1 Introduction

Quantum coherence is an essential ingredient for a plethora of physical phenomena in quantum optics, quantum information, solid state physics, and nanoscale thermodynamics [1, 2]. As a kind of quantum resource, some basic characterizations of coherence are studied such as quantification and manipulation. The quantification of quantum coherence is initiated in Ref. [3] and some coherence measures such as the l1 norm of coherence [3], the relative entropy of coherence [3], intrinsic randomness of coherence [4], coherence concurrence [5], distillable coherence [6], robustness of coherence [7], geometric coherence [8], coherence number [9] are proposed from different aspects.

The quantum coherence manipulation is the conversion between quantum states under incoherent operations [10]. For pure states, one can be converted into another by incoherent operations if and only if their coherence vectors satisfy a majorization relation [11], which is a counterpart of the celebrated Nielson theorem in entanglement manipulation [12]. Then a necessary and sufficient condition for the transformation between two pure state ensembles is demonstrated [13], but the conversion between ensembles is different from the manipulation of mixed states. The conversion from pure states to mixed states is studied and the necessary and sufficient condition has been provided in terms of a sequence of inequalities about a given coherence measure in Ref. [14].

The quantification and manipulation of coherence are not separate because the coherence is not increased under the incoherent operations. On one hand, some necessary and sufficient conditions of the coherence manipulation is presented in terms of coherence measures [13, 14]. On the other hand, some coherence measures are introduced based on the coherence manipulation [14, 15]. Generally, both the transformation between mixed states under incoherent operations and the quantification of coherence in terms of extremization for mixed states are complex and difficult because of the infinite pure state decompositions of mixed states.

Here we focus on the conversion between pure states and X state and calculate its coherence measure. X state is a class of mixed states with density matrix in X shape. Its quantum correlation such as quantum discord [16,17,18] and one-way quantum deficit [19, 20], and quantum coherence such as coherence concurrence [21] are investigated broadly. Attributing to the symmetry of X state formally, the study of its quantumness seems to be more feasible than general mixed states.

In the following, we first find an optimal pure state decomposition of X state that can major any pure state decomposition of X state. Then the necessary and sufficient condition for pure states which can be converted to X state is given analytically. We also find the optimal pure state associated with X state such that any pure state which can be converted to X state should be majorized by it. Then we construct the incoherent operations converting the pure state to X state. This process is illustrated by an explicit example in a three dimensional system. At last we calculate the coherence measure for X states.

2 The Conversion from Pure States to X States Under Incoherent Operations

2.1 Some Basic Concepts

The resource theory of coherence is composed of two basic elements: free states and free operations. Let H be a finite-dimensional Hilbert space with dim(H) = d. Take a set of basis \(\{|i\rangle \}_{i=1}^{d}\), we call the diagonal quantum state \(\rho ={\sum }_{i = 1}^{d} \lambda _{i}|i\rangle \langle i|\) under this set of basis as the incoherent state. This set of incoherent states is labeled by ∇. Φ is an operation if and only if there exists finite bounded linear operators Kn satisfying \({\sum }_{n} {K}_{n}^{\dag } K_{n}=I\) and \({\Phi }(\rho )={\sum }_{n} K_{n}\rho {K}_{n}^{\dag }\), where I is the identity operation. Φ is an incoherent operation if it fulfills \(K_{n}\delta K_{n}^{\dag }/Tr(K_{n}\delta K_{n}^{\dag })\in \nabla \) for all δ ∈∇ and for all n [22]. We denote when the quantum state ρ can be transformed to \(\rho ^{\prime }\) by incoherent operations.

The quantum coherence is degreed by a nonnegative function named the coherence measure. A coherence measure should satisfy five conditions as follows [3]: (A1) C(ρ) = 0 for all ρ ∈∇; (A2) monotonicity: C(ε(ρ)) ≤ C(ρ) for any incoherent operation ε; (A3) strong monotonicity: \({\sum }_{n}p_{n}C(K_{n}\rho K_{n}^{\dagger }/p_{n})\leq C(\rho )\) with \(p_{n}=Tr(K_{n}\rho K_{n}^{\dagger })\) and \(\rho _{n}=K_{n}\rho K_{n}^{\dagger }/p_{n}\); (A4) convexity: \(C(\rho )\leq {\sum }_{i} p_{i}C(\rho _{i})\) for any \(\rho ={\sum }_{i}p_{i}\rho _{i}\); (A5) only maximally coherent states reach the maximum: C(ρ) is maximal only for ρ = |Φd〉〈Φd|, where \(|{\Phi }_{d}\rangle =\frac {1}{\sqrt {d}}{\sum }_{i=1}^{d}e^{\mathrm {i}\theta _{i}}|i\rangle \) with real 𝜃i and i the imaginary unit. C(⋅) will be called as a coherence monotone if it satisfies all above conditions but (A4).

In this paper, we fix the reference basis as \(\{|i\rangle \}_{i=1}^{d}\). For any pure state \(|\phi \rangle = {\sum }_{i=1}^{d} \phi _{i} |i\rangle \), its coherence vector is (|ϕ1|2,|ϕ2|2,⋯ ,|ϕd|2)T. Let \(\mathcal {R}^{\downarrow }(|\phi \rangle )=(|\phi _{1}|^{2\downarrow }, |\phi _{2}|^{2\downarrow }, \cdots , |\phi _{d}|^{2\downarrow })^{T}\) be the vector obtained by rearranging the entries of the coherence vector in descending order |ϕ1|2↓≥|ϕ2|2↓≥⋯ ≥|ϕd|2↓. Based on the vector \(\mathcal {R}^{\downarrow }(|\phi \rangle )\), a series of coherence measures are introduced in Ref. [14].

Definition 1

For any pure state \(|\phi \rangle ={\sum }_{i=1}^{d} \phi _{i} |i\rangle \), let

$$ C_{f_{l}}(|\phi\rangle\langle\phi|)=\sum\limits_{i=l}^{d}|\phi_{i}|^{2\downarrow}, $$
(1)

then \(C_{f_{l}}(\rho )=\min \limits {\sum }_{n} \mu _{n} C_{f_{l}}(|\phi _{n}\rangle \langle \phi _{n}|)\) is a coherence measure, where the minimization is taken over all pure state decompositions of \(\rho ={\sum }_{n} \mu _{n} |\phi _{n}\rangle \langle \phi _{n} |\). We call \(C_{f_{l}}(\rho )\) the l-th order coherence of ρ, l = 2,⋯ ,d.

Suppose x = (x1, x2,⋯ , xd)T and y = (y1, y2,⋯ , yd)T are two probability distributions in H with coordinates in the decreasing order. x is majorized by y denoted as x ≺y, if \({\sum }_{i=1}^{k} x_{i} \leq {\sum }_{i=1}^{k} y_{i}\) for 1 ≤ k ≤ d. By the majorization, the necessary and sufficient condition for the conversion between two pure states is presented [11].

Lemma 1

For any two pure states \(|\phi \rangle ={\sum }_{i=1}^{d} \phi _{i} |i\rangle \) and \(|\psi \rangle ={\sum }_{i=1}^{d} \psi _{i} |i\rangle \), if and only if (|ψ1|2↓,|ψ2|2↓,⋯ ,|ψd|2↓)T ≺ (|ϕ1|2↓,|ϕ2|2↓,⋯ ,|ϕd|2↓)T.

Furthermore, the necessary and sufficient condition for the conversion between pure states and mixed states is shown by a series of inequalities in terms of the l-th order coherence in Ref. [14].

Lemma 2

For any pure state |ϕ〉 and any mixed state ρ, if and only if there exists a pure state ensemble {pi,|ψi〉} of ρ such that

$$ C_{f_{l}}(|\phi\rangle\langle\phi|)\geq \underset{i}{\sum} p_{i} C_{f_{l}}(|\psi_{i}\rangle\langle\psi_{i}|),\ \ l=2,\cdots, d. $$
(2)

By Lemma 2, if the pure state |ϕ〉 can be converted to the mixed state ρ, then one should find a pure state decomposition of ρ such that the average coherence measured by the l-th order coherence in (1) satisfies the d − 1 inequalities in (2). So generally it is not easy to determine whether a pure state can be converted to a given mixed state.

2.2 The Transformation from Pure States to X States Under Incoherent Operations

Now we study the transformation from pure states to X states by incoherent operations. The d dimensional X states denoted as Xd are quantum states with density matrices in X shape under the reference basis,

$$ \begin{array}{@{}rcl@{}} X_{d}= \left( \begin{array}{cccccccc} \rho_{11} &0 &0 &{\cdots} &0 &0 & \rho_{1,d}\\ 0&\rho_{22} &0 &{\cdots} &0 &\rho_{{2},{d-1}} & 0\\ 0& 0 &\rho_{33} &{\cdots} &\rho_{{3},{d-2}} & 0 & 0\\ \cdots& {\cdots} &{\cdots} &{\cdots} &{\cdots} & {\cdots} & \cdots\\ 0& 0 &\rho_{{d-2},3} &{\cdots} &\rho_{{d-2},{d-2}} & 0 & 0\\ 0&\rho_{{d-1},2} &0 &{\cdots} &0 &\rho_{{d-1},{d-1}} & 0\\ \rho_{d,1} &0 &0 &{\cdots} &0 &0 & \rho_{dd} \end{array}\right). \end{array} $$
(3)

Now we show an optimal pure state decomposition for X states.

Theorem 1

For quantum state Xd in (3), let \(\mathfrak {D}=\{\mu _{ki}, |\chi _{ki}\rangle \} \cup \left \{\frac {1-(-1)^{d}}{2} \rho _{[d/2]+1,}\right .\) \(\left .{[d/2]+1}, |[d/2]+1\rangle \vphantom {\frac {1-(-1)^{d}}{2}} \right \}\) be the pure state decomposition of Xd, where

$$ \begin{array}{@{}rcl@{}} \begin{array}{rcl} |\chi_{1i}\rangle \langle \chi_{1i}|&=&\left( \begin{array}{ccccccc} \frac{1+z_{i}}{2}&\frac{\rho_{i,d-i+1}}{\rho_{ii}+\rho_{d-i+1,d-i+1}}\\ \frac{\rho_{i,d-i+1}^{*}}{\rho_{ii}+\rho_{d-i+1,d-i+1}}&\frac{1-z_{i}}{2} \end{array} \right),\\ |\chi_{2i}\rangle \langle \chi_{2i}|&=&\left( \begin{array}{ccccccc} \frac{1-z_{i}}{2}&\frac{\rho_{i,d-i+1}}{\rho_{ii}+\rho_{d-i+1,d-i+1}}\\ \frac{\rho_{i,d-i+1}^{*}}{\rho_{ii}+\rho_{d-i+1,d-i+1}}&\frac{1+z_{i}}{2} \end{array} \right), \end{array} \end{array} $$
(4)

are pure states on the subspace spanned by {|i〉〈i|,|d − i + 1〉〈d − i + 1|}, \(z_{i}=\sqrt {1-4|\frac {\rho _{i,d-i+1}}{\rho _{ii}+\rho _{d-i+1,d-i+1}}|^{2}}\), [d/2] is the integer part of d/2. Let μki = τiλki with τi = ρii + ρd−i+ 1,d−i+ 1 for k = 1,2, \(\lambda _{1i}=\frac {1}{2}+\frac {\rho _{ii}-\rho _{d-i+1,d-i+1}}{2\sqrt {(\rho _{ii}+\rho _{d-i+1,d-i+1})^{2}-4|\rho _{i,d-i+1}|^{2}}}\) and λ2i = 1 − λ1i, i = 1,2,⋯ ,[d/2]. Then any pure state decomposition {pj,|ψj〉} of Xd is majorized by the pure state decomposition \(\mathfrak {D}\),

$$ \underset{j}{\sum} p_{j} \mathcal{R}^{\downarrow}(|\psi_{j}\rangle)\prec \sum\limits_{i=1}^{[d/2]} \sum\limits_{k=1}^{2} \mu_{ki} \mathcal{R}^{\downarrow}(|\chi_{ki}\rangle) + \frac{1-(-1)^{d}}{2} \rho_{[d/2]+1,[d/2]+1} \mathcal{R}^{\downarrow}(|[d/2]+1\rangle). $$

Proof

First we write the state Xd in the form of direct sum,

$$ X_{d}=\tau_{1}\rho_{1}\bigoplus \tau_{2}\rho_{2}\bigoplus\cdots\bigoplus \tau_{[d/2]}\rho_{[d/2]}\bigoplus \frac{1-(-1)^{d}}{2} \rho_{[d/2]+1,[d/2]+1} |[d/2]+1\rangle \langle [d/2]+1|, $$

with \(\rho _{i}=\left (\begin {array}{ccccccc} \frac {\rho _{ii}}{\rho _{ii}+\rho _{d-i+1,d-i+1}}&\frac {\rho _{i,d-i+1}}{\rho _{ii}+\rho _{d-i+1,d-i+1}}\\ \frac {\rho _{i,d-i+1}^{*}}{\rho _{ii}+\rho _{d-i+1,d-i+1}}&\frac {\rho _{d-i+1,d-i+1}}{\rho _{ii}+\rho _{d-i+1,d-i+1}} \end {array} \right )\) on the subspace spanned by {|i〉〈i|,|d − i + 1〉〈d − i + 1|} and τi = ρii + ρd−i+ 1,d−i+ 1, i = 1,2,⋯ ,[d/2]. Then ρi can be decomposed as \(\{\lambda _{ki},\ |\chi _{ki}\rangle \}_{k=1}^{2}\) with |χki〉 in (4) such that for any pure state decomposition {qsi,|ψsi〉} with \(|\psi _{si}\rangle ={\sum }_{j} {\psi }_{j}^{(si)}|j\rangle \) of ρi [15], it has

$$ \underset{s}{\sum} q_{si} \mathcal{R}^{\downarrow}(|\psi_{si} \rangle)\prec\sum\limits_{k=1}^{2} \lambda_{ki} \mathcal{R}^{\downarrow}(|\chi_{ki}\rangle) $$

for i = 1,2,⋯ ,[d/2]. Therefore

$$ \begin{array}{@{}rcl@{}} \begin{array}{rcl} {\sum}_{s} q_{si} |\psi_{1}^{(si)}|^{2\downarrow}&\leq&\lambda_{1i} \frac{1+z_{i}}{2}+\lambda_{2i} \frac{1+z_{i}}{2}=\frac{1+z_{i}}{2}, \end{array} \end{array} $$

which is equivalent to

$$ \begin{array}{@{}rcl@{}} \begin{array}{rcl} {\sum}_{s} q_{si} |\psi_{2}^{(si)}|^{2\downarrow}&\geq& \frac{1-z_{i}}{2}, \end{array} \end{array} $$
(5)

for i = 1,2,⋯ ,[d/2].

Combining Lemma 2 and Definition 1, for any pure state decomposition {pj,|ψj〉} of Xd with \(|\psi _{j}\rangle ={\sum }_{i} {\psi }_{i}^{(j)} |i\rangle \), we have

$$ \begin{array}{rcl} {\sum}_{j} p_{j} C_{f_{2}}(|\psi_{j}\rangle\langle \psi_{j}|)&\geq&C_{f_{2}}(X_{d})\\ &=&{\sum}_{i=1}^{[d/2]} \tau_{i} C_{f_{2}}(\rho_{i}) + \frac{1-(-1)^{d}}{2} \rho_{[d/2]+1,[d/2]+1} C_{f_{2}}(|[d/2]+1\rangle) \\ &=&{\sum}_{i=1}^{[d/2]} \tau_{i} \min{\sum}_{s} q_{si} C_{f_{2}}(|\psi_{si} \rangle\langle\psi_{si}|)\\ &\geq& {\sum}_{i=1}^{[d/2]} \tau_{i} \frac {1-z_{i}}{2}. \end{array} $$
(6)

The first inequality is the convexity of the coherence measure, the second equality is the additivity of the coherence measure under the direct sum operation [23], the third equality is the definition of the coherence measure \(C_{f_{2}}\), and the last inequality is due to the (5). Therefore

$$ \begin{array}{@{}rcl@{}} &&\underset{j}{\sum} p_{j}|\psi_{1}^{(j)}|^{2\downarrow}\leq \sum\limits_{i=1}^{[d/2]} \tau_{i} \frac{1+z_{i}}{2} + \frac{1-(-1)^{d}}{2} \rho_{[d/2]+1,[d/2]+1},\\ &&\underset{j}{\sum} p_{j}|\psi_{1}^{(j)}|^{2\downarrow} + \underset{j}{\sum} p_{j}|\psi_{2}^{(j)}|^{2\downarrow}\leq 1. \end{array} $$

Hence \({\sum }_{j} p_{j} \mathcal {R}^{\downarrow }(|\psi _{j}\rangle )\) ≺ \( {\sum }_{i=1}^{[d/2]} {\sum }_{k=1,2}\mu _{ki} \mathcal {R}^{\downarrow }(|\chi _{ki}\rangle )\)\( + \frac {1-(-1)^{d}}{2} \rho _{[d/2]+1,[d/2]+1} \mathcal {R}^{\downarrow }(|[d/2]\) + 1〉). □

We call the pure state decomposition \(\mathfrak {D}\) in Theorem 1 the optimal as all other pure state decompositions are majoried by it. Let the vector \(\textbf {v}={\sum }_{i=1}^{[d/2]} {\sum }_{k=1,2} \mu _{ki} \mathcal {R}^{\downarrow }(|\chi _{ki}\rangle ) + \frac {1-(-1)^{d}}{2} \rho _{[d/2]+1,[d/2]+1} \mathcal {R}^{\downarrow }(|[d/2]+1\rangle )\). In fact, the vector v has only two nonzero entries

$$ \begin{array}{@{}rcl@{}} \textbf{v}=(v_{1},v_{2},0,\cdots,0)^{T} \end{array} $$
(7)

with \(v_{1}=\frac {1}{2} + \frac {1-(-1)^{d}}{4}\rho _{[d/2]+1,[d/2]+1} +\frac {1}{2}{\sum }_{i=1}^{[d/2]}\sqrt {({\rho _{ii}+\rho _{d-i+1,d-i+1}})^{2}- 4|{\rho _{i,d-i+1}}|^{2}}\), \(v_{2}={\sum }_{i=1}^{[d/2]}\frac {\rho _{ii}+\rho _{d-i+1,d-i+1}}{2}-\frac {1}{2}{\sum }_{i=1}^{[d/2]}\sqrt {({\rho _{ii}+\rho _{d-i+1,d-i+1}})^{2}- 4|{\rho _{i,d-i+1}}|^{2}}\). Then all pure state decompositions of Xd states are majorized by the vector v. By the vector v and the optimal pure state decomposition \(\mathfrak {D}\) of Xd states, we derive the necessary and sufficient condition for the pure state converted to Xd states.

Theorem 2

A pure state with \(|\phi \rangle ={\sum }_{i=1}^{d} \phi _{i} |i\rangle \) if and only if

$$ |\phi_{1}|^{2\downarrow}\leq \frac{1}{2} +\frac{1}{2}\sum\limits_{i=1}^{[d/2]}\sqrt{({\rho_{ii}+\rho_{d-i+1,d-i+1}})^{2}- 4|{\rho_{i,d-i+1}}|^{2}} +\frac{1-(-1)^{d}}{4}\rho_{[d/2]+1,[d/2]+1}. $$
(8)

Proof

On one hand, if any pure state |ϕ〉 can be converted to Xd state, according to Lemma 2 and Theorem 1, there exists a pure state decomposition {pj,|ψj〉} of Xd state such that \(C_{f_{2}}(|\phi \rangle \langle \phi |)\geq {\sum }_{j} p_{j} C_{f_{2}}(|\psi _{j}\rangle \langle \psi _{j}|)\geq {\sum }_{i=1}^{[d/2]} \tau _{i}\frac {1-z_{i}}{2}\). It implies \(|\phi _{1}|^{2\downarrow }\leq {\sum }_{i=1}^{[d/2]}\tau _{i}\frac {1+z_{i}}{2} + \frac {1-(-1)^{d}}{2} \rho _{[d/2]+1,[d/2]+1} =\frac {1}{2} +\frac {1}{2}{\sum }_{i=1}^{[d/2]}\sqrt {({\rho _{ii}+\rho _{d-i+1,d-i+1}})^{2}- 4|{\rho _{i,d-i+1}}|^{2}} +\frac {1-(-1)^{d}}{4}\rho _{[d/2]+1,[d/2]+1}\).

On the other hand, if the pure state |ϕ〉 satisfies (8), then \(C_{f_{2}}(|\phi \rangle \langle \phi |)= 1- |\phi _{1}|^{2\downarrow }\geq 1- \left (\frac {1}{2} +\frac {1}{2}{\sum }_{i=1}^{[d/2]}\sqrt {({\rho _{ii}+\rho _{d-i+1,d-i+1}})^{2}- 4|{\rho _{i,d-i+1}}|^{2}} +\frac {1-(-1)^{d}}{4}\rho _{[d/2]+1,[d/2]+1}\right )= {\sum }_{i=1}^{[d/2]} \tau _{i}\frac {1-z_{i}}{2}={\sum }_{ki} \mu _{ki} C_{f_{2}}(|\chi _{ki}\rangle \langle \chi _{ki}|) \). Furthermore \(C_{f_{l}}(|\phi \rangle \langle \phi |)\geq {\sum }_{ki} \mu _{ki} C_{f_{l}}(|\chi _{ki}\rangle \langle \chi _{ki}|)+ \frac {1-(-1)^{d}}{2} \rho _{[d/2]+1,[d/2]+1} C_{f_{l}} (|[d/2]+1\rangle \langle [d/2]+1|)=0\) for l = 3,4,⋯ ,d. Therefore the optimal pure state decomposition \(\mathfrak {D}\) of Xd satisfies the (2) in Lemma 2 for the pure state |ϕ〉. So (8) is the necessary and sufficient condition for pure states to be converted to Xd states. □

Theorem 2 shows whether a pure state can be converted to Xd state is only decided by its largest magnitude under the reference basis. Let R(ρ) be the set of pure state that can be converted into the given state ρ by incoherent operations. As to Xd state, the pure state \(|\phi \rangle ={\sum }_{i} \phi _{i} |i\rangle \) is in the set of R(Xd) if and only if (8) holds true. Denote

$$ |{\Xi}\rangle=(\eta_{1}, \eta_{2}, 0,\cdots,0)^{T} $$
(9)

with \(\eta _{1}=\sqrt {v_{1}}\), \(\eta _{2}=\sqrt {1-{{v}_{1}^{2}}}\) with v1 defined in (7). Then we can express the necessary and sufficient condition of pure states that can be converted to the Xd states by the majorization relation.

Corollary 1

Any pure state if and only if \(\mathcal {R}^{\downarrow }(|\phi \rangle )\prec \mathcal {R}^{\downarrow }(|{\Xi }\rangle )\).

Now we consider the incoherent operations transforming pure states to X states. We accomplish this transformation by two steps. First we convert the pure state |ϕ〉 to the intermediate pure state |Ξ〉 by an incoherent operation Φ1. The Kraus operators of the incoherent operation Φ1 depends on the input state |ϕ〉 and output state |Ξ〉 which have been constructed in Ref. [11]. Then we convert the intermediate pure state |Ξ〉 to the given Xd state under an incoherent operation Φ2 with Kraus operators

$$ \begin{array}{rcl} &K_{1i}&=\sqrt{\mu_{1i}}\left( \frac{\sqrt{\frac{1+z_{i}}{2}}}{\eta_{1}} |i\rangle\langle 1|+ \frac{\sqrt{\frac{1-z_{i}}{2}}}{\eta_{2}} |d-i+1\rangle\langle 2|\right), \\ &K_{2i}&=\sqrt{\mu_{2i}}\left( \frac{\sqrt{\frac{1-z_{i}}{2}}}{\eta_{2}} |i\rangle\langle 2|+ \frac{\sqrt{\frac{1+z_{i}}{2}}}{\eta_{1}} |d-i+1\rangle\langle 1|\right),\\ &K_{3}&= \frac{1-(-1)^{d}}{2} \frac{\sqrt{\rho_{[d/2]+1,[d/2]+1}}}{\eta_{1}}|[d/2]+1\rangle\langle 1|,\\ &K_{0}&={\sum}_{i=3}^{d}|i\rangle\langle i| \end{array} $$

for i = 1,2,⋯ ,[d/2]. In this way any pure state fulfills the (8) can be converted to the X state by incoherent operators Φ1 and Φ2.

2.3 An Explicit Example in a Three Dimensional System

Now we illustrate the transformation from pure states to X states by an explicit three dimensional example.

Example 1

Given a three dimensional X state represented by the 3 × 3 Hermitian matrix \( X_{3}=\left (\begin {array}{ccc} \rho _{11} & 0 & \rho _{13}\\ 0 & \rho _{22} & 0\\ \rho _{13}^{*} & 0 & \rho _{33} \end {array} \right )\), by Theorem 2, the pure state |ϕ〉 = ϕ1|1〉 + ϕ2|2〉 + ϕ3|3〉 can be converted into the X3 state if and only if |ϕ〉 satisfies that

$$ |\phi_{1}|^{2\downarrow} \leq \frac{1}{2}\left( 1+\rho_{22} + \sqrt{(\rho_{11}+\rho_{33})^{2}-4|\rho_{13}|^{2}}\right). $$

Now we construct the incoherent operations Φ1 and Φ2 such that |ϕ〉 |Ξ〉 and |Ξ〉 X3, where |Ξ〉 = η1|1〉 + η2|2〉 + η3|3〉, \(\eta _{1}=\sqrt {\frac {1}{2}(1+\rho _{22} + \sqrt {(\rho _{11}+\rho _{33})^{2}-4|\rho _{13}|^{2}})}\), \(\eta _{2}=\sqrt {1-{\eta _{1}^{2}}}\), η3 = 0, respectively. Without loss of generality, we assume |ϕ1|≥|ϕ2|≥|ϕ3|.

(1) If ϕ2≠ 0, ϕ3 = 0, let

$$ A=\left( \begin{array}{ccc} a&1-a&0\\ 1-a&a&0\\ 0&0&1 \end{array} \right) $$

(0 ≤ a ≤ 1) be a doubly stochastic matrix such that \((|\phi _{1}|^{2}, |\phi _{2}|^{2}, |\phi _{3}|^{2})^{T} = A({\eta _{1}^{2}}, {\eta _{2}^{2}}, {\eta _{3}^{2}})^{T}\). Define

$$ K_{1}=\left( \begin{array}{ccc} \frac{\sqrt{a}\eta_{1}}{\phi_{1}}&0&0\\ 0&\frac{\sqrt{a}\eta_{2}}{\phi_{2}}&0\\ 0&0&\frac{1}{\sqrt{2}} \end{array} \right), K_{2}=\left( \begin{array}{ccc} 0&\frac{\sqrt{1-a}\eta_{1}}{\phi_{2}}&0\\ \frac{\sqrt{1-a}\eta_{2}}{\phi_{1}}&0&0\\ 0&0&\frac{1}{\sqrt{2}} \end{array} \right). $$

It is easy to check that the incoherent operation Φ1 with Kraus operators K1 and K2 converts the pure state |ϕ〉 into the pure state |Ξ〉.

(2) If ϕ3≠ 0, let A be a doubly stochastic matrix such that \((|\phi _{1}|^{2}, |\phi _{2}|^{2}, |\phi _{3}|^{2})^{T} = A\left ({\eta _{1}^{2}}, {\eta _{2}^{2}}, {\eta _{3}^{2}}\right )^{T}\). Then A can be reduced to a T transform for some indices i, j. Denote

$$ \begin{array}{rcl} K_{1}&=&\sqrt{t} \text{diag}\left( \frac{\eta_{1}}{\phi_{1}}, \frac{\eta_{2}}{\phi_{2}}, \frac{\eta_{3}}{\phi_{3}}\right),\\ K_{2}&=&\sqrt{1-t} \text{diag}\left( \frac{\eta_{1}}{\phi_{\pi(1)}}, \frac{\eta_{2}}{\phi_{\pi(2)}}, \frac{\eta_{3}}{\phi_{\pi(3)}}\right)P_{\pi}, \end{array} $$

with π a permutation corresponding to T and Pπ the matrix corresponding to π, 0 ≤ t ≤ 1. According to the assumption \((|\phi _{1}|^{2}, |\phi _{2}|^{2}, |\phi _{3}|^{2})^{T} = A\left ({\eta }_{1}^{2}, {\eta }_{2}^{2}, {\eta }_{3}^{2}\right )^{T}\), it follows \(K_{1}^{\dagger }K_{1}+K_{2}^{\dagger }K_{2}=I\). Then one can prove the incoherent operation \({\Phi }_{1}(\cdot )={\sum }_{j=1}^{2}K_{j}(\cdot )K_{j}^{\dagger }\) transforms |ϕ〉 to |Ξ〉.

Now we construct the incoherent operation Φ2 converting the pure state |Ξ〉 to X3 state. For Φ2, we construct Kraus operators as follows,

$$ \begin{array}{rcl} K_{1}&=&\frac{\sqrt{\mu_{11}}\sqrt{\frac{1+z}{2}}}{\eta_{1}}|1\rangle\langle 1|+\frac{\sqrt{\mu_{11}}\sqrt{\frac{1-z}{2}}}{\eta_{2}}|3\rangle\langle 2|,\\ K_{2}&=&\frac{\sqrt{\mu_{21}}\sqrt{\frac{1-z}{2}}}{\eta_{2}}|1\rangle\langle 2|+\frac{\sqrt{\mu_{21}}\sqrt{\frac{1+z}{2}}}{\eta_{1}}|3\rangle\langle 1|,\\ K_{3}&=&\frac{\sqrt{\rho_{22}}}{\eta_{1}}|2\rangle\langle 1|,\\ K_{4}&=&|3\rangle\langle 3|, \end{array} $$

where \(z=\sqrt {1-4|\frac {\rho _{13}}{\rho _{11}+\rho _{33}}|^{2}},\mu _{11}=\lambda _{11}({\rho _{11}+\rho _{33}})\), μ21 = λ21(ρ11 + ρ33), \(\lambda _{11}=\frac {1}{2}+\frac {\rho _{11}-\rho _{33}}{2\sqrt {(\rho _{11}+\rho _{33})^{2}-4|\rho _{13}|^{2}}}\). Then we can prove that the incoherent operation Φ2 with Kraus operators \(\{K_{j}\}_{j=1}^{4}\) converts the pure state |Ξ〉 to X3 state.

3 Coherence Measure of X States

In the context of coherence manipulation, some coherence measures or monotone have been introduced [14, 15]. Now we calculate the coherence existed in the X states measured by these quantifiers.

Theorem 3

The l-th order coherence of Xd state is

$$ C_{f_{2}}(X_{d})=\frac{1}{2}\sum\limits_{i=1}^{[d/2]} \rho_{ii}+\rho_{d-i+1,d-i+1} - \sqrt{(\rho_{ii}+\rho_{d-i+1,d-i+1})^{2}-4|\rho_{i,d-i+1}|^{2}} $$

and \(C_{f_{l}}(X_{d})=0\) for l ≥ 3.

Proof

For the l-th order coherence, we show the pure state decomposition \(\mathfrak {D}\) is the optimal pure state decomposition for Xd state. First, for l ≥ 3, it has \(C_{f_{l}} (|\chi _{ki}\rangle )=C_{f_{l}}(|[d/2]+1\rangle )=0\). Therefore \(C_{f_{l}}(X_{d})=0\) by the definition of l-th order coherence for l ≥ 3.

Second, for l = 2, by (6) in the proof in Theorem 1, for any pure state decomposition {pj,|ψj〉} of given state Xd, we have

$$ \underset{j}{\sum} p_{j} C_{f_{2}}(|\psi_{j}\rangle)\geq \sum\limits_{i=1}^{[d/2]} \sum\limits_{k=1}^{2}\mu_{ki}C_{f_{2}} (|\chi_{ki}\rangle)+\frac{1-(-1)^{d}}{2} \rho_{[d/2]+1,[d/2]+1} C_{f_{2}}(|[d/2]+1\rangle). $$

Therefore,

$$ \begin{array}{rcl} C_{f_{2}}(X_{d})&=&\min {\sum}_{j} p_{j} C_{f_{2}}(|\psi_{j}\rangle)\\ &=&{\sum}_{i=1}^{[d/2]} {\sum}_{k=1}^{2}\mu_{ki}C_{f_{2}} (|\chi_{ki}\rangle)+\frac{1-(-1)^{d}}{2} \rho_{[d/2]+1,[d/2]+1} C_{f_{2}}(|[d/2]+1\rangle)\\ &=&\frac{1}{2}{\sum}_{i=1}^{[d/2]} \rho_{ii}+\rho_{d-i+1,d-i+1} - \sqrt{(\rho_{ii}+\rho_{d-i+1,d-i+1})^{2}-4|\rho_{i,d-i+1}|^{2}}. \end{array} $$

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Another coherence monotone related to the coherence manipulation is introduced as Cm(ρ) = inf|ϕ〉∈R(ρ)F(|ϕ〉), where F(|ϕ〉) is a coherence measure satisfying (A1) − (A3) and (A5), R(ρ) is the set of pure states that can be converted into ρ by incoherent operations [15]. By Corollary 1, one can prove the pure state |Ξ〉 defined in (9) is the optimal one for Cm of Xd thanks to the equivalence \(\mathcal {R}^{\downarrow }(|\phi \rangle )\prec \mathcal {R}^{\downarrow }(|{\Xi }\rangle )\) if and only if F(|ϕ〉) ≥ F(|Ξ〉). So Cm(Xd) = F(|Ξ〉).

4 Conclusions

Overall, we have investigated the transformation from pure states to X states under incoherent operations. We derive an optimal pure state decomposition of X state such that all pure state decompositions of X state are majorized by it. Then the necessary and sufficient condition for the pure states to be converted to X state is demonstrated. We find the optimal pure state associated with X state such that any pure state which can be converted to X state should be majorized by it. The incoherent operations transforming pure states to X states are also analyzed. An explicit example is given in a three dimensional system. The coherence measure is also calculated for X states. We hope these results can promote the study of coherence manipulation.