 2α )) ]} 1 α . (39) λ>0,σ σ∈IB(Hε) i=1 ∑n † [Proof]. To prove the results of the Theorem 2, we need = min ||ρε − λ P iU σUP i||tr {| ⟩}n λ>0,σ to use the results of the Propositions 1 to 5. Let i i=1 i=1 be an orthonormal basis for the Hilbert space HR, and P ∑n and ε(ρ) be defined in Eqs. (27) and (31), respectively. = min ||ρε − λ P iσP i||tr λ>0,σ Since C(ρ, E) is a POVM coherence measure satisfying i=1 conditions (P1) to (P4) if C(ρε, P ) is a unitarily invari- = Ctr(ρε, P ), ant block coherence measure satisfying conditions (B1) to (B4), we only need to prove the unitary invariance E- where we have used the facts that trace norm is unitarily { ∈ S } { † ∈ S } q. (32) and show that Cl1 (ρ, E), CT,α(ρ, E), Ctr(ρ, E), invariant and σ : σ (H) = U σU : σ (H) . 6

(4). It is easy to see that Cw(ρ, E) has the form of Eq. The coherence with respect to POVM measurements has (38). Next we show that Cw(ρε, P ) is unitarily invariant. operational significance. Similar to the robustness of co- Note that herence and the maximum relative entropy of coherence which characterize success probability of subchannel dis- ∑n crimination [4, 8], it would be also an interesting issue Cw(ρε, P ) = min{s ≥ 0|ρε ≥ (1 − s) P iσP i}, σ to explore the operational meanings of the POVM co- i=1 herence measures given in Theorem 2. Our results may where σ is any density operator on Hε. highlight further investigations on the coherence of quan- For any unitary U on Hε, we have tum states and the applications in quantum information processing. † † Cw(UρεU ,UPU ) ∑n † † † = min{s ≥ 0|UρεU ≥ (1 − s) UP iU σUP iU } ACKNOWLEDGMENTS σ i=1 ∑n † This work was supported by the National Science = min{s ≥ 0|ρε ≥ (1 − s) PU σUP i} σ Foundation of China (Grant No. 11675113), the Key i=1 ∑n Project of Beijing Municipal Commission of Education (Grant No. KZ201810028042), Beijing Natural Science = min{s ≥ 0|ρε ≥ (1 − s) P iσP i} σ Foundation (Grant No. Z190005), Young Talent fund i=1 of University Association for Science and Technology in = Cw(ρε, P ), Shaanxi (Grant No. 20190111), and Academy for Multi- disciplinary Studies, Capital Normal University. which completes the proof. (5). It is easy to see that CR,α(ρ, E) has the form of Eq. (39). Similarly to the proof of (3), one can show that Cw(ρε, P ) is unitarily invariant. APPENDIX We remark that the coherence measure Cl1 (ρ, P ) was proposed in [23]. In [20] the authors conjectured that

Cl1 (ρ, E) is a well defined POVM coherence measure sat- A. Proof of Proposition 1 isfying the conditions (P1)-(P4). Combining with our re- sult of proposition 1, we have strictly proved in Theorem From the definition of BI state and the properties of 2 that Cl1 (ρ, E) is indeed a well defined POVM coherence trace norm, Cl1 (ρ, P ) satisfies the condition (B1). It sat- measure. isfies the conditions (B4) and (B5) due to the properties of trace norm. Since (B3) and (B4) imply (B2), we only need to prove that Cl (ρ, P ) fulfills (B3). IV. SUMMARY 1 For any BI channel ϕ with BI decomposition ϕBI = ∑ † {Kl}l, K Kl = Id, each Kl has the form [20], We have established an alternative framework for l l quantifying the coherence with respect to projective ∑n measurements, and provided several coherence measures Kl = Pfl(i)MlPi, (A1) with respect to projective measurements. We then ob- i=1 tained several coherence measures with respect to gen- { }n eral POVM measurements, from which a conjecture has where fl(i) is a function on i i=1,Ml is a matrix on H. † † been verified concerning the coherence measure Cl1 (ρ, E). Denote pl =tr(KlρKl ), ρl = KlρKl /pl. We have

∑ ∑ || † || plCl1 (ρl,P ) = PiKlρKl Pj tr ̸ ∑l ∑ l,i=j † || ′ ′ || = PiKl Pi ρPj Kl Pj tr (A2) ̸ ′̸ ′ l,i=∑j i =j † ≤ || ′ ′ || PiKlPi ρPj Kl Pj tr ′̸ ′ l,ij,i∑=j † || ′ ′ ′ ′ || = Pfl(i )KlPi ρPj Kl Pfl(j ) tr (A3) l,i′≠ j′ 7 ∑ ∑ † || ′ ′ ′ | ′ ′ ⟩⟨ ′ ′ | ′ || = Pfl(i )Kl si j k ψi j k ψi j k Kl Pfl(j ) tr (A4) ′̸ ′ l,i∑=j k † ≤ ′ ′ || ′ | ′ ′ ⟩⟨ ′ ′ | ′ || si j k Pfl(i )Kl ψi j k ψi j k Kl Pfl(j ) tr lk,i′≠ j′ ∑ ∑ √ † † ′ ′ ⟨ ′ ′ | ′ | ′ ′ ⟩⟨ ′ ′ | ′ | ′ ′ ⟩ = si j k ψi j k Kl Pfl(i )Kl ψi j k ψi j k Kl Pfl(j )Kl ψi j k (A5) k,i′≠ j′ l ∑ √∑ √∑ † † ≤ s ′ ′ ⟨ψ ′ ′ |K P ′ K |ψ ′ ′ ⟩ ⟨ψ ′ ′ |K ′ P ′ K ′ |ψ ′ ′ ⟩ (A6) i j k i j k l fl(i ) l i j k i j k l fl′ (j ) l i j k k,i′≠ j′ l l′ ∑ √ ∑ √ ∑ † † = s ′ ′ ⟨ψ ′ ′ | K P ′ K |ψ ′ ′ ⟩ ⟨ψ ′ ′ | K ′ P ′ K ′ |ψ ′ ′ ⟩ i j k i j k l fl(i ) l i j k i j k l fl′ (j ) l i j k k,i′≠ j′ l l′ ∑ √ √ ≤ ′ ′ ⟨ ′ ′ | | ′ ′ ⟩ ⟨ | | ⟩ si j k ψi j k Im ψi j k ψi′j′k Im ψi′j′k (A7) ′̸ ′ k,i∑=j ∑ ′ ′ || ′ ′ || = si j k = Pi ρPj tr = Cl1 (ρ, P ). k,i′≠ j′ i′≠ j′

In Eq. (A2) we have used the property that {Kl}l is a We now prove that ∑ † ′ ′ ′ BI decomposition, that is, PiKl( i Pi ρPi )Kl Pj = 0 ∑ 1 { α α }α − for any i ≠ j. In Eq. (A3) we have used PiKlPi′ = i tr[(Piρ Pi) ] 1 DT,α(ρ) = . (A12) ′ ′ ′ ′ ′ − PiPfl(i )KlPi = δi,fl(i )Pfl(i )KlPi which is a result of α 1 Eq. (A1). In Eq. (A4) we∑ have used the singular val- To go ahead, we need the lemmas below. ue decomposition, Pi′ ρPj′ = si′j′k|ψi′j′k⟩⟨ψ ′ ′ | with k i j k Lemma 1. H¨olderinequality. {s ′ ′ } the singular values, {|ψ ′ ′ ⟩} ({|ψ ′ ′ ⟩} ) a set i j k k i j k k i j k k Suppose {a }n , {b }n , are all positive real numbers, of orthonormal vectors. In Eq. (A5) we have taken into i i=1 i i=1 √ then ||| ⟩⟨ ||| ⟨ | ⟩⟨ | ⟩ account the fact that ψ φ tr = ψ ψ φ φ for any ∈ | ⟩ | ⟩ 1) when α (0, 1), pure states ψ and φ . In∑ Eq.√ (A6) we√∑ have√ used∑ the ≤ ′ n n n Cauchy-Schwarz inequality albl al ′ bl ∑ ∑ 1 ∑ 1 l l l − − ≤ α α 1 α 1 α with al ≥ 0 and bl ≥ 0. In Eq. (A7) we have used the aibi ( ai ) ( bi ) , (A13) ∑ † ′ ≤ ′ ≤ i=1 i=1 i=1 fact that l Kl Pfl(i )Kl Im since Pfl(i ) Im and ∑ † K Kl = Im. 1 1 1 1 l l α 1−α α 1−α and the equality holds if and only if ai /bi = aj /bj for any i, j; B. Proof of Proposition 2 2) when α > 1,

n n n ∑ ∑ 1 ∑ 1 − − For α > 0, the quantum Tsallis relative entropy is ≥ α α 1 α 1 α aibi ( ai ) ( bi ) , (A14) defined as i=1 i=1 i=1 tr(ρασ1−α) − 1 || ∈ S 1 1 1 1 DT,α(ρ σ) = , ρ, σ (H), α 1−α α 1−α α − 1 and the equality holds if and only if ai /bi = aj /bj supp(ρ) ⊂ supp(σ) when α ≥ 1, (A8) for any i, j. Lemma 2 (Ref. [34]). For r × r positive semidefinite where supp(ρ) = {|ψ⟩|ρ|ψ⟩ ̸= 0} is the support of ρ. matrices M and N, it holds that It is shown that for α > 0 [33], ∑r ∑r || ≥ || ⇔ ↓ ↓ ≤ ≤ ↓ ↓ DT,α(ρ σ) 0,DT,α(ρ σ) = 0 ρ = σ. (A9) λr+1−j(M)λj (N) tr(MN) λj (M)λj (N), j=1 j=1 Also, Dα(ρ||σ) is monotonic under CPTP maps for α ∈ (0; 2] [33], (A15) { ↓ } DT,α(ϕ(ρ)||ϕ(σ)) ≤ DT,α(ρ||σ). (A10) where λj (M) j are the eigenvalues of M in decreasing order. Define Now for α ∈ (0, 1) and σ ∈ IB(H), we have

DT,α(ρ) = min DT,α(ρ||σ). (A11) α 1−α σ∈IB(H) tr(ρ σ ) 8

∑n 1 { α α }α α 1−α = tr[(Piρ Pi) ] . (A20) = tr[ρ (PiσPi) ] i=1 In above derivation, we have used Lemma 1 and Lemma ∑n ∑n 1−α α 1−α α 1−α 1 α 2. Again, when σ takes the value in Eq. (A18), Eq. = q tr(ρ σ ) ≤ { [tr(ρ σ )] α } , i i i (A11) achieves Eq. (A12). As a result we get Eq. (A12). i=1 i=1 From Eqs. (A9) and (A11) we see that DT,α(ρ) ≥ 0 (A16) and DT,α(ρ) = 0 if and only if ρ ∈ IB(H). Then from Eq. (A12) we have where qi =tr(PiσPi), σi = PiσPi/qi, the H¨olderin- equality has been used, and the equality holds if and ∑ α 1 α ≥ { tr[(Piρ Pi) α ]} − 1 only if there exists constant C 0 such that qi = i ≥ 0, α 1−α 1 − C[tr(ρ σi )] α for any i. Furthermore, α 1 α 1−α namely, tr(ρ σi ) α 1−α ∑ = tr(ρ P σ P ) 1 i i i α α − i tr[(Piρ Pi) ] 1 ∑mi ≥ 0, ≤ ↓ α ↓ 1−α α − 1 λj (Piρ Pi)λj (σi ) j=1 with the equality holding if and only if ρ ∈ IB(H), which ∑mi ↓ ↓ proves that CT,α(ρ, P ) satisfies (B1). = λ (P ραP )(λ (σ ))1−α j i i j i For any ϕBI ∈ CBI(H), from Eqs. (A10) and (A11) we j=1 have ∑mi ∑mi ↓ 1 ↓ − 1 − ≤ { α α }α{ 1 α 1−α }1 α ∗ [λj (Piρ Pi)] [(λj (σi)) ] DT,α(ρ) = min DT,α(ρ||σ) = DT,α(ρ||σ ) ∈I j=1 j=1 σ B(H) ∗ α 1 α ≥ DT,α(ϕBI(ρ)||ϕBI(σ )) = {tr[(Piρ Pi) α ]} , (A17) ≥ min DT,α(ϕBI(ρ)||σ) = DT,α(ϕBI(ρ)), (A21) ∈I where the Lemma 1 and Lemma 2 have been used. It is σ B(H) easy to check that when ∗ where σ ∈ I (H) such that min ∈I D (ρ||σ) = ∑ B σ B(H) T,α n α 1 ∗ (P ρ P ) α DT,α(ρ||σ ). σ = ∑ i=1 i i (A18) n α 1 From Eq. (A12), Eq. (A21) is equivalent to i=1 tr[(Piρ Pi) α ] ∑ 1 { α α }α − Eq. (A11) achieves Eq. (A12). As a result we get Eq. i tr[(Piρ Pi) ] 1 (A12). − ∑ α 1 For α > 1, we have α α 1 α { tr[(P (ϕ (ρ)) P ) α ]} − 1 ≤ i i BI i , tr(ρασ1−α) α − 1 ∑ α 1−α = tr[ρ (PiσPi) ] which is further equivalent to i ∑ ∑ 1 − − α 1 α α 1 α tr[(Piρ Pi) α ] − 1 = qi tr(ρ σi ) i α − 1 ∑i ∑ 1 1 α 1−α α α α α ≥ { α } tr[(P (ϕBI(ρ)) Pi) ] − 1 [tr(ρ σi )] , (A19) ≤ i i . i α − 1 and the equality holds if and only if there exists a con- We then proved that C (ρ, P ) satisfies (B2). 1−α 1 T,α ≥ α α stant C 0 such that qi = C[tr(ρ σi )] for any i. Now we prove that CT,α(ρ, P ) also satisfies (B5). Sup- Moreover, pose ρ = p1ρ1 ⊕ p2ρ2 as described in (B5). Then α 1−α tr(ρ σi ) ∑n − α 1 α 1 α tr[(P ρ P ) α ] = tr(ρ Piσi Pi) i i ∑mi i=1 ↓ ↓ − ∑ ∑ α 1 α α 1 α 1 ≥ λ (Piρ Pi)λ − (σ ) α α j mi+1 j i = p1 tr[(Pk1 ρ1 Pk1 ) ] + p2 tr[(Pk2 ρ2 Pk2 ) ] j=1 k1 k2 ∑mi ∑n ∑n ↓ α ↓ 1−α 1 1 = λ (P ρ P )(λ (σ )) α α α α j i i mi+1−j i = p1 tr[(Piρ1 Pi) ] + p2 tr[(Piρ2 Pi) ]. (A22) j=1 i=1 i=1 ∑mi ∑mi ↓ α 1 α ↓ 1−α 1 1−α ≥ { [λ (P ρ P )] α } { [(λ (σ )) ] 1−α } Substituting (A22) into Eq. (17), we then proved that j i i mi+1−j i j=1 j=1 CT,α(ρ, P ) satisfies (B5). 9

C. Proof of Corollary 1 = p1C(ρ1) + p2C(ρ2,P ),

where we have used the facts that σ1, σ2 ∈ S(H), {q1, q2} Set α = 1 + ε. Consider the Taylor expansions around λq1 λq2 is a probability distribution, λ1 = and λ2 = . ε = 0, p1 p2

M 1+ε = M + εM ln M + o(ε2), E. Proof of Proposition 4 ln(M + εN) = ln M + o(ε),

1 2 = 1 − ε + o(ε ), It can be proved that Cw(ρ, P ) fulfills the condition- 1 + ε s (B1), (B3) and (B4) by using a similar way adopted where M, N are Hermitian matrices, o(ε) denotes the in Ref. [31]. Here we equivalently prove that Cw(ρ, P ) infinitesimal term with the order ε or higher around ε = fulfills (B1), (B2) and (B5). (B1) is evidently satisfied. α 2 To prove (B2), suppose {Kl}l ∈ CBI(H) with {Kl}l a BI 0. We have Piρ Pi = Pi(ρ+ερ ln ρ+o(ε ))Pi. Therefore, decomposition. Then there exists σ ∈ IB(H) such that 1 α α tr[(Piρ Pi) ] ρ ≥ [1 − Cw(ρ, P )]σ, 2 ∑ ∑ = tr[(P ραP )1−ε+o(ε )] † ≥ − † i i KlρKl [1 Cw(ρ, P )] KlσKl . α α α 2 = tr[(Piρ Pi) − ε(Piρ Pi) ln(Piρ Pi) + o(ε )] l l ∑ † = tr[PiρPi + εPi(ρ ln ρ)Pi − ε(PiρPi) ln(PiρPi) ∈ I Since∑ l KlσKl B(H), we obtain +o(ε2)]. † ≤ Cw( l KlρKl ,P ) Cw(ρ, P ), which proves that (B2) is satisfied. Applying the L’Hospital’s rule to Eq. (17), we have To prove (B5), let us consider again ρ = p1ρ1 ⊕ p2ρ2 as described in (B5). Then there exists σ ∈ IB(H) such lim CT,α(ρ, P ) α→1 that ∑ d α 1/α ≥ − = lim tr[(Piρ Pi) ] ρ [1 Cw(ρ, P )]σ, α→1 dα ∑ ∑ i ≥ − ∑ Pk1 ρPk1 [1 Cw(ρ, P )] Pk1 σPk1 , = tr[P (ρ ln ρ)P − (P ρP ) ln(P ρP )] k k i i i i i i ∑1 ∑1 i ∑ Pk ρPk ≥ [1 − Cw(ρ, P )] Pk σPk . − 2 2 2 2 = tr(ρ ln ρ) tr[(PiρPi) ln(PiρPi)] k2 k2 i ∑ ∑ Denote k Pk1 σPk1 = q1σ1, k Pk1 σPk1 = q2σ2, with = (ln 2)Crel(ρ, P ). 1 1 {∑q1, q2} a probability distribution,∑ σ1, σ2 ∈ IB(H). Since P ρP = p ρ , P ρP = p ρ , we have k1 k1 k1 1 1 k2 k2 k2 2 2 D. Proof of Proposition 3 [1 − Cw(ρ, P )]q1 ρ1 ≥ σ1, p1 Obviously, the condition (B1) is satisfied. (B2) is al- − || || ≥ [1 Cw(ρ, P )]q2 so satisfied as a consequence of the fact that M tr ρ2 ≥ σ2, p2 ||ϕ(M)||tr for any CPTP map ϕ and any Hermitian ma- − trix M [35]. Concerning (B5), we consider ρ = p1ρ1 ⊕ [1 Cw(ρ, P )]q1 Cw(ρ1,P ) ≤ 1 − , p2ρ2 as described in (B5). Any σ ∈ IBI(H) can be writ- p1 ten as [1 − Cw(ρ, P )]q2 Cw(ρ2,P ) ≤ 1 − , p2 σ = q1σ1 ⊕ q2σ2, (A23) p1Cw(ρ1,P ) + p2Cw(ρ2,P ) ≤ Cw(ρ, P ). (A24) ≥ ≥ ∈ S with q1 0, q2 0, q1 + q2 = 1, and σ1, σ2 (H), ′ ′ ∈ I Conversely, there exist σ1, σ2 B(H) such that σ1Pk = σ2Pk = 0 for any k1 and k2. It follows that 2 1 ′ ρ1 ≥ [1 − Cw(ρ1,P )]σ , ⊕ 1 C(p1ρ1 p2ρ2,P ) ≥ − ′ ρ2 [1 Cw(ρ2,P )]σ2. = min ||p1ρ1 ⊕ p2ρ2 − λ(q1σ1 ⊕ q2σ2)||tr λ>0,q1,σ1,σ2 It follows that λq1 λq2 = min (p1||ρ1 − σ1||tr + p2||ρ2 − σ2||tr) p1ρ1 ⊕ p2ρ2 λ>0,q1,σ1,σ2 p p 1 2 ≥ − ′ − ′ p1[1 Cw(ρ1,P )]σ1 + p2[1 Cw(ρ2,P )]σ2, || − λq1 || = p1 min ρ1 σ1 tr Cw(ρ, P ) ≤ p1Cw(ρ1,P ) + p2Cw(ρ2,P ). (A25) λ1>0,σ1 p1 λq2 Eqs. (A24) and (A25) imply (B5), which completes the +p2 min ||ρ2 − σ2||tr λ2>0,σ2 p2 proof. 10

1−α 1−α 1 ∗ α − F. Proof of Proposition 5 ≤ {tr[(ϕBI(ρ)) 2α ϕBI(σ )(ϕBI(ρ)) 2α ) ]} 1 α 1−α 1−α 1 α − ≤ max {tr[(ϕBI(ρ)) 2α σ(ϕBI(ρ)) 2α ) ]} 1 α . ∈I This proof is a generalization of the proof for the The- σ B(H) ∈ 1 ∈ S orem 1 in Ref. [14]. For α [ 2 , 1), σ, ρ (H), the sandwiched R´enyi relative entropy is defined as [36, 37], This proves that CR,α(ρ, P ) satisfies (C2). Next we prove CR,α(ρ, P ) satisfies (C5). Consider ρ = 1−α 1−α α ln tr[(ρ 2α σρ 2α ) ] p1ρ1 ⊕ p2ρ2 as described in (B5). As any σ ∈ IBI(H) can F (σ||ρ) = . α α − 1 be written as Eq. (A23), it follows that

∈ 1 || ≥ 1−α 1−α α It is shown that [37, 38] for α [ 2 , 1),Fα(σ ρ) 0, max tr[(ρ 2α σρ 2α ) ] ∈I where the equality holds if and only if σ = ρ. This is σ B(H) − − − 1 α 1 α equivalent to that { 1 α α 2α 2α α = max (p1 q1 ) max tr[(ρ1 σ1ρ1 ) ] q1,q2 σ1 1−α 1−α α − − tr[(ρ 2α σρ 2α ) ] ≤ 1, − 1 α 1 α 1 α α 2α 2α α } +(p2 q2 ) max tr[(ρ2 σ2ρ2 ) ] σ2 and to that { 1−α α 1−α α } = max p1 q1 t1 + p2 q2 t2 1−α 1−α 1 q1,q2 α − {tr[(ρ 2α σρ 2α ) ]} 1 α ≤ 1, 1 1 1−α 1−α −1 α−1 −1 α−1 1−α = p1 p2 t1t2(p1 t1 + p2 t2 ) , with the equality holding if and only if σ = ρ. This says where that C (ρ, P ) satisfies (C1). R,α 1−α 1−α ∈ 1 2α 2α α For α [ 2 , 1), it has been shown that [37, 39] for t1 = max tr[(ρ1 σ1ρ1 ) , σ, ρ ∈ S(H), and any CPTP map ϕ, σ1 1−α 1−α 2α 2α α t2 = max tr[(ρ2 σ2ρ2 ) ], Fα(ϕ(σ)||ϕ(ρ)) ≤ Fα(σ||ρ). σ2

This implies and the Lemma 1 (note here t1 > 0 and t2 > 0) has been taken into account. 1−α 1−α α tr[(ϕ(ρ)) 2α ϕ(σ)(ϕ(ρ)) 2α ) ] Consequently, 1−α 1−α 2α 2α α ≥ tr[(ρ σρ ) ], 1−α 1−α 1 2α 2α α 1−α − − max ({tr[(ρ σρ ) ]} ) 1 α 1 α α 1 ∈I {tr[(ϕ(ρ)) 2α ϕ(σ)(ϕ(ρ)) 2α ) ]} 1−α σ B(H) − − 1 1 α 1 α 1 1−α 1−α − α − α 1 α ≥ {tr[(ρ 2α σρ 2α ) ]} 1 α . = { max tr[(ρ 2α σρ 2α ) ]} σ∈IB(H) ∗ ∈ I 1 1 1 1 For any BI map ϕBI, there exists σ B(H) such that 1−α 1−α −1 α−1 −1 α−1 = p1p2t1 t2 (p1 t1 + p2 t2 ) 1−α 1−α 1 1 1 − − { 2α 2α α } 1−α 1 α 1 α max tr[(ρ σρ ) ] = p1t1 + p2t2 . σ∈IB(H) 1−α 1−α 1 ∗ α − = {tr[(ρ 2α σ ρ 2α ) ]} 1 α This shows that CR,α(ρ, P ) satisfies (C5).

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