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 measure (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 quantum information 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 Hilbert space H. Under a fixed fj igd orthonormal basis i i=1 of H, the state ρ is called in- coherent if hijρjji = 0 for any i =6 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 fj igd proposed in [23] in a lager Hilbert space. The Naimark i i=1 corresponds to a rank-1 projective measurement fj ih jgd h j j i 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 jiihijρjjihjj = 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- fj igd jective measurements, not necessarily rank-1. With this mal basis i i=1 as the coherence with respect to the fj ih jgd 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 (POVMs) 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 0 if ⊗ 0 AA0 ⊗ 0 8 0 6 0 A. Block incoherent states and block incoherent (Pi Pi0 )ρ (Pj Pj0 ) = 0; (i; i ) = (j; j ); (5) channels 0 0 0 0 where (i; i ) =6 (j; j ) means that i =6 j or i =6 j : 0 0 We denote the set of all states on HAA by S(HAA ) The block coherence theory was introduced in [23]. We 0 0 and the set of all channels on S(HAA ) by C(HAA ): A adopt the framework proposed in [20] for quantifying the 0 0 AA C AA block coherence. Consider a quantum system A associat- quantum channel 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 0 0 AA f AA g ⊕n ϕ = Kl l such that has partition H = i=1πi intoP orthogonal subspaces πi n 0 0 0 of dimension dimπi = mi, i=1 mi = m. Correspond- ⊗ 0 AA AA AA y ⊗ 0 f gn (Pi Pi0 )Kl ρ (Kl ) (Pj Pj0 ) = 0 (6) ingly, one gets a projective measurement P = Pi i=1, 0 0 with each projector satisfying Pi(H) = πi. A state ρ on for all l and (i; i ) =6 (j; j ). We denote the set of all block H is called block incoherent (BI) with respect to P if AA0 AA0 incoherent channels on C(H ) by CBI(H ) and call AA0 f AA0 g 8 6 such an expression ϕ = Kl l a block incoherent PiρPj = 0; i = j; (1) 0 decomposition of ϕAA . or Xn 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. Xn (B1) Faithfulness: C(ρ; P ) ≥ 0 with equality if ρ 2 IB(H) = f PiρPijρ 2 S(H)g: (3) IB(H): i=1 (B2) Monotonicity: C(ϕBI(ρ); P ) ≤ C(ρ; P ) for any 2 C A quantum channel is a completely positive and trace ϕBI BI(H): P ≤ preserving (CPTP) linear map of quantum states [27]. (B3) Strong monotonicity: l plC(ρl; P ) C(ρ; P ) for any block incoherent decomposition ϕ = fK g 2 A quantum channel ϕ is oftenP expressed by the Kraus BI l l y C y y operators fK g satisfying K K = I , where I is BI(H) of ϕBI, pl = tr(KPlρKl ), ρl = KPlρKl =pl. l l l l l m m ≤ the identity operator on H and y 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 fρjgj and any probability distribution fpjgj: an expression of Kraus operators ϕ = fKlgl such that This framework coincides with the one in the standard f gn coherence theory [1] if all Pi i=1 are rank-1. Note that y 8 8 6 PiKlρKl Pj = 0; l; i = j (4) (B3) and (B4) together imply (B2). The framework of the standard coherence theory [1] for any ρ 2 IB(H). Such an expression ϕ = fKlgl 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 2 S(H), and for the case of standard coherence theory [16]. For bipartite f g [ f g f g [ 0 any partition P = Pk1 k1 Pk2 k2 such that k1 k1 systems, let A be another quantum system associating n 0 0 fk g = fkg , fk g \ fk g = ? and ρ P = with the m -dimensional complex Hilbert space H . Par- 2 k2 k=1 1 k1 2 k2 1 k2 0 0 0 0 ⊕n ρ2Pk1 = 0 for any k1 and k2: titioning H = i=1πi into orthogonal subspaces πi of P 0 0 0 0 n 0 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- 0 f 0gn0 jective measurement P = Pi i=1 with each projector ing the block coherence.