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Entropy-preserving maps on quantum states
Linear Algebra and its Applications 467 (2015) 243–253
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Entropy-preserving maps on quantum states ✩ Kan He ∗ , Qing Yuan, Jinchuan Hou College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, 030024, P.R. China
a r t i c l e
i n f o
Article history: Received 9 June 2013 Accepted 7 November 2014 Available online 27 November 2014 Submitted by C.-K. Li MSC: 47B49 47N50 Keywords: Quantum entropy Quantum states Preservers
a b s t r a c t In the paper, we give a characterization of surjective maps on quantum states preserving quantum entropy of convex combinations. Let S(H) be the set of all quantum states (positive operators with the unit trace) on a complex Hilbert space H with dim H = n < ∞ and S(ρ) the quantum entropy of the quantum state ρ. For arbitrary a surjective map φ : S(H) → S(H), we claim that φ preserves quantum entropy of convex combinations of quantum states, i.e., satisfies S(tρ + (1 −t)σ) = S(tφ(ρ) +(1 −t)φ(σ)) for arbitrary ρ, σ ∈ S(H) and t ∈ [0, 1] if and only if there exists a unitary or anti-unitary operator U on H such that φ(ρ) = U ρU ∗ for all ρ ∈ S(H). © 2014 Elsevier Inc. All rights reserved.
1. Introduction In the mathematical framework of quantum information theory, quantum states are positive operators with trace 1 on a complex Hilbert space H and denote by S(H) the set of all quantum states on H, which is a convex subset of the space of trace-class operators T (H). If dim H = n < ∞, then T (H) is identical with B(H), i.e., the n × n complex matrix algebra. A pure state is a rank one state, i.e., a rank one projection. ✩ This work supported by Program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi and National Science Foundation of China (11201329, 11171249). * Corresponding author. E-mail addresses: [email protected] (K. He), [email protected] (J. Hou).
http://dx.doi.org/10.1016/j.laa.2014.11.013 0024-3795/© 2014 Elsevier Inc. All rights reserved.
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Denote by P1 (H) the set of all pure states. In quantum information, it is important to understand, characterize, and construct different classes of maps on quantum states. For instance, all quantum channels and quantum operations are completely positive linear maps; in quantum error correction, one has to construct the recovery map for a given channel; to study the entanglement of states, one constructs entanglement witnesses, which are special types of positive maps; see [15]. Therefore, it is helpful to know the characterizations of maps on quantum states leaving invariant some important subsets or quantum properties. Such questions have attracted the attention of many researchers (see Ref. [1–3,5–7,9–14] and their references). For example, Kadison shows that φ is a convex isomorphism from S(H) onto S(H), i.e. for any states ρ1 , ρ2 and any non-negative scalar p ≤ 1, φ satisfies φ pρ1 + (1 − p)ρ2 = pφ(ρ1 ) + (1 − p)φ(ρ2 ) if and only if there is a unitary or anti-unitary operator U on H such that φ(ρ) = U ρU ∗ for all ρ ∈ S(H) (see Ref. [2]). For a quantum state ρ, the quantum entropy of ρ is defined as follows: S(ρ) = − tr(ρ log2 ρ). Also quantum entropy is called von Neumman entropy, and plays an important role in the theory of quantum information (see Ref. [2,4,15,16]). Assume that {pi }ni=1 are eigenvalues of ρ ∈ S(H) with dim H = n < ∞ and the Shannon entropy H({pi }ni=1 ) = −Σni=1 pi log2 pi , where 0 log2 0 = 0 and 1 log2 1 = 0, then quantum entropy of ρ equals to the Shannon entropy H({pi }ni=1 ). Some important topics in quantum information is developed by quantum entropy. The relative entropy of quantum states ρ, σ is defined as follows S(ρσ) = tr(ρ log2 ρ) − tr(ρ log2 σ). Let i = 1, 2, . . . , m, given a probability distribution pi , for any collection ρi of quantum states, the corresponding Holevo bound is χ(pi , ρi ) = S Σi pi ρi − Σi pi S(ρi ). In [12], Molnár gives a characterization of bijective maps on quantum states preserving relative entropy. Furthermore, without assumption of bijectivity, Molnár [13] gives a characterization of general maps on quantum states preserving relative entropy. Molnár and Nagy [14] show that a map on quantum states preserving Holevo bound is implemented by a unitary or anti-unitary operator. Recalled that a quantum channel is a completely positive and trace preserving linear map. Li and Busch [9] give a characterization of quantum channels preserving entropy. A map φ : S(H) → S(H) preserves
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quantum entropy of convex combinations if φ satisfies for arbitrary ρ, σ ∈ S(H) and t ∈ [0, 1], S tρ + (1 − t)σ = S tφ(ρ) + (1 − t)φ(σ) . An open problem is to characterize general maps on quantum states preserving quantum entropy of convex combinations. In the paper, we will give a characterization of subjective maps on quantum states preserving quantum entropy of convex combinations. Let S(H) be the set of all quantum states on a complex Hilbert space H with dim H = n < ∞. For arbitrary a surjective map φ : S(H) → S(H), we show that φ preserves quantum entropy of convex combinations of quantum states if and only if there exists a unitary or anti-unitary operator U such that φ(ρ) = U ρU ∗ for all ρ ∈ S(H) (see Theorem 2.1). 2. Maps preserving quantum entropy of convex combinations of quantum states In the section, we will determine the structure of surjective maps on quantum states preserving quantum entropy of convex combinations. The following is our main result. Theorem 2.1. Let S(H) be the set of all quantum states on a complex Hilbert space H with dim H = n and 1 < n < ∞. For a surjective map φ : S(H) → S(H), the following statements are equivalent: (I) φ satisfies for arbitrary ρ, σ ∈ S(H) and t ∈ [0, 1], S tρ + (1 − t)σ = S tφ(ρ) + (1 − t)φ(σ) ;
(2.1)
(II) there exists a unitary or anti-unitary operator U such that φ(ρ) = U ρU ∗ for all ρ ∈ S(H). Before the proof of Theorem 2.1, we need the following lemmas. Lemma 2.2. (See Main theorem, [4].) Let H be a complex Hilbert space with dim H = n < ∞ and ρ, σ ∈ S(H), then ρ = U σU ∗ for some unitary operator U if and only if S(tρ + (1 − t) nI ) = S(tσ + (1 − t) nI ) for arbitrary t ∈ [0, 1]. Here let us introduce the majorization order, assume that two real vectors a (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ), we have there exist permutations π, ω {1, 2, . . . , n} → {1, 2, . . . , n} such that aπ(1) ≥ aπ(2) ≥ · · · ≥ aπ(n) and bω(1) ≥ bω(2) · · · ≥ bω(n) . We call a ≺ b if Σki=1 aπ(i) ≤ Σki=1 bω(i) for k = 1, 2, . . . , n and Σni=1 aπ(i) Σni=1 bω(i) . Write a↓ = (aπ(1) , aπ(2) , . . . , aπ(n) ) and b↓ = (bω(1) , bω(2) , . . . , bω(n) ).
= : ≥ =
Lemma 2.3. (See Lemma 2.3, [9].) With the above assumption, and ai ≥ 0, bi ≥ 0 for i = 1, 2, . . . , n. if a ≺ b and H(a) = H(b), then a↓ = b↓ .
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Lemma 2.4. (See Theorem 11.8, [15].) Let H be a complex Hilbert space with dim H = n < ∞ and ρ ∈ S(H), then S(ρ) = log2 n if and only if ρ = nI , where I is the identity on H. Lemma 2.5. (See Theorem 11.8, [15].) Quantum entropy is non-negative, i.e., S(ρ) ≥ 0 for arbitrary ρ ∈ S(H). S(ρ) = 0 if and only if ρ is a pure state. Lemma 2.6. (See Theorem 11.8, [15].) Let ρi s be quantum states, i = 1, 2, . . . , k and pi s are non-negative scalars with Σki=1 pi = 1, then S Σki=1 pi ρi ≤ H(pi ) + Σki=1 pi S(ρi ), where the equality holds true if and only if ρi ρj = 0, for all 1 ≤ i, j ≤ k. Lemma 2.7. (See Eq. (11.79), [15].) Suppose ρi s and pi s satisfy assumptions in Lemma 2.6, then S Σki=1 pi ρi ≥ Σki=1 pi S(ρi ), where the equality holds true if and only if ρi = ρj , 1 ≤ i, j ≤ k. Lemma 2.8. (See [15].) The quantum entropy is unitary similarity invariant, i.e., S(ρ) = S(U ρU ∗ ) for each ρ ∈ S(H) and any unitary operator U . In the following lemma, we give a sufficient and necessary condition of equivalence of quantum states associated with quantum entropy, which is a key to completing the proof of Theorem 2.1. Lemma 2.9. For ρ, σ ∈ S(H) with dim H = n < ∞, then the following statements are equivalent: (i) ρ = σ; (ii) S(tρ + (1 − t) nI ) = S(tσ + (1 − t) nI ) for arbitrary t ∈ [0, 1] and there exists a number t0 ∈ (0, 1) such that S t0 ρ + (1 − t0 )P = S t0 σ + (1 − t0 )P
(2.2)
for all pure states P ∈ P1 (H). Proof of Lemma 2.9. Checking (i) ⇒ (ii) is straightforward, so we will only deal with (ii) ⇒ (i).
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Since ρ, σ satisfy S(tρ + (1 − t) nI ) = S(tσ + (1 − t) nI ) for arbitrary t ∈ [0, 1], from Lemma 2.2, we have that there exists a unitary operator U such that ρ = U σU ∗ , i.e, ρ, σ has the same eigenvalues and their multiplies. So we assume that eigenvalues of ρ are λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0, and so are eigenvalues of σ. In order to prove that ρ = σ, it is enough to show that ρ and σ have the same eigenvector associated with each λi (i = 1, 2, . . . , n). Assume that the unit vector x1 is the eigenvector of ρ with respect to λ1 . Let P1 = x1 ⊗ x1 be a pure state, ρP1 = P1 ρ, ρx1 = λ1 x1 , we have that eigenvalues of t0 ρ +(1 −t0 )P1 are t0 λ1 + 1 − t0 , t0 λ2 , . . . , t0 λn and t0 λ1 + 1 − t0 ≥ t0 λ2 ≥ · · · ≥ t0 λn ≥ 0. By Corollary 4.3.3 and Theorem 4.3.4 in Pa182 in [8], then there exist non-negative scalars di (t0 ) (1 ≤ i ≤ n) dependent on t0 , satisfying 0 ≤ di (t0 ) ≤ 1 − t0 and Σni=1 di (t0 ) = 1 − t0 such that t0 λi + di (t0 ) (1 ≤ i ≤ n) are eigenvalues of t0 σ + (1 − t0 )P1 . Write λ t0 ρ + (1 − t0 )P1 = (t0 λ1 + 1 − t0 , t0 λ2 , . . . , t0 λn ) and λ t0 σ + (1 − t0 )P1 = t0 λ1 + d1 (t0 ), t0 λ2 + d2 (t0 ), . . . , t0 λn + dn (t0 ) . Since 0 ≤ di (t0 ) ≤ 1 − t0 and Σni=1 di (t0 ) = 1 − t0 , λ t0 σ + (1 − t0 )P1 ≺ λ t0 ρ + (1 − t0 )P1 . Furthermore, from Eq. (2.2) in (ii), we have that H λ t0 ρ + (1 − t0 )P1 = S t0 ρ + (1 − t0 )P1 = S t0 σ + (1 − t0 )P1 = H λ t0 σ + (1 − t0 )P1 . It follows from Lemma 2.3 that λ(t0 σ + (1 − t0 )P1 ) = λ↓ (t0 ρ + (1 − t0 )P1 ). So there exists a permutation τ : {1, 2, . . . , n} → {1, 2, . . . , n} such that t0 λ1 + 1 − t0 = t0 λτ (1) + dτ (1) (t0 ), t0 λj = t0 λτ (j) + dτ (j) (t0 ),
and
for j = 2, 3, . . . , n.
Since λ1 is the maximal eigenvalue and dτ (1) (t0 ) ≤ 1 − t0 , the above equation implies that λ1 = λτ (1) ,
dτ (1) (t0 ) = 1 − t0
and dτ (j) (t0 ) = 0 for j = 2, 3, . . . , n.
It follows that x1 is the eigenvector of σ with respect to λ1 . Now let H = [x1 ] ⊕ H2 , we have that ρ=
λ1 0
0 ρ2
and σ =
λ1 0
0 σ2
.
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Furthermore, let x2 be the eigenvector of σ with respect to λ2 , taking P = x2 ⊗ x2 in Eq. (2.2) again, similar to the above discussion, we have x2 is also the eigenvector of σ with respect to λ2 . Reduplicating the discussion, we have ρ, σ have the same eigenvector with respect to λi for each i. In sum, ρ and σ have the same eigenvalues and associated eigenvectors, i.e., ρ = σ. Completing the proof. 2 Next to prove Theorem 2.1. Proof of Theorem 2.1. (II) ⇒ (I) is straightforward, let us check (I) ⇒ (II). The proof will be divided into the following two cases: dim H > 2 and dim H = 2. Case 1. dim H > 2. The proof is divided to the following claims. Claim 1. S(ρ) = S(φ(ρ)) for all ρ ∈ S(H), φ( nI ) = directions.
I n
and φ preserves pure states in both
Taking t = 1 in Eq. (2.1), we have S(ρ) = S(φ(ρ)) for all ρ ∈ S(H). If ρ = nI , then by Lemma 2.4, S(ρ) = log2 n. It follows that S(φ(ρ)) = S(ρ) = log2 n. Applying Lemma 2.4 again, we have φ(ρ) = nI . If ρ is a pure state, by Lemma 2.5, S(ρ) = 0. It follows S(φ(ρ)) = 0. By Lemma 2.5 again, we have φ(ρ) is a pure state. One can deal with the converse case similarly. So φ preserves pure states in both directions. Claim 2. φ is injective. To show that φ an injective map, i.e., ρ = σ ⇒ φ(ρ) = φ(σ). It is equivalent to prove φ(ρ) = φ(σ) ⇒ ρ = σ. If φ(ρ) = φ(σ), by Lemma 2.7, we have for any t ∈ (0, 1), S(tφ(ρ) + (1 − t)φ(σ)) = tS(φ(ρ)) + (1 − t)S(φ(σ)). Since S(ρ) = S(φ(ρ)) and Eq. (2.1), we have for any t ∈ (0, 1), tS(ρ) + (1 − t)S(σ) = tS φ(ρ) + (1 − t)S φ(σ) = S tφ(ρ) + (1 − t)φ(σ) = S tρ + (1 − t)σ . By Lemma 2.7 again, we have that ρ = σ. Claim 3. φ preserves orthogonality in both directions, i.e., ρσ = 0 ⇔ φ(ρ)φ(σ) = 0 for all ρ, σ ∈ S(H). If ρσ = 0, by Lemma 2.6, we have for any t ∈ (0, 1), S(tρ + (1 − t)σ) = H(t, 1 − t) + tS(ρ) + (1 − t)S(σ). By Claim 1 and Eq. (2.1), we have
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H(t, 1 − t) + tS(ρ) + (1 − t)S(σ) = H(t, 1 − t) + tS φ(ρ) + (1 − t)S φ(σ) = S tφ(ρ) + (1 − t)φ(σ) = S tρ + (1 − t)σ . By Lemma 2.6 again, we have φ(ρ)φ(σ) = 0. Similar we can show φ(ρ)φ(σ) = 0 ⇒ ρσ = 0. So φ preserves orthogonality in both directions. Now φ : S(H) → S(H) is a bijective map and preserves pure states in both directions and their orthogonality in both directions. Claim 4. Theorem 2.1 holds true in Case 1. Since dim H > 2, such a map has been characterized by L. Molnár in [10], that is, there is a unitary or anti-unitary operator U on H such that φ(P ) = U P U ∗ for all pure states P . Let ψ(ρ) = U ∗ φ(ρ)U for every ρ ∈ S(H). ψ and φ have the same properties since quantum entropy is unitary similarity invariant, and ψ satisfies ψ(P ) = U ∗ U P U ∗ U = P for all pure states P ∈ P1 (H). In order to complete the proof, it suffices to show that ψ(ρ) = ρ for all ρ ∈ S(H). In fact, as ψ satisfies Eq. (2.1) and ψ( nI ) = nI , we have that, for any pure state P and t ∈ [0, 1], S tρ + (1 − t)P = S tψ(ρ) + (1 − t)ψ(P ) = S tψ(ρ) + (1 − t)P and I I I = S tψ(ρ) + (1 − t)ψ = S tψ(ρ) + (1 − t) . S tρ + (1 − t) n n n It follows from Lemma 2.9 that ψ(ρ) = ρ. Completing the proof in the case of dim H > 2. Case 2. dim H = 2. If dim H = 2, let us identify S(H) with the subset of 2 × 2 density matrices S(H) =
1 1+z 2 x − iy
x + iy 1−z
(x, y, z) ∈ R3 : x2 + y 2 + z 2 ≤ 1 .
Note that (x, y, z) satisfies x2 + y 2 + z 2 = 1 if and only if the corresponding matrix is a rank one projection. Let H be a complex Hilbert space with dim H = 2, we still have that S(ρ) = S(φ(ρ)) for all ρ ∈ S(H) and φ is injective. Furthermore, φ preserves orthogonality and pure states in both directions. Since S(ρ) = S(φ(ρ)) and dim H = 2, then ρ and φ(ρ) have the same eigenvalues for any ρ ∈ S(H), which are λ, 1 − λ. Since φ preserves pure states in both directions, we have that there is a unitary operator V such that
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φ(E11 ) = φ
1 0 0 0
=V
1 0 0 0
V ∗.
Let ψ(ρ) = V ∗ φ(ρ)V , ψ has the same property of φ and ψ
1 0 0 0
=
For any ρ ∈ S(H) and assume that ρ = ψ(ρ) = Taking σ =
1 0 00
1 2
1 2
1 + z x − iy
1 0
0 0
1+z x+iy x−iy 1−z
.
x + iy 1 − z
and .
in Eq. (2.1), we have that for any t ∈ [0, 1],
1 S tρ + (1 − t) 0
0 0
1 0 = S tψ(ρ) + (1 − t) . 0 0
Since qubit states with the same binary entropy have the same eigenvalues and eigenpolynomial, we have that λ2 − λ −
x2 + y 2 + (1 − z)2 2 1 − z t + t = λI − tρ + (1 − t)E11 4 2 = λI − tψ(ρ) + (1 − t)E11 = λ2 − λ −
x 2 + y 2 + (1 − z )2 2 1 − z t + t. 4 2
It follows that z = z and |x +iy| = |x +iy |, so there exists a function θ : S(H) → [0, 2π) such that eiθ(ρ) (x + iy) = x + iy for any ρ ∈ S(H).
a b1 1 1 Now for arbitrary ρ, σ, assume that ρ = b¯1 1−a and b1 = x1 +iy , , write a1 = 1+z 2 1 1 2
iθ a2 b2 a1 e 1 b1 x2 +iy2 1+z2 σ = b¯ 1−a , write a2 = 2 and b2 = , ψ(ρ) = , ψ(σ) = 2 e−iθ1 b¯1 1−a1 2 2
iθ2 a2 e b2 . Then by Eq. (2.1) again, S(tρ + (1 − t)σ) = S(tψ(ρ) + (1 − t)ψ(σ)). So −iθ2 ¯ e
b2 1−a2
tb1 + (1 − t)b2 ta1 + (1 − t)a2 S tb¯1 + (1 − t)b¯2 1 − (ta1 + (1 − t)a2 ) ta1 + (1 − t)a2 teiθ1 b1 + (1 − t)eiθ2 b2 . =S te−iθ1 b¯1 + (1 − t)e−iθ2 b¯2 1 − (ta1 + (1 − t)a2 )
So we have λ2 − λ + −(a1 − a2 )2 − (b1 − b2 )(b¯1 − b¯2 ) t2 + a1 − a2 − 2a1 a2 + 2(a1 )2 − b2 (b¯1 − b¯2 ) − b¯2 (b1 − b2 ) t + a2 (1 − a2 ) − |b2 |2
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= λI − tρ + (1 − t)σ = λI − tψ(ρ) + (1 − t)ψ(σ) = λ2 − λ + −(a1 − a2 )2 − eiθ1 b1 − eiθ2 b2 e−iθ1 b¯1 − e−iθ2 b¯2 t2 + a1 − a2 − 2a1 a2 + 2(a1 )2 − eiθ2 b2 e−iθ1 b¯1 − e−iθ2 b¯2 − e−iθ2 b¯2 eiθ1 b1 − eiθ2 b2 t + a2 (1 − a2 ) − |b2 |2 . It follows that (b1 − b2 )(b¯1 − b¯2 ) = eiθ1 b1 − eiθ2 b2 e−iθ1 b¯1 − e−iθ2 b¯2 . So b1 b¯2 ei(θ1 −θ2 ) + b2 b¯1 ei(θ2 −θ1 ) = b1 b¯2 + b2 b¯1 .
(2.3)
In Eq. (2.3), if b1 , b2 are nonzero real numbers, we have cos(θ1 − θ2 ) = 1
and
sin(θ1 − θ2 ) = 0.
Since θ2 , θ1 ∈ [0, 2π), θ2 − θ1 = 0. So θ2 = θ1 . By the arbitrarity of b1 , b2 , we have the following observation: θ(ρ) is a constant θ0 for ρ ∈ S(H) when b(ρ) is the nonzero real number, where ρ=
r b(ρ)
b(ρ) 1−r
.
Finally if b1 = x + iy with nonzero x, y and b2 is a nonzero real number in Eq. (2.3), we have x cos(θ1 − θ0 ) − x = y sin(θ1 − θ0 ). It follows that either cos(θ1 − θ0 ) = 1
and
sin(θ1 − θ0 ) = 0,
x2 − y 2 x2 + y 2
and
sin(θ1 − θ0 ) =
(2.4)
or cos(θ1 − θ0 ) = For ρ =
a b(ρ) ∈ ¯ 1−a b(ρ) i(θ(ρ)−θ0 )
means that e
−2xy . x2 + y 2
(2.5)
S(H), if Eq. (2.4) occurs, then ei(θ(ρ)−θ0 ) b(ρ) = b(ρ); Eq. (2.5) b(ρ) = ¯b(ρ). Next to show that either Eq. (2.4) holds true for all
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states ρ, or Eq. (2.5) holds true for all states ρ. Assume on the contrary that there exist states ρ, σ such that b(ρ) = b1 = x1 + iy1 and b(σ) = b2 = x2 + iy2 such that Eq. (2.4) holds for b1 and Eq. (2.5) holds for b2 . Without loss of generality, from the above observation, let yi = 0 for i = 1, 2. It follows from Eq. (2.3) that b1 b2 + b¯2 b¯1 = b1 b¯2 ei(θ1 −θ0 −(θ2 −θ0 )) + b2 b¯1 e−i(θ1 −θ0 −(θ2 −θ0 )) = b1 b¯2 + b2 b¯1 . This is a contradiction. In sum, if Eq. (2.4) holds true for all states ρ, we have a eiθ(ρ) b(ρ) = ψ(ρ) = 1−a e−iθ(ρ)¯b(ρ) e−iθ0 e−i(θ(ρ)−θ0 )¯b(ρ) i θ0 −i θ0 e 2 0 a b(ρ) e 2 0 θ0 θ0 = . ¯b(ρ) 1 − a 0 e−i 2 0 ei 2
a
∗
So ψ(ρ) = W ρW , where W =
ei
θ0 2
0
eiθ0 ei(θ(ρ)−θ0 ) b(ρ) 1−a
0 e−i
θ0 2
. If Eq. (2.5) holds true for all states ρ, we
have a eiθ(ρ) b(ρ) = 1−a e−iθ(ρ)¯b(ρ) e−iθ0 e−i(θ(ρ)−θ0 )¯b(ρ) i θ0 ¯b(ρ) e−i θ20 e 2 0 a 0 θ0 θ0 = . 0 e−i 2 b(ρ) 1 − a 0 ei 2
ψ(ρ) =
a
eiθ0 ei(θ(ρ)−θ0 ) b(ρ) 1−a
So ψ(ρ) = W ρT W ∗ , where ρT is the transpose of the matrix ρ and W = Let U = V W , we complete the proof. 2
ei
θ0 2
0
0 e−i
θ0 2
.
Acknowledgements Authors are very grateful to Professor Chi-Kwong Li for some suggestions that help to improve the presentation of the paper. References [1] E. Alfsen, F. Shultz, Unique decompositions, faces, and automorphisms of separable states, J. Math. Phys. 51 (2010) 052201. [2] I. Bengtsson, K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press, Cambridge, 2006. [3] S. Friedland, C.-K. Li, Y.-T. Poon, N.-S. Sze, The automorphism group of separable states in quantum information theory, J. Math. Phys. 52 (2011) 042203. [4] K. He, J.-C. Hou, M. Li, A von Neumann entropy condition of unitary equivalence of quantum states, Appl. Math. Lett. 25 (2012) 1153–1156. [5] K. He, J.-C. Hou, X.-F. Qi, Characterizing sequential isomorphisms on Hilbert space effect algebras, J. Phys. A 42 (2009) 345203. [6] K. He, J.-C. Hou, C.-K. Li, A geometric characterization of invertible quantum measurement maps, J. Funct. Anal. 264 (2013) 464–478.
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Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Entropy-preserving maps on quantum states ✩ Kan He ∗ , Qing Yuan, Jinchuan Hou College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, 030024, P.R. China
a r t i c l e
i n f o
Article history: Received 9 June 2013 Accepted 7 November 2014 Available online 27 November 2014 Submitted by C.-K. Li MSC: 47B49 47N50 Keywords: Quantum entropy Quantum states Preservers
a b s t r a c t In the paper, we give a characterization of surjective maps on quantum states preserving quantum entropy of convex combinations. Let S(H) be the set of all quantum states (positive operators with the unit trace) on a complex Hilbert space H with dim H = n < ∞ and S(ρ) the quantum entropy of the quantum state ρ. For arbitrary a surjective map φ : S(H) → S(H), we claim that φ preserves quantum entropy of convex combinations of quantum states, i.e., satisfies S(tρ + (1 −t)σ) = S(tφ(ρ) +(1 −t)φ(σ)) for arbitrary ρ, σ ∈ S(H) and t ∈ [0, 1] if and only if there exists a unitary or anti-unitary operator U on H such that φ(ρ) = U ρU ∗ for all ρ ∈ S(H). © 2014 Elsevier Inc. All rights reserved.
1. Introduction In the mathematical framework of quantum information theory, quantum states are positive operators with trace 1 on a complex Hilbert space H and denote by S(H) the set of all quantum states on H, which is a convex subset of the space of trace-class operators T (H). If dim H = n < ∞, then T (H) is identical with B(H), i.e., the n × n complex matrix algebra. A pure state is a rank one state, i.e., a rank one projection. ✩ This work supported by Program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi and National Science Foundation of China (11201329, 11171249). * Corresponding author. E-mail addresses: [email protected] (K. He), [email protected] (J. Hou).
http://dx.doi.org/10.1016/j.laa.2014.11.013 0024-3795/© 2014 Elsevier Inc. All rights reserved.
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Denote by P1 (H) the set of all pure states. In quantum information, it is important to understand, characterize, and construct different classes of maps on quantum states. For instance, all quantum channels and quantum operations are completely positive linear maps; in quantum error correction, one has to construct the recovery map for a given channel; to study the entanglement of states, one constructs entanglement witnesses, which are special types of positive maps; see [15]. Therefore, it is helpful to know the characterizations of maps on quantum states leaving invariant some important subsets or quantum properties. Such questions have attracted the attention of many researchers (see Ref. [1–3,5–7,9–14] and their references). For example, Kadison shows that φ is a convex isomorphism from S(H) onto S(H), i.e. for any states ρ1 , ρ2 and any non-negative scalar p ≤ 1, φ satisfies φ pρ1 + (1 − p)ρ2 = pφ(ρ1 ) + (1 − p)φ(ρ2 ) if and only if there is a unitary or anti-unitary operator U on H such that φ(ρ) = U ρU ∗ for all ρ ∈ S(H) (see Ref. [2]). For a quantum state ρ, the quantum entropy of ρ is defined as follows: S(ρ) = − tr(ρ log2 ρ). Also quantum entropy is called von Neumman entropy, and plays an important role in the theory of quantum information (see Ref. [2,4,15,16]). Assume that {pi }ni=1 are eigenvalues of ρ ∈ S(H) with dim H = n < ∞ and the Shannon entropy H({pi }ni=1 ) = −Σni=1 pi log2 pi , where 0 log2 0 = 0 and 1 log2 1 = 0, then quantum entropy of ρ equals to the Shannon entropy H({pi }ni=1 ). Some important topics in quantum information is developed by quantum entropy. The relative entropy of quantum states ρ, σ is defined as follows S(ρσ) = tr(ρ log2 ρ) − tr(ρ log2 σ). Let i = 1, 2, . . . , m, given a probability distribution pi , for any collection ρi of quantum states, the corresponding Holevo bound is χ(pi , ρi ) = S Σi pi ρi − Σi pi S(ρi ). In [12], Molnár gives a characterization of bijective maps on quantum states preserving relative entropy. Furthermore, without assumption of bijectivity, Molnár [13] gives a characterization of general maps on quantum states preserving relative entropy. Molnár and Nagy [14] show that a map on quantum states preserving Holevo bound is implemented by a unitary or anti-unitary operator. Recalled that a quantum channel is a completely positive and trace preserving linear map. Li and Busch [9] give a characterization of quantum channels preserving entropy. A map φ : S(H) → S(H) preserves
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quantum entropy of convex combinations if φ satisfies for arbitrary ρ, σ ∈ S(H) and t ∈ [0, 1], S tρ + (1 − t)σ = S tφ(ρ) + (1 − t)φ(σ) . An open problem is to characterize general maps on quantum states preserving quantum entropy of convex combinations. In the paper, we will give a characterization of subjective maps on quantum states preserving quantum entropy of convex combinations. Let S(H) be the set of all quantum states on a complex Hilbert space H with dim H = n < ∞. For arbitrary a surjective map φ : S(H) → S(H), we show that φ preserves quantum entropy of convex combinations of quantum states if and only if there exists a unitary or anti-unitary operator U such that φ(ρ) = U ρU ∗ for all ρ ∈ S(H) (see Theorem 2.1). 2. Maps preserving quantum entropy of convex combinations of quantum states In the section, we will determine the structure of surjective maps on quantum states preserving quantum entropy of convex combinations. The following is our main result. Theorem 2.1. Let S(H) be the set of all quantum states on a complex Hilbert space H with dim H = n and 1 < n < ∞. For a surjective map φ : S(H) → S(H), the following statements are equivalent: (I) φ satisfies for arbitrary ρ, σ ∈ S(H) and t ∈ [0, 1], S tρ + (1 − t)σ = S tφ(ρ) + (1 − t)φ(σ) ;
(2.1)
(II) there exists a unitary or anti-unitary operator U such that φ(ρ) = U ρU ∗ for all ρ ∈ S(H). Before the proof of Theorem 2.1, we need the following lemmas. Lemma 2.2. (See Main theorem, [4].) Let H be a complex Hilbert space with dim H = n < ∞ and ρ, σ ∈ S(H), then ρ = U σU ∗ for some unitary operator U if and only if S(tρ + (1 − t) nI ) = S(tσ + (1 − t) nI ) for arbitrary t ∈ [0, 1]. Here let us introduce the majorization order, assume that two real vectors a (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ), we have there exist permutations π, ω {1, 2, . . . , n} → {1, 2, . . . , n} such that aπ(1) ≥ aπ(2) ≥ · · · ≥ aπ(n) and bω(1) ≥ bω(2) · · · ≥ bω(n) . We call a ≺ b if Σki=1 aπ(i) ≤ Σki=1 bω(i) for k = 1, 2, . . . , n and Σni=1 aπ(i) Σni=1 bω(i) . Write a↓ = (aπ(1) , aπ(2) , . . . , aπ(n) ) and b↓ = (bω(1) , bω(2) , . . . , bω(n) ).
= : ≥ =
Lemma 2.3. (See Lemma 2.3, [9].) With the above assumption, and ai ≥ 0, bi ≥ 0 for i = 1, 2, . . . , n. if a ≺ b and H(a) = H(b), then a↓ = b↓ .
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Lemma 2.4. (See Theorem 11.8, [15].) Let H be a complex Hilbert space with dim H = n < ∞ and ρ ∈ S(H), then S(ρ) = log2 n if and only if ρ = nI , where I is the identity on H. Lemma 2.5. (See Theorem 11.8, [15].) Quantum entropy is non-negative, i.e., S(ρ) ≥ 0 for arbitrary ρ ∈ S(H). S(ρ) = 0 if and only if ρ is a pure state. Lemma 2.6. (See Theorem 11.8, [15].) Let ρi s be quantum states, i = 1, 2, . . . , k and pi s are non-negative scalars with Σki=1 pi = 1, then S Σki=1 pi ρi ≤ H(pi ) + Σki=1 pi S(ρi ), where the equality holds true if and only if ρi ρj = 0, for all 1 ≤ i, j ≤ k. Lemma 2.7. (See Eq. (11.79), [15].) Suppose ρi s and pi s satisfy assumptions in Lemma 2.6, then S Σki=1 pi ρi ≥ Σki=1 pi S(ρi ), where the equality holds true if and only if ρi = ρj , 1 ≤ i, j ≤ k. Lemma 2.8. (See [15].) The quantum entropy is unitary similarity invariant, i.e., S(ρ) = S(U ρU ∗ ) for each ρ ∈ S(H) and any unitary operator U . In the following lemma, we give a sufficient and necessary condition of equivalence of quantum states associated with quantum entropy, which is a key to completing the proof of Theorem 2.1. Lemma 2.9. For ρ, σ ∈ S(H) with dim H = n < ∞, then the following statements are equivalent: (i) ρ = σ; (ii) S(tρ + (1 − t) nI ) = S(tσ + (1 − t) nI ) for arbitrary t ∈ [0, 1] and there exists a number t0 ∈ (0, 1) such that S t0 ρ + (1 − t0 )P = S t0 σ + (1 − t0 )P
(2.2)
for all pure states P ∈ P1 (H). Proof of Lemma 2.9. Checking (i) ⇒ (ii) is straightforward, so we will only deal with (ii) ⇒ (i).
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Since ρ, σ satisfy S(tρ + (1 − t) nI ) = S(tσ + (1 − t) nI ) for arbitrary t ∈ [0, 1], from Lemma 2.2, we have that there exists a unitary operator U such that ρ = U σU ∗ , i.e, ρ, σ has the same eigenvalues and their multiplies. So we assume that eigenvalues of ρ are λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0, and so are eigenvalues of σ. In order to prove that ρ = σ, it is enough to show that ρ and σ have the same eigenvector associated with each λi (i = 1, 2, . . . , n). Assume that the unit vector x1 is the eigenvector of ρ with respect to λ1 . Let P1 = x1 ⊗ x1 be a pure state, ρP1 = P1 ρ, ρx1 = λ1 x1 , we have that eigenvalues of t0 ρ +(1 −t0 )P1 are t0 λ1 + 1 − t0 , t0 λ2 , . . . , t0 λn and t0 λ1 + 1 − t0 ≥ t0 λ2 ≥ · · · ≥ t0 λn ≥ 0. By Corollary 4.3.3 and Theorem 4.3.4 in Pa182 in [8], then there exist non-negative scalars di (t0 ) (1 ≤ i ≤ n) dependent on t0 , satisfying 0 ≤ di (t0 ) ≤ 1 − t0 and Σni=1 di (t0 ) = 1 − t0 such that t0 λi + di (t0 ) (1 ≤ i ≤ n) are eigenvalues of t0 σ + (1 − t0 )P1 . Write λ t0 ρ + (1 − t0 )P1 = (t0 λ1 + 1 − t0 , t0 λ2 , . . . , t0 λn ) and λ t0 σ + (1 − t0 )P1 = t0 λ1 + d1 (t0 ), t0 λ2 + d2 (t0 ), . . . , t0 λn + dn (t0 ) . Since 0 ≤ di (t0 ) ≤ 1 − t0 and Σni=1 di (t0 ) = 1 − t0 , λ t0 σ + (1 − t0 )P1 ≺ λ t0 ρ + (1 − t0 )P1 . Furthermore, from Eq. (2.2) in (ii), we have that H λ t0 ρ + (1 − t0 )P1 = S t0 ρ + (1 − t0 )P1 = S t0 σ + (1 − t0 )P1 = H λ t0 σ + (1 − t0 )P1 . It follows from Lemma 2.3 that λ(t0 σ + (1 − t0 )P1 ) = λ↓ (t0 ρ + (1 − t0 )P1 ). So there exists a permutation τ : {1, 2, . . . , n} → {1, 2, . . . , n} such that t0 λ1 + 1 − t0 = t0 λτ (1) + dτ (1) (t0 ), t0 λj = t0 λτ (j) + dτ (j) (t0 ),
and
for j = 2, 3, . . . , n.
Since λ1 is the maximal eigenvalue and dτ (1) (t0 ) ≤ 1 − t0 , the above equation implies that λ1 = λτ (1) ,
dτ (1) (t0 ) = 1 − t0
and dτ (j) (t0 ) = 0 for j = 2, 3, . . . , n.
It follows that x1 is the eigenvector of σ with respect to λ1 . Now let H = [x1 ] ⊕ H2 , we have that ρ=
λ1 0
0 ρ2
and σ =
λ1 0
0 σ2
.
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Furthermore, let x2 be the eigenvector of σ with respect to λ2 , taking P = x2 ⊗ x2 in Eq. (2.2) again, similar to the above discussion, we have x2 is also the eigenvector of σ with respect to λ2 . Reduplicating the discussion, we have ρ, σ have the same eigenvector with respect to λi for each i. In sum, ρ and σ have the same eigenvalues and associated eigenvectors, i.e., ρ = σ. Completing the proof. 2 Next to prove Theorem 2.1. Proof of Theorem 2.1. (II) ⇒ (I) is straightforward, let us check (I) ⇒ (II). The proof will be divided into the following two cases: dim H > 2 and dim H = 2. Case 1. dim H > 2. The proof is divided to the following claims. Claim 1. S(ρ) = S(φ(ρ)) for all ρ ∈ S(H), φ( nI ) = directions.
I n
and φ preserves pure states in both
Taking t = 1 in Eq. (2.1), we have S(ρ) = S(φ(ρ)) for all ρ ∈ S(H). If ρ = nI , then by Lemma 2.4, S(ρ) = log2 n. It follows that S(φ(ρ)) = S(ρ) = log2 n. Applying Lemma 2.4 again, we have φ(ρ) = nI . If ρ is a pure state, by Lemma 2.5, S(ρ) = 0. It follows S(φ(ρ)) = 0. By Lemma 2.5 again, we have φ(ρ) is a pure state. One can deal with the converse case similarly. So φ preserves pure states in both directions. Claim 2. φ is injective. To show that φ an injective map, i.e., ρ = σ ⇒ φ(ρ) = φ(σ). It is equivalent to prove φ(ρ) = φ(σ) ⇒ ρ = σ. If φ(ρ) = φ(σ), by Lemma 2.7, we have for any t ∈ (0, 1), S(tφ(ρ) + (1 − t)φ(σ)) = tS(φ(ρ)) + (1 − t)S(φ(σ)). Since S(ρ) = S(φ(ρ)) and Eq. (2.1), we have for any t ∈ (0, 1), tS(ρ) + (1 − t)S(σ) = tS φ(ρ) + (1 − t)S φ(σ) = S tφ(ρ) + (1 − t)φ(σ) = S tρ + (1 − t)σ . By Lemma 2.7 again, we have that ρ = σ. Claim 3. φ preserves orthogonality in both directions, i.e., ρσ = 0 ⇔ φ(ρ)φ(σ) = 0 for all ρ, σ ∈ S(H). If ρσ = 0, by Lemma 2.6, we have for any t ∈ (0, 1), S(tρ + (1 − t)σ) = H(t, 1 − t) + tS(ρ) + (1 − t)S(σ). By Claim 1 and Eq. (2.1), we have
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H(t, 1 − t) + tS(ρ) + (1 − t)S(σ) = H(t, 1 − t) + tS φ(ρ) + (1 − t)S φ(σ) = S tφ(ρ) + (1 − t)φ(σ) = S tρ + (1 − t)σ . By Lemma 2.6 again, we have φ(ρ)φ(σ) = 0. Similar we can show φ(ρ)φ(σ) = 0 ⇒ ρσ = 0. So φ preserves orthogonality in both directions. Now φ : S(H) → S(H) is a bijective map and preserves pure states in both directions and their orthogonality in both directions. Claim 4. Theorem 2.1 holds true in Case 1. Since dim H > 2, such a map has been characterized by L. Molnár in [10], that is, there is a unitary or anti-unitary operator U on H such that φ(P ) = U P U ∗ for all pure states P . Let ψ(ρ) = U ∗ φ(ρ)U for every ρ ∈ S(H). ψ and φ have the same properties since quantum entropy is unitary similarity invariant, and ψ satisfies ψ(P ) = U ∗ U P U ∗ U = P for all pure states P ∈ P1 (H). In order to complete the proof, it suffices to show that ψ(ρ) = ρ for all ρ ∈ S(H). In fact, as ψ satisfies Eq. (2.1) and ψ( nI ) = nI , we have that, for any pure state P and t ∈ [0, 1], S tρ + (1 − t)P = S tψ(ρ) + (1 − t)ψ(P ) = S tψ(ρ) + (1 − t)P and I I I = S tψ(ρ) + (1 − t)ψ = S tψ(ρ) + (1 − t) . S tρ + (1 − t) n n n It follows from Lemma 2.9 that ψ(ρ) = ρ. Completing the proof in the case of dim H > 2. Case 2. dim H = 2. If dim H = 2, let us identify S(H) with the subset of 2 × 2 density matrices S(H) =
1 1+z 2 x − iy
x + iy 1−z
(x, y, z) ∈ R3 : x2 + y 2 + z 2 ≤ 1 .
Note that (x, y, z) satisfies x2 + y 2 + z 2 = 1 if and only if the corresponding matrix is a rank one projection. Let H be a complex Hilbert space with dim H = 2, we still have that S(ρ) = S(φ(ρ)) for all ρ ∈ S(H) and φ is injective. Furthermore, φ preserves orthogonality and pure states in both directions. Since S(ρ) = S(φ(ρ)) and dim H = 2, then ρ and φ(ρ) have the same eigenvalues for any ρ ∈ S(H), which are λ, 1 − λ. Since φ preserves pure states in both directions, we have that there is a unitary operator V such that
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φ(E11 ) = φ
1 0 0 0
=V
1 0 0 0
V ∗.
Let ψ(ρ) = V ∗ φ(ρ)V , ψ has the same property of φ and ψ
1 0 0 0
=
For any ρ ∈ S(H) and assume that ρ = ψ(ρ) = Taking σ =
1 0 00
1 2
1 2
1 + z x − iy
1 0
0 0
1+z x+iy x−iy 1−z
.
x + iy 1 − z
and .
in Eq. (2.1), we have that for any t ∈ [0, 1],
1 S tρ + (1 − t) 0
0 0
1 0 = S tψ(ρ) + (1 − t) . 0 0
Since qubit states with the same binary entropy have the same eigenvalues and eigenpolynomial, we have that λ2 − λ −
x2 + y 2 + (1 − z)2 2 1 − z t + t = λI − tρ + (1 − t)E11 4 2 = λI − tψ(ρ) + (1 − t)E11 = λ2 − λ −
x 2 + y 2 + (1 − z )2 2 1 − z t + t. 4 2
It follows that z = z and |x +iy| = |x +iy |, so there exists a function θ : S(H) → [0, 2π) such that eiθ(ρ) (x + iy) = x + iy for any ρ ∈ S(H).
a b1 1 1 Now for arbitrary ρ, σ, assume that ρ = b¯1 1−a and b1 = x1 +iy , , write a1 = 1+z 2 1 1 2
iθ a2 b2 a1 e 1 b1 x2 +iy2 1+z2 σ = b¯ 1−a , write a2 = 2 and b2 = , ψ(ρ) = , ψ(σ) = 2 e−iθ1 b¯1 1−a1 2 2
iθ2 a2 e b2 . Then by Eq. (2.1) again, S(tρ + (1 − t)σ) = S(tψ(ρ) + (1 − t)ψ(σ)). So −iθ2 ¯ e
b2 1−a2
tb1 + (1 − t)b2 ta1 + (1 − t)a2 S tb¯1 + (1 − t)b¯2 1 − (ta1 + (1 − t)a2 ) ta1 + (1 − t)a2 teiθ1 b1 + (1 − t)eiθ2 b2 . =S te−iθ1 b¯1 + (1 − t)e−iθ2 b¯2 1 − (ta1 + (1 − t)a2 )
So we have λ2 − λ + −(a1 − a2 )2 − (b1 − b2 )(b¯1 − b¯2 ) t2 + a1 − a2 − 2a1 a2 + 2(a1 )2 − b2 (b¯1 − b¯2 ) − b¯2 (b1 − b2 ) t + a2 (1 − a2 ) − |b2 |2
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= λI − tρ + (1 − t)σ = λI − tψ(ρ) + (1 − t)ψ(σ) = λ2 − λ + −(a1 − a2 )2 − eiθ1 b1 − eiθ2 b2 e−iθ1 b¯1 − e−iθ2 b¯2 t2 + a1 − a2 − 2a1 a2 + 2(a1 )2 − eiθ2 b2 e−iθ1 b¯1 − e−iθ2 b¯2 − e−iθ2 b¯2 eiθ1 b1 − eiθ2 b2 t + a2 (1 − a2 ) − |b2 |2 . It follows that (b1 − b2 )(b¯1 − b¯2 ) = eiθ1 b1 − eiθ2 b2 e−iθ1 b¯1 − e−iθ2 b¯2 . So b1 b¯2 ei(θ1 −θ2 ) + b2 b¯1 ei(θ2 −θ1 ) = b1 b¯2 + b2 b¯1 .
(2.3)
In Eq. (2.3), if b1 , b2 are nonzero real numbers, we have cos(θ1 − θ2 ) = 1
and
sin(θ1 − θ2 ) = 0.
Since θ2 , θ1 ∈ [0, 2π), θ2 − θ1 = 0. So θ2 = θ1 . By the arbitrarity of b1 , b2 , we have the following observation: θ(ρ) is a constant θ0 for ρ ∈ S(H) when b(ρ) is the nonzero real number, where ρ=
r b(ρ)
b(ρ) 1−r
.
Finally if b1 = x + iy with nonzero x, y and b2 is a nonzero real number in Eq. (2.3), we have x cos(θ1 − θ0 ) − x = y sin(θ1 − θ0 ). It follows that either cos(θ1 − θ0 ) = 1
and
sin(θ1 − θ0 ) = 0,
x2 − y 2 x2 + y 2
and
sin(θ1 − θ0 ) =
(2.4)
or cos(θ1 − θ0 ) = For ρ =
a b(ρ) ∈ ¯ 1−a b(ρ) i(θ(ρ)−θ0 )
means that e
−2xy . x2 + y 2
(2.5)
S(H), if Eq. (2.4) occurs, then ei(θ(ρ)−θ0 ) b(ρ) = b(ρ); Eq. (2.5) b(ρ) = ¯b(ρ). Next to show that either Eq. (2.4) holds true for all
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states ρ, or Eq. (2.5) holds true for all states ρ. Assume on the contrary that there exist states ρ, σ such that b(ρ) = b1 = x1 + iy1 and b(σ) = b2 = x2 + iy2 such that Eq. (2.4) holds for b1 and Eq. (2.5) holds for b2 . Without loss of generality, from the above observation, let yi = 0 for i = 1, 2. It follows from Eq. (2.3) that b1 b2 + b¯2 b¯1 = b1 b¯2 ei(θ1 −θ0 −(θ2 −θ0 )) + b2 b¯1 e−i(θ1 −θ0 −(θ2 −θ0 )) = b1 b¯2 + b2 b¯1 . This is a contradiction. In sum, if Eq. (2.4) holds true for all states ρ, we have a eiθ(ρ) b(ρ) = ψ(ρ) = 1−a e−iθ(ρ)¯b(ρ) e−iθ0 e−i(θ(ρ)−θ0 )¯b(ρ) i θ0 −i θ0 e 2 0 a b(ρ) e 2 0 θ0 θ0 = . ¯b(ρ) 1 − a 0 e−i 2 0 ei 2
a
∗
So ψ(ρ) = W ρW , where W =
ei
θ0 2
0
eiθ0 ei(θ(ρ)−θ0 ) b(ρ) 1−a
0 e−i
θ0 2
. If Eq. (2.5) holds true for all states ρ, we
have a eiθ(ρ) b(ρ) = 1−a e−iθ(ρ)¯b(ρ) e−iθ0 e−i(θ(ρ)−θ0 )¯b(ρ) i θ0 ¯b(ρ) e−i θ20 e 2 0 a 0 θ0 θ0 = . 0 e−i 2 b(ρ) 1 − a 0 ei 2
ψ(ρ) =
a
eiθ0 ei(θ(ρ)−θ0 ) b(ρ) 1−a
So ψ(ρ) = W ρT W ∗ , where ρT is the transpose of the matrix ρ and W = Let U = V W , we complete the proof. 2
ei
θ0 2
0
0 e−i
θ0 2
.
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