In quantum theory, the uncertainty relation is a fundamental consequence of the superposition principle and the incompatible nature of observables. The uncertainty relation is a powerful tool in many quantum information tasks, such as quantum key distribution,^[1,2] quantum random number generation,^[3,4] entanglement witness,^[5] and Einstein–Podolsky–Rosen (EPR) steering.^[6,7] The original formulation of this concept was given by Heisenberg in 1927,^[8] and then further generalized by Robertson^[9] for two arbitrary observables in 1929, in which a lower bound of total variance of two observables was given. Such an uncertainty relation based on variance generally relies on the definition of observables. Deutsch^[10] and Maassen et al.^[11] proposed an uncertainty relation based on the Shannon entropy of measurement outcomes. Moreover, the majorization entropic uncertainty relation^[12] and the strong majorization entropic uncertainty relation^[13] were considered based on the Rényi entropy and Tsallis entropy of the measurement outcomes by using the majorization technique. Kurzyk et al. discussed entropic uncertainty relations in terms of the Tsallis entropies for states with a fixed amount of entanglement.^[14] Several other discussions on the uncertainty relation have been published.^[15–19]

Quantum coherence is one of the most important physical resources in quantum mechanics, which can be used in quantum optics,^[20] quantum information and quantum computation,^[21] thermodynamics,^[22,23] and low temperature thermodynamics.^[24–26] Many considerable efforts have recently been made to quantify the coherence of quantum states from the perspective of resource theory.^[27–50] The authors of Ref. [27] proposed a rigorous framework to quantify coherence. Consequently, various coherence measures have been defined based on this framework, such as l₁-norm of coherence,^[27] relative entropy of coherence (REOC),^[27] quantum coherence via skew information (SIC),^[28] and the geometric coherence (GC).^[29]

As quantum coherence is a basis-dependent notion, we may ask if coherence respects some uncertainty relations for two or more incompatible bases. A concept of quantum uncertainty relation of quantum coherence was proposed in Refs. [51]–[53], and the sum of quantum coherence based on two measurement bases was called the quantum uncertainty relation of quantum coherence. Generally, the uncertainty relation of a measurement contains classical (predictable) and quantum (unpredictable) parts that originate from classical noise and quantum effect, respectively.^[54] The quantum uncertainty relation of quantum coherence was considered as a quantum part (or genuine quantum uncertainty relation).^[52] For the qubit case, quantum uncertainty relations based on two measurement bases were considered by using REOC, l₁ norm of coherence, and coherence of formation.^[52] They gave the lower bounds of the sum of the quantum coherence for a qubit state based on two measurement bases. These bounds are related to the maximum eigenvalue of the state and the maximum overlap of the two measurement bases. However, one will want to know the following question. (i) For a fixed single qubit state and two pairs of measurement bases which have the same maximum overlap, will they give the same lower bound? In the first place of this work, we try to answer this question. We first consider quantum uncertainty relations of the quantum coherence through a different method from Ref. [52]. We will obtain some lower bounds with parameters, and their minimal bounds. Moreover, we can find that for two pairs of measurement bases which have the same maximum overlap, the quantum uncertainty relations and lower bounds with parameters are different, but the minimal bounds are the same.

Any quantum system inevitably suffers from environmental effect. Therefore, it is important to investigate how the noisy environment influences the entropic uncertainty. Dynamics of the entropic uncertainty relations under noisy channels were explored.^[55–58] Naturally, we believe that it is also important to investigate another question, i.e., (ii) how does the noisy environment influences the quantum uncertainty relation of the quantum coherence? In the next place of the paper, we will consider dynamics of quantum uncertainty relations of the quantum coherence and their lower bounds under the amplitude damping channel (ADC). We find that ADC changes the uncertainty relations and their lower bounds, and their tendencies depend on the initial state. In this paper, we only consider the influence under the amplitude damping channel, and other Markovian channels can be considered by the similar method.

This paper is organized as follows. We briefly recall some notions we are going to use in our analysis in Section 2. In Section 3 we discuss quantum uncertainty relations of the quantum coherence in single state. In Section 4 we discuss dynamics of quantum uncertainty relations of the quantum coherence under ADC, and we summarize our results in Section 5.

In this section, we review some notions related to quantifying the quantum coherence. Considering a finite-dimensional Hilbert space H with d = dim(H). Let {|i⟩, i = 1, 2, …, d} be a particular basis of H. A state is called an incoherent state if and only if its density operator is diagonal in this basis, and the set of all the incoherent states is usually denoted as Δ. Baumgratz et al.^[27] have proposed that the quantum coherence can be measured by a function C that maps a state ρ to a non-negative real value, moreover, C must satisfy the following properties: C(ρ)≥ 0 and C(ρ) = 0 if and only if ρ ∈ Δ; C(ρ)≥ C(Φ(ρ)), where Φ is any incoherent completely positive and trace preserving map; C(ρ)≥ ∑_ip_iC(ρ_i), where , , for all K_i with and ; ∑_ip_iC(ρ_i)≥ C(∑_ip_iρ_i) for any ensemble {p_i,ρ_i}.

In accordance with the criterion, several coherence measures have been studied. It has been shown that l₁ norm of coherence,^[27] relative entropy of coherence,^[27] the quantum coherence via skew information,^[28] and geometric coherence^[29] satisfy the above four conditions.

The relative entropy of coherence^[27] is defined as

l₁ norm of coherence^[27] is defined as

The quantum coherence via skew information^[28] of ρ is defined as

The geometric coherence is defined by Streltsov et al.^[29] as follows:

In this section, we consider quantum uncertainty relations of the above four coherence measures for single-qubit states. Considering a single qubit state ρ with spectral decomposition

For convenience, we first show the following lemma.

Given a single qubit state ρ, we can find a corresponding value ω. Therefore, for convenience, we replace θ + ω with θ in the following. Hence, we have |⟨x₀|φ⟩|² = sin²(θ), furthermore, |⟨x₀|φ_⊥⟩|² = |⟨ x₁|φ⟩|² = cos²(θ), and |⟨x₁|φ_⊥⟩|² = sin² (θ). Similarly, we have |⟨y₀|φ⟩|² = |⟨y₁|φ_⊥⟩|² = sin²(θ + ε), and |⟨y₀|φ_⊥⟩|² = |⟨y₁|φ⟩|² = cos²(θ + ε).

After the projective measurements X and Y, the states of the system are given, respectively, by

Based on the above preliminaries, we consider quantum uncertainty relations of the above quantum coherence measures. Instead of optimizing the uncertainty relations over all possible states ρ, we will optimize them over θ. We first present quantum uncertainty relations of REOC as follows.

Theorem 1 gives a lower bound B_r1(θ) with parameters θ, the value θ depends on orthogonal base X and eigenvector |φ⟩. We obtain the minimum lower bound B_r1 by optimizing the value θ. In Ref. [52], Yuan et al. have given a quantum uncertainty relation of REOC

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) The comparison between Br1 and Br2 for the cases of (a) p = 0.65, (b) p = 0.8, and (c) p = 0.95; (d) Br1(θ), Br1, Br2, and are compared for the case of and p = 0.75.

According to Eq. (10) and Fig. 1(d), we can answer the question (i) in Section 1 aiming to REOC. For two pairs of measurement bases which have the same overlap, i.e., the same values ε, we can obtain different uncertainties of the quantum coherence and different lower bounds B_r1(θ), but the same lower B_r1 and B_r2 can be obtained.

Now, we consider a quantum uncertainty relation of SIC based on bases X and Y, and give a lower bound of . In Fig. 2(a), we compare and B_SI for p = 0.75 and .

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) For and p = 0.75, (a) and BSI, (b) and Bl1, and (c) and Bg are compared.

According to Eq. (15) and Fig. 2(a), we can answer the question (i) in Section 1 aiming to SIC. For two pairs of measurement bases which have the same overlap, i.e., the same values ε, we can obtain different red uncertainty relations of SIC, but the same lower B_SI will be obtained.

In Ref. [52], Yuan et al. have given a quantum uncertainty relation of l₁-norm of coherence based on two fixed bases, and a lower bound of has been obtained. This bound can be saturated in some special cases. In the following, we will show this bound from our perspective. Our perspective will provide a simple method to prove the lower bound. In Fig. 2(b), we compare with B_l1.

In the following, we will show f(α) = |sin(α)| + |sin(α + β)|≥ |sin(β)|, where . It is obvious that f is a period function, and the minimal positive period is T = π. Considering α∈[0,π]. When α ∈ [0,π − β], in this case, α + β ∈ [β, π], hence f(α) = sin(α) + sin(α + β), f″(α) = − sin(α) − sin(α + β) ≤ 0, then f is a concave function. Therefore, f has a minimal value of f_min = sin(β) for θ = 0 or θ = π − β. When θ ∈ [π − β, π], f(θ) = sin(θ) − sin(θ + β). In the same way, we can find f_min = sin(β) for θ = π. Replacing α and β with 2θ and 2ε, respectively, we can prove our result. Moreover, inequality can be saturated when θ = 0, or θ = π − ε.

According to Eq. (18) and Fig. 2(b), we can answer the question (i) in Section 1 aiming to the l₁ norm of coherence. For two pairs of measurement bases which have the same overlap, i.e., the same values ε, we can obtain different uncertainties of l₁ norm of coherence, but the same lower bound B_l1 can be obtained. In addition to REOC and l₁ norm of coherence, the uncertainty relation of the coherence of formation has been discussed in Ref. [52] for single qubit state. By using the way similar to Ref. [52], we can also answer the question (i) in Section 1 aiming to the coherence of formation. We may find an optimal lower bound than the previous one if we find a suitable inequality about entropy. This is a problem to be solved.

Next, we consider a quantum uncertainty relation of GC based on bases X and Y, and give a lower bound of . In Fig. 2(c), we compare and B_g for p = 0.75 and .

According to Eq. (20) and Fig. 2(c), we can answer the question (i) in Section 1 aiming to GC. For two pairs of measurement bases which have the same overlap, i.e., the same values ε, we can obtain different uncertainty relations of GC, but the same lower bound B_g can be obtained.

In this section, to answer the question (ii) in Section 1, we discuss dynamics of quantum uncertainty relations of quantum coherence under amplitude damping channel for single-qubit states. Our task is to study dynamics of left-hand sides (LHSs) and right-hand sides (RHSs) of Eqs. (10), (15), (18), and (20) under ADC. In fact, we can consider other Markovian channels by a similar method. ADC can be characterized by the Kraus’ operators , , where parameters q ∈ [0,1]. We consider single-qubit mixed states with the form

We first compare B_r1, B_SI, B_l1, and B_g with , , , and . Due to Eqs. (10), (15), (18), and (20), we find B_r1, B_SI, B_l1, and B_g are increasing functions with respect to p for a fixed value ε. Now, we consider D − C = (q − 1)[t² + z² − 1 + q(1 − z)²]. In the case of t² ≤ 2z − 2z², we have D − C ≥ 0 for any q ∈ [0,1]. Hence, p′ > p holds. Therefore, , , , and , and inequalities become equalities when q = 1. In the case of , if , then D ≥ C, i.e., p′ ≥ p, hence, , , , and . If , then D ≤ C, i.e, p′≤ p, hence, , , , and .

Next, we discuss the variation of , , , and with the coefficient q. In the case of t² ≤ 2z − 2z², holds for any q ∈ [0,1]. , , , and will be reduced with q increasing. In the case of , if , then , that is to say that p′ will be increased with q increasing, hence, , , , and will be increased with q increasing. If , then , that is to say that p′ will be decreased with q increasing, hence, , , , and will be decreased with q increasing.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) . (a), (b), (c), and (d) represent the variation of , , , and with the coefficient q, respectively. Red “+” is initial state ρ1, and green “°” is initial state ρ2.

In the following, we investigate dynamics of LHSs of Eqs. (10), (15), (18), and (20) under ADC. It is difficult to consider all cases. For simplicity, we only consider some special cases. We treat the LHSs of Eqs. (10), (15), (18), and (20) as some functions with respect to q, and treat other parameters t, z, ε, and θ as constants. We first consider l₁-norm of coherence for single state with the form of Eq. (24).

We consider some special cases as follows:

(I) In the case of cos(2θ) ≥ 0, sin(2θ)≤ 0, cos(2θ + 2 ε)≥ 0, and sin(2θ + 2 ε) ≤ 0. That is to say . Here, we assume , the set is empty set if . In this case,

We denote . M is clearly non-negative. When M ≥ 1, holds for all q ∈ [0,1], hence f increases as q increases. When M ∈ [0,1], if q ∈ [0,M²], and if q ∈[M²,1]. Therefore, f first increases and then decreases as q increases.

(II) In the case of cos(2θ) < 0, sin(2θ) > 0, cos(2θ + 2 ε) < 0, and sin(2θ + 2 ε) > 0. That is to say . Here, we assume , the set is empty set if . In this case,

Similar to case (I), when M ≥ 1, holds for all q in [0,1], hence f increases as q increases. When M ∈ [0,1], if q ∈ [0,M²], and if q ∈ [M²,1]. Therefore, f first increases and then decreases as q increases.

(III) In the case of cos(2θ)< 0, sin(2θ) > 0, cos(2θ + 2 ε) > 0, and sin(2θ + 2 ε) < 0. That is to say . Here, , the set is a empty set if . In this case,

We denote , M₁ is clearly non-negative. When M₁ ≥ 1, holds for all q ∈ [0,1], hence f increases as q increases. When M₁ ∈ [0,1], if , and if . Therefore, f first increases and then decreases as q increases.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. (color online) The variations of function f and the lower bounds under ADC. The red solid line represents the value of f, and the green dotted line represents the lower bounds. U represents the value of function f or its lower bound. (a)–(f) correspond to (A)–(F) in Example 2, respectively.

(A) Given a state ρ₁ with parameters , and a two-tuples . In this case M = 5.759 > 1.

(B) Given a state ρ₂ with parameters , and a two-tuples . In this case, M = 0.4564 < 1.

(C) Given a state ρ₃ with parameters , and a two-tuples , In this case M = 5.759 > 1.

(D) Given a state ρ₄ with parameters , and a two-tuples . In this case, M = 0.4811 < 1.

(E) Given a state ρ₅ with parameters , and a two-tuples . In this case, M₁ = 0.2887 < 1.

(F) Given a state ρ₆ with parameters , and a two-tuples . In this case, M₁ = 1.3693 > 1.

Now, we consider dynamic of the LHS of Eq. (10) for the state with the form of Eq. (24) under ADC. We denote , B_r = 2[1 − H(p′)](1 − c), where , and D = 1+ qt² − q[1 − z² + (1 − q)(1 − z)²]. , . We draw g and B_r as a function of q. In Fig. 5, we discuss the dynamics of the function g and B_r for some initial states with respect to q.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. (color online) The variations of the function g and the lower bounds under ADC. and p = 0.75. The red line represents the value of function g, and the green dotted line represents the lower bounds. U represents the value of function g or its lower bound.

It is a difficult task to calculate LHSs of Eqs. (15) and (20) for state with the form of Eq. (24). However, we can discuss their dynamics by fixing the parameters ε and θ. In this case, we find LHSs of Eqs. (15) and (20) have the same tendency with RHSs of Eqs. (15) and (20) under ADC.

In this paper, we discussed quantum uncertainty relations of quantum coherence through a different method from Ref. [52]. We obtained some lower bounds with parameters, and their minimal bounds. Moreover, we found that for two pairs of measurement bases which have the same maximum overlap, the different quantum uncertainty relations and lower bounds with parameters maybe obtained, but the same minimal bounds can be obtained. This result answered the question (i) posed in Section 1. In the next place, we considered dynamics of quantum uncertainty relations of quantum coherence and their lower bounds under the amplitude damping channel (ADC). We found that ADC changed the uncertainty relations and their lower bounds, and their tendencies depended on the initial state. These discussions answered a part of the question (ii) posed in Section 1.

There are many further issues that need to be solved in the future. We can discuss the uncertainty relations for other coherence measures. We only considered dynamics of of quantum uncertainty relations of quantum coherence under ADC. Dynamics of quantum uncertainty relations of quantum coherence under any incoherent operator^[27] (strictly incoherent operation and maximally incoherent operation^[45]) is an interesting subject. Quantum uncertainty relations based on many fixed bases are also an interesting subject for future work.