Project supported by the China Scholarship Council (Grant No. 201806305050).
Abstract
Coherence is a fundamental ingredient for quantum physics and a key resource for quantum information theory. Baumgratz, Cramer and Plenio established a rigorous framework (BCP framework) for quantifying coherence [Baumgratz T, Cramer M and Plenio M B Phys. Rev. Lett.113 140401 (2014)]. In the present paper, under the BCP framework we provide two classes of coherence measures based on the sandwiched Rényi relative entropy. We also prove that we cannot get a new coherence measure f(C(·)) by a function f acting on a given coherence measure C.
Quantum coherence is one of the most fundamental features of quantum physics. Recently, Baumgratz, Cramer, and Plenio established a rigorous framework (BCP framework) for quantifying coherence.[1] The BCP framework has been widely accepted and triggered rapidly growing research for quantifying coherence (recent reviews see Refs. [2–4]).
To construct a coherence measure, we first define the incoherent states and incoherent operations as in Ref. [1]. For a fixed orthonormal basis of the d-dimensional Hilbert space H, a quantum state σ on H is named to be incoherent with respect to if σ is diagonal when expressed in . We denote the set of all incoherent states by , and the set of density operators by . A quantum operation ϕ, or called a CPTP (completely positive trace preserving) map, can be expressed by a set of Kraus operators {Kn}n satisfying , where I being the identity operator on H, and operate a state ρ as . A quantum operation ϕ will be called an incoherent operation (ICPTP) ΦI if it admits that a set of Kraus operators {Kn}n and is diagonal for any n and any incoherent state σ. Notice that the definitions of incoherent state, incoherent operation, and the coherence measure all depend on the fixed orthonormal basis , we call this basis the reference basis.
The BCP framework consists of the following postulates (C1)–(C4) that any quantifier of coherence C should fulfill:
Non-negativity
MonotonicityC does not increase under the operation of any incoherent operation ΦI,
Strong monotonicity for any incoherent operation ΦI = {Kn}n with being diagonal for any n and any incoherent state σ,
ConvexityC is a convex function of the state, i.e.,
where pn > 0, ∑npn = 1, .
We call a quantifier C satisfying (C1)–(C4) together a coherence measure. Note that (C3) + (C4) implies (C2).[1]
Yu, Zhang, Xu and Tong proposed condition (C5), and showed that (C1)–(C4) is equivalent to (C1) + (C2) + (C5).[5]
Additivity on block-diagonal states
where p1 > 0, p2 > 0, p1 + p2 = 1, .
BCP framework draws strong attention and many discussions, but it is not the unique framework for quantifying coherence, and other potential candidates have been investigated (see, e.g., Refs. [6–16]).
So far, some coherence measures under the BCP framework have been found out for different applications and backgrounds, such as relative entropy of coherence,[1] the l1 norm of coherence,[1] geometric measure of coherence,[17] modified trace norm of coherence,[5] robustness of coherence,[18,19] coherence measure via quantum skew information,[20] coherence measures based on Tsallis relative entropy,[21–24] coherence weight.[25] For a coherence measure defined only for all pure states, it can be extended to mixed states via the convex roof construction.[6,12,26,27] Also, a coherence measure defined on all pure states is determined by its majorization property on the modulus square of coefficients of pure states.[7,8,28–30] Although the convex roof construction and majorization on pure states together provide a powerful way to construct coherence measures, the coherence measures obtained in such a way generally speaking are only in the form of optimization and it is difficult to obtain analytical expressions.[2]
Coherence measures have many fruitful applications in exploring the properties of coherence, such as the freezing of coherence[4,31–34] and coherence estimating based on experimental data.[35]
In this paper, we provide two classes of coherence measures based on the sandwiched Rényi relative entropy in Section 2, the geometric measure of coherence[17] is a special case of them. Also in Section 3, we discuss whether or not one can get a new coherence measure through a function of a given coherence measure.
2. Coherence measures based on sandwiched Rényi relative entropy
In this section, we propose two classes of coherence measures based on the sandwiched Rényi relative entropy.
For , it has been shown that[39] for and for any CPTP map ϕ,
This implies
For any ICPTP map ΦI, there exists such that
This proves that Cs1,α(ρ) satisfies (C2).
Next we prove Cs1,α(ρ) satisfies (C5). Suppose that ρ is block-diagonal in the reference basis ,
with p1 > 0, p2 > 0, p1 + p2 = 1; ρ1 and ρ2 are density operators. For any diagonal state σ, we can write it as
with the diagonal states σ1 and σ2 having the same rows (columns) with ρ1 and ρ2, respectively, q1 ≥ 0, q2 ≥ 0, q1 + q2 = 1.
It follows
where we have denoted
and have used the Hölder inequality in Appendix A (note that t1 > 0, t2 > 0) such that
the equality holds when
Consequently,
This shows that Cs1,α(ρ) satisfies (C5), and then we complete this proof.
We remark that when corresponds to the geometric measure of coherence.[17]
We remark that a coherence quantifier based on sandwiched Rényi relative entropy was also investigated in Refs. [7,15], but the quantifier is not a coherence measure in the sense of satisfying (C1)–(C4), i.e., under the BCP framework, we will give a proof for this assertion in Example 3.
Examples 1 and 2 show that Cs,α and Cs1,β will not be equivalent even if α = β.
3. Linearization theorem of coherence measures
Constructing a coherence measure under the BCP framework in general is not an easy task because of the stringent requirements of (C1)–(C4). (C5) is more easy to check than (C3) + (C4) in many cases such as in the proof of Theorem 1 in the present work. Different coherence measures have their advantages in different applications. An attractive idea for constructing new coherence measures is to ask whether or not there exists a function f such that f[C(ρ)] still is a coherence measure for a given coherence measure C. The answer of this question is essentially negative. In fact we have the following theorem.
Suppose the dimension of system d > 2, for the state
with p1 > 0, p2 > 0, p1 + p2 = 1; ρ1 and ρ2 are density operators, dimρ1 ≥ 1, dimρ2 ≥ 1, dimρ1 + dim ρ2 = d. Since C(ρ) is coherence measure, (C5) leads to
Similarly, f(C(ρ)) is also a coherence measure, then (C5) leads to
As a result, we obtain
Without loss of generality, suppose dimρ2 ≥ 2, then there exist ρ1 and ρ2 such that C(ρ1) = 0, C(ρ2) = μ > 0. The above equation yields
with f(μ) > 0.
Let p2μ = x, then
We complete this proof.
We make a note that Eq. (33) is a functional equation (see, e.g., Ref. [40]).
Since coherence measures C and λ C (λ > 0) have no essential difference, from Theorem 3 we can say that it is impossible to obtain a new coherence measure f(C(ρ)) by a function f.
Note that in the proof of Theorem 3 we only use C(ρ) and f(C(ρ)) satisfying (C1) + (C5) while (C2) is unnecessary.
4. Conclusions
In summary, under the BCP framework for quantifying coherence, we have proposed two classes of coherence measures based on sandwiched Rényi relative entropy. We also prove that it is essentially impossible to obtain a new coherence measure f(C(ρ)) by a function f acting on a given coherence measure C.
There are many open questions for future investigations after the present work. For example, the monotonicity of Cs1,α and Cs,α in α, the ordering of magnitude for them and other coherence measures, the operational interpretations for them, potential applications in quantum information processing, and also the counterparts for quantifying coherence of Gaussian states as performed in Refs. [41,42].
