Appendix A Solutions of Exercises - Springer978-3-319-45241...Appendix A Solutions of Exercises...

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Appendix A Solutions of Exercises Exercise 1.1 P ρ M (1) = 9 16 , P ρ M (2) = 1 16 , and P ρ M (3) = 3 8 . Exercise 1.2 A = E 1 E 2 , where E 1 = 1 2 11 11 and E 2 = 1 2 1 1 1 1 . Exercise 1.3 We can calculate it in two ways. (1) 1 × 1 2 + (1) × 1 2 = 0. (2) Tr ρ A = 0. Exercise 1.4 We can calculate it in two ways. (1) 1 2 × 1 2 + (1) 2 × 1 2 = 1. (2) Tr ρ A 2 = 1. Exercise 1.5 Since 1 2 1 4 1 4 1 2 = 3 4 1 2 1 2 1 2 1 2 + 1 4 1 2 1 2 1 2 1 2 , we have 1 2 1 4 1 4 1 2 1/2 = ( 3 4 ) 1/2 1 2 1 2 1 2 1 2 + ( 1 4 ) 1/2 1 2 1 2 1 2 1 2 = 2 3 1 2 1 2 1 2 1 2 + 2 1 2 1 2 1 2 1 2 = 1 3 + 1 1 3 1 1 3 1 1 3 + 1 . Hence, the POVM { M i } 3 i =1 is given as M 1 = 1 3 + 1 1 3 1 1 3 1 1 3 + 1 1 2 1 2 1 2 1 2 1 2 1 3 + 1 1 3 1 1 3 1 1 3 + 1 = 1 3 1 3 1 3 1 3 , © Springer International Publishing AG 2017 M. Hayashi, A Group Theoretic Approach to Quantum Information, DOI 10.1007/978-3-319-45241-8 205

Transcript of Appendix A Solutions of Exercises - Springer978-3-319-45241...Appendix A Solutions of Exercises...

  • Appendix ASolutions of Exercises

    Exercise 1.1PρM(1) = 916 , PρM(2) = 116 , and PρM(3) = 38 .

    Exercise 1.2

    A = E1 − E2, where E1 = 12(1 11 1

    )and E2 = 12

    (1 −1

    −1 1).

    Exercise 1.3We can calculate it in two ways. (1) 1 × 12 + (−1) × 12 = 0. (2) Tr ρA = 0.

    Exercise 1.4We can calculate it in two ways. (1) 12 × 12 + (−1)2 × 12 = 1. (2) Tr ρA2 = 1.

    Exercise 1.5

    Since

    ( 12

    14

    14

    12

    )= 34

    ( 12

    12

    12

    12

    )+ 14

    ( 12 − 12

    − 12 12

    ), we have

    ( 12

    14

    14

    12

    )−1/2= (3

    4)−1/2

    ( 12

    12

    12

    12

    )+ (1

    4)−1/2

    ( 12 − 12

    − 12 12

    )

    = 2√3

    ( 12

    12

    12

    12

    )+ 2

    ( 12 − 12

    − 12 12

    )=

    (1√3

    + 1 1√3

    − 11√3

    − 1 1√3

    + 1

    ).

    Hence, the POVM {Mi }3i=1 is given as

    M1 =(

    1√3

    + 1 1√3

    − 11√3

    − 1 1√3

    + 1

    )1

    2

    ( 12

    12

    12

    12

    ) ( 1√3

    + 1 1√3

    − 11√3

    − 1 1√3

    + 1

    )

    =( 1

    313

    13

    13

    ),

    © Springer International Publishing AG 2017M. Hayashi, A Group Theoretic Approach to Quantum Information,DOI 10.1007/978-3-319-45241-8

    205

  • 206 Appendix A: Solutions of Exercises

    M2 =(

    1√3

    + 1 1√3

    − 11√3

    − 1 1√3

    + 1

    )1

    4

    (1 00 0

    ) ( 1√3

    + 1 1√3

    − 11√3

    − 1 1√3

    + 1

    )

    =(

    13 + 12√3 − 16

    − 16 13 − 12√3

    ),

    M3 =(

    1√3

    + 1 1√3

    − 11√3

    − 1 1√3

    + 1

    )1

    4

    (0 00 1

    ) ( 1√3

    + 1 1√3

    − 11√3

    − 1 1√3

    + 1

    )

    =(

    13 − 12√3 − 16

    − 16 13 + 12√3

    ).

    Exercise 1.6

    Since XX† = 12(1 00 1

    ), the Schmidt rank is 1. The Schmidt coefficients are

    ( 1√2, 1√

    2).

    Exercise 1.7

    Y ⊗ Z |X〉〉A,B =∑k, j

    ∑k ′, j ′

    yk,k ′ z j, j ′xk ′, j ′ |k〉 ⊗ | j〉 = |Y X ZT 〉〉A,B . (A.1)

    Exercise 1.8

    TrB |X〉〉A,B A,B〈〈Y | = TrB∑k, j

    xk, j |k〉 ⊗ | j〉∑k ′, j ′

    yk ′, j ′ 〈k ′| ⊗ 〈 j ′| (A.2)

    =∑k ′,k, j

    xk, j yk ′, j |k〉〈k ′| = XY †. (A.3)

    Exercise 1.9√p|0〉|0〉 + √1 − p|1〉|1〉.

    Exercise 1.10Define the POVM Mi := X piρTi (XT )−1. Using Exercise 1.8, we have

    TrB Mi |X〉〉〈〈X | = TrB X piρTi (XT )−1|X〉〉〈〈X |= TrB |XX−1 piρi (X†)−1〉〉〈〈X | = XX−1 piρi (X†)−1X† = piρi .

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  • Appendix A: Solutions of Exercises 207

    Exercise 1.11Choose the system HB as the space spanned by the CONSs {|i〉}i . Then, we define|X〉〉 as ∑i √pi |xi 〉|i〉. Define the PVM E as Ei := |i〉〈i |. So, we can check thedesired condition.

    Exercise 1.12Since TrA 1d |Ui 〉〉〈〈Ui ||ρA ⊗ I = TrA(ρA ⊗ I ) 1d |Ui 〉〉〈〈Ui | = TrA 1d |ρAUi 〉〉〈〈Ui | =1dU

    †i ρAUi , we have Tr

    1d |Ui 〉〉〈〈Ui |ρA ⊗ ρmix,B = TrB 1dU †i ρAUiρmix,B = 1d2 .

    Exercise 2.1Let x1 and x2 be two elements of the domain of f . For an arbitrary number λ ∈ [0, 1],the linearity of f implies that f (λx1(1−λ)x2) = λ f (x1)+(1−λ) f (x2). Since g is aconcave function,wehave g(λ f (x1)+(1−λ) f (x2)) ≤ λg( f (x1))+(1−λ)g( f (x2)).Combining these two inequalities, we obtain g( f (λx1(1 − λ)x2)) ≤ λg( f (x1)) +(1 − λ)g( f (x2)).

    Exercise 2.2For a square matrix X , we choose two Hermitian matrices X1 and X2 such thatX = X1 + X2i . Then, we have |Tr Xρ|2 = (Tr X1ρ)2 + (Tr X2ρ)2. Exercise 2.1guarantees the convexity of the functions (Tr X1ρ)2 and (Tr X2ρ)2. Since a sum ofconvex functions is a convex function, we obtain the desired statement.

    Exercise 2.3We fix two elements x1, x2 ∈ X and λ ∈ (0, 1). For any � > 0, we choose a′ suchthat fa′(λx1 + (1 − λ)x2) ≥ supa fa(λx1 + (1 − λ)x2) − �. Since

    fa′(λx1 + (1 − λ)x2) ≤ λ fa′(x1) + (1 − λ) fa′(x2)≤ λ(sup

    afa(x1)) + (1 − λ)(sup

    afa(x2)),

    we have

    supa

    fa(λx1 + (1 − λ)x2) − � ≤ λ(supa

    fa(x1)) + (1 − λ)(supa

    fa(x2)).

    Since � > 0 is arbitrary, we have

    supa

    fa(λx1 + (1 − λ)x2) ≤ λ(supa

    fa(x1)) + (1 − λ)(supa

    fa(x2)),

    which is the desired statement.

    Exercise 2.4Let {pi } be an arbitrary distribution on {1, . . . , d}. So, we have∑ki=1 p↓i ≥ kd , whichimplies the desired statement.

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  • 208 Appendix A: Solutions of Exercises

    Exercise 2.5

    Iρ(A : B) = H(ρA) + H(ρB) − H(ρ)= − Tr ρ(log ρA) ⊗ IB − Tr ρIA ⊗ (log ρB) + Tr ρ log ρ= − Tr ρ log(ρA ⊗ ρB) + Tr ρ log ρ = D(ρ‖ρA ⊗ ρB).

    Exercise 2.6We focus on the eigenvectors of the density matrix ρ whose eigenvalue is not λ. Wemove such eigenvectors cyclically and obtain another density matrix. We take theaverage among such density matrices with equal weight. Such the averaged densitymatrix has von Neumann entropy (1 − λ) log dimH + h(λ). Hence, the concavityof von Neumann entropy yields the desired statement.

    Exercise 2.7The (2.44) can be shown as follows.

    D(ρ‖σA ⊗ σB) = Tr ρ(log ρ − logσA ⊗ σB)= Tr ρ(log ρ − (logσA) ⊗ IB − IA ⊗ (logσB))= Tr ρ(log ρ − (logσA) ⊗ IB − IA ⊗ (log ρB))

    + Tr ρ(IA ⊗ (log ρB) − IA ⊗ (logσB))= Tr ρ(log ρ − log(σA ⊗ ρB)) + Tr ρB(log ρB − logσB)= D(ρ‖σA ⊗ ρB) + D(ρB‖σB). (A.4)

    Exercise 2.8We denote the spectral decomposition E by {Ei }. We define a unitary representationf(θ1, . . . , θn) := ∑nj=1 eiθ j E j of the group Rn . The pinching ΛE with respect tothe PVM E satisfies that ΛE (ρ) = ρ. Due to Theorem 2.9, the image ΛE (ρ) of thepinching is the point closest to ρ among the invariant density matrices with respectto this representation in the sense of relative entropy. That is, Theorem 2.9 yields(2.45).

    Exercise 2.9Since 〈Φ|Fm ⊗ I |Φ〉 = Tr Fmρ, we have

    F2e (ρ,Λ) = 〈Φ|(Λ ⊗ id)(|Φ〉〈Φ|)|Φ〉= 〈Φ|

    ∑m

    (Fm ⊗ I )|Φ〉〈Φ|(Fm ⊗ I )†|Φ〉

    =∑m

    |〈Φ|Fm ⊗ I |Φ〉|2 =∑m

    |Tr Fmρ|2. (A.5)

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  • Appendix A: Solutions of Exercises 209

    Exercise 2.10Since 〈Φ|(Fm ⊗ I )|Φ〉 = Tr Fmρ, we have

    F2e (ρ,Λ) = 〈Φ|(Λ ⊗ id)(|Φ〉〈Φ|)|Φ〉= 〈Φ|

    ∑m

    (Fm ⊗ I )|Φ〉〈Φ|(Fm ⊗ I )†|Φ〉

    =∑m

    〈Φ|(Fm ⊗ I )|Φ〉〈Φ|(Fm ⊗ I )†|Φ〉 =∑m

    |Tr Fmρ|2.

    Exercise 2.11Based on Exercise 1.11, we choose the reference systemHR , a purification |X〉〉 of ρonHB , and a PVM E = {Ei }i onHR such that pi |xi 〉〈xi = TrB Ei |X〉〉〈〈X |. Hence,

    F2e (ρ,Λ) = 〈Φ|(Λ ⊗ id)(|Φ〉〈Φ|)|Φ〉≤ F2(ΛE (|Φ〉〈Φ|),ΛE ((id ⊗ Mi )(Λ ⊗ id)(|Φ〉〈Φ|)))=

    ∑j

    p j 〈x j |Λ(|x j 〉〈x j |)|x j 〉. (A.6)

    Exercise 2.12H1+s(ρx) is calculated to be 1s log((

    1+‖x‖2 )

    1+s + ( 1−‖x‖2 )1+s).Exercise 2.13To show 2.82, it is enough to show that

    D1+s(ρ‖ρA ⊗ σB) ≥ D1+s(ρ‖ρA ⊗ σ∗B(1 + s)) (A.7)

    for any state σB . This inequality is equivalent to

    esD1+s (ρ‖ρA⊗σB ) ≤ esD1+s (ρ‖ρA⊗σ∗B (1+s)) (A.8)

    for s ∈ (−1, 0) and

    esD1+s (ρ‖ρA⊗σB ) ≥ esD1+s (ρ‖ρA⊗σ∗B (1+s)) (A.9)

    for s ∈ (0,∞).(A.8) is equivalent to the following inequality

    Tr ρ1+sρ−sA ⊗ σ−sB = TrB(TrA ρ1+sρ−sA )σ−sB ≤ (TrB(TrA ρ1+sρ−sA )1

    1+s )1+s . (A.10)

    We apply matrix Hölder inequality (2.21) to the case when A = (TrA ρ1+sρ−sA ),B = σ−sB , p = 11+s > 0 and q = 1−s > 0 with s ∈ (−1, 0). Since the RHS of (2.21)equals the RHS of (A.10), we obtain (A.10).

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  • 210 Appendix A: Solutions of Exercises

    Similarly, (A.9) is equivalent to the following inequality

    TrB(TrA ρ1+sρ−sA )σ

    −sB ≥ (TrB(TrA ρ1+sρ−sA )

    11+s )1+s . (A.11)

    We obtain (A.11) by applying matrix reverse Hölder inequality (2.22) to the casewhen A = (TrA ρ1+sρ−sA ), B = σ−sB , p = 11+s > 0 and q = 1−s > 0 with s ∈ (0,∞).

    Exercise 3.1Due to Theorem 3.1, it is sufficient to show that TrB |Ψ 〉〈Ψ | � TrB |Φ〉〈Φ|. Thisrelation is trivial from the definition of majorization.

    Exercise 3.2Firstly, Bob applies the above measurement on the composite system HB ⊗ HB ′ ,and sends the outcome to Alice. Then, Alice applies unitary on the system HAdependently of the measurement outcome. Then, the final state of the compositesystem HA ⊗ HC is a maximally entangled state.

    Exercise 3.3The commutation relation 3.1 guarantees that

    (XF(t1) ⊗ · · · ⊗ XF(tr ))(ZF(s1) ⊗ · · · ⊗ ZF(sr )) 1qr/2

    ∑x∈Frq

    |x〉

    = ω∑r

    i=1 trsi tiF

    (ZF(s1) ⊗ · · · ⊗ ZF(sr ))(XF(t1) ⊗ · · · ⊗ XF(tr )) 1qr/2

    ∑x∈Frq

    |x〉

    = ω∑r

    i=1 trsi tiF

    (ZF(s1) ⊗ · · · ⊗ ZF(sr )) 1qr/2

    ∑x∈Frq

    |x〉.

    Hence, the definition (3.12) guarantees the desired statement.

    Exercise 3.4Using |+〉 := 1√

    2(|0〉 + |1〉) and |−〉 := 1√

    2(|0〉 − |1〉), we have |uζ0〉 = 1√2 (|0〉

    |+〉⊗(r−1) + |1〉|−〉⊗(r−1)) when ζ is a star graph.

    Exercise 3.5We have |uζ0〉 = 12 (|0, 0,+〉+|0, 1,−〉+|1, 0,+〉−|1, 1,−〉) when ζ is 1D Clustergraph with r = 3.

    Exercise 3.6We have |uζ0〉 = 12 (|0, 0,+〉+|0, 1,−〉+|1, 0,−〉−|1, 1,+〉) when ζ is Ring graphwith r = 3.

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  • Appendix A: Solutions of Exercises 211

    Exercise 3.7Using 3.25

    EG,2(|Φ〉〈Φ|) = − log max|a,b〉〈a,b|〈a, b|Φ〉〈Φ|a, b〉

    = − log max|a,b〉〈a,b| |〈a|∑i

    √pi |i〉〈i |b〉|2

    = − log |maxi

    √pi |2 = − logmax

    ipi .

    Exercise 3.8Consider the bipartite cut between the vertex connected to all of other vertex (1stvertex) and the remaining vertexes. Under this bipartite cut, we have EG,2:1,(2,...,r) =log 2. Then, we find that EG,2 is not greater than log 2 due to (3.26). We employ theconcrete form obtained in Exercise 3.4. This bound can be attained when we choosethe separable state |0〉|+〉⊗(r−1). So, we conclude that EG,2 = log 2.

    Exercise 3.9We employ the concrete form obtained in Exercise 3.5. Consider the bipartite cutbetween the first vertex and the remaining vertexes. Under this bipartite cut, we haveEG,2:1,(2,3) = log 2. Then, we find that EG,2 is not greater than log 2 due to (3.26).This bound can be attained when we choose the separable state |+, 0,+〉. So, weconclude that EG,2 = log 2.

    Exercise 3.10We employ the concrete form obtained in Exercise 3.6. To optimize the quantity|〈a, b, c|uζ0〉|, we can restrict our vector |a〉, |b〉, |c〉 to real vectors. So, without lossof generality, we can assume that |a〉 = cos θ|0〉 + sin θ|1〉. Then,

    〈a|uζ0〉 =1

    2(cos θ(|0,+〉 + |1,−〉) sin θ(|0,−〉 − |1,+〉))

    = 12(|0〉(cos θ|+〉 + sin θ|−〉)|1〉(cos θ|−〉 − sin θ|+〉).

    Since cos θ|+〉+ sin θ|−〉 is orthogonal to cos θ|−〉− sin θ|+〉, maxb,c〈a, b, c|uζ0〉 =12 . So, maxa,b,c〈a, b, c|uζ0〉 = 12 . That is, EG,2 = 2 log 2.

    Exercise 4.1Since themaximumvalue is attainedwhenM isMdλ|λ;λ〉〈λ;λ|, Example 4.1 guaranteesthat the desired maximum value is

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  • 212 Appendix A: Solutions of Exercises

    ∫ 10

    �dPλ1,λ,�

    d�d� =

    ∫ 10

    �d�

    2λ+12λ1

    d�d� =

    ∫ 10

    �2λ + 12λ1

    �2λ+12λ1

    −1d�

    =∫ 10

    2λ + 12λ1

    �2λ+12λ1 d� =

    2λ+12λ1

    2λ+12λ1

    + 1 =2λ + 1

    2λ + 2λ1 + 1 .

    Exercise 4.2Using (4.23), we have

    maxM̂

    DF (M̂) = dλd2λ

    = 2λ + 14λ + 1

    ∼= 12

    + 18λ

    . (A.12)

    Exercise 4.3Using (4.23), we have the same asymptotic characterization as (A.12).

    Exercise 4.4

    maxM̂

    DF (M̂) = dλd2λ

    = λZ2λZ

    = 12

    (A.13)

    Exercise 4.5

    maxM̂

    DF (M̂) = dλd2λ

    = (n + r − 1)!(r − 1)!(2n)!(r − 1)!n!(2n + r − 1)! =

    (n + r − 1)!(2n)!n!(2n + r − 1)!

    = n + r − 12n + r − 1 · · ·

    n + 12n + 1

    ∼= 12r−1

    (1 + r − 12n

    + r − 22n

    · · · + 12n

    ) = 12r−1

    (1 + r(r − 1)4n

    )

    Exercise 4.6

    maxM̂

    DF (M̂) = dλd2λ

    = λrZ

    (2λZ )r= 1

    2r

    Exercise 4.7 (4.24) can be calculated as follows.

    (2λ + 1)(et − 1)(et (λ+λ1+1) − e−t (λ+λ1))(2(λ + λ1) + 1)(et (λ1+1) − e−tλ1)(et (λ+1) − e−tλ)

    ∼= (2λ + 1)(1 − e−t )

    2(λ + λ1) + 1∼= 1 − e

    −t

    2(1 + 1

    4λ). (A.14)

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  • Appendix A: Solutions of Exercises 213

    Exercise 4.8The RHS (4.25) can be calculated in (A.14).

    Exercise 4.9The RHS of (4.26) equals 1−e

    −t2 .

    Exercise 4.10(4.37) shows that

    limn→∞

    1

    n|F−1[ϕn](−nx)|2 =

    ∣∣∣∫ 1

    −11√2e−i xydy

    ∣∣∣2 =∣∣∣√2 sin x

    x

    ∣∣∣2.Exercise 4.11cos π2n+2 ∼= 1 − 12 ( π2n+2 )2 + 124 ( π2n+2 )4 ∼= 1 − π

    2

    81n2 + π

    2

    41n3 .

    Exercise 4.121E + 14E3 .

    Exercise 5.1When a coset contains (a, 0, 0), it is {(a + b, b, b)}b∈Fq . The coset does not contain(c, 0, 0) for c �= a nor (0, a′, 0), (0, 0, a′). So, we can choose the representatives{J (x)}x∈Fq satisfying the requirement.

    Exercise 5.2Assume that the coset x contains (a, b, c). When b = c, JML(x) = (a − b, 0, 0).When a = c, JML(x) = (0, b − a, 0). When a = b, JML(x) = (0, 0, c − a). Whena > b > c, JML(x) = (a − c, b − c, 0). Other cases can be given in the same way.

    Exercise 5.3We have three patterns for correct decoding as follows.

    (1) The noise is (0, 0, 0). This probability is (1 − p)3.(2) The noise is (a, 0, 0), (0, a, 0), or (0, 0, a). This probability is 3(q − 1) pq−1

    (1 − p)2 = 3p(1 − p)2.(3) The noise is (a, b, 0), etc. Among this pattern, q2 − 1− 3(q − 1) = q2 − 3q + 2

    cases are correctable. Each case has probability ( pq−1 )2(1 − p) = p2(1−p)

    (q−1)2 . The

    probability of correctly decoding with this pattern is (q2 − 3q + 2) p2(1−p)(q−1)2 =

    (q−2)p2(1−p)q−1 .

    Totally, the whole correctly decoding probability is (1 − p)3 + 3p(1 − p)2 +(q−2)p2(1−p)

    q−1 = (1 − p)2(1 + 2p) + (q−2)p2(1−p)

    q−1 .

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  • 214 Appendix A: Solutions of Exercises

    Exercise 5.4Then, the element of CG,1 whose first, second, third, and fourth entries are zero islimited to (0, 0, 0, 0, 0, 0, 0)T . Hence, we can choose representatives whose first,second, third, and fourth entries are zero. Since F7q/CG,1 is the three-dimensionalspace, all of elements of F7q/CG,1 are given as [(0, 0, 0, 0, a, b, c)T ]a,b,c∈Fq .

    Exercise 5.5Any non-zero element of CG,1 has at least three non-zero entries. When x ∈ F7q hasonly one non-zero entry, any other element of [x] has at least two non-zero entries.Hence, we can choose decoder satisfying the required condition.

    Exercise 5.6We have at least the following two patterns for correct decoding as follows.

    (1) The noise is (0, 0, 0). This probability is (1 − p)7.(2) The noise has only one non-zero entry. This probability is 7(q−1) pq−1 (1− p)6 =

    7p(1 − p)6.Hence, the whole correctly decoding probability is larger than (1− p)7 +7p(1−

    p)6 = (1 − p)6(1 + 6p).

    Exercise 5.7Since all of column vectors ofG2 are given as linear combinations of column vectorsof G1, the code space CH,1 contains the code space CH,2.

    Exercise 5.8Since G1GT3 = 0, we have CG,1 ⊂ C⊥G,3. Since the dimension of CG,1 is 4 and thedimension of C⊥G,3 is 7 − 3 = 4, CG,1 = C⊥G,3. Since G2GT4 = 0, we can show thatCG,2 = C⊥G,4 in the same way.

    Exercise 5.9The desired statement can be shown in the same way as Exercise 5.6.

    Exercise 5.10The torsion condition is shown as follows. CG,2 ⊂ CG,1 = C⊥G,3.

    Exercise 5.11Due to Exercise 5.6 and Exercise 5.9, there exist classical decoders {sx }x∈Xr /C⊥G,3 and{t y}y∈Xr /C⊥G,2 such that

    1 −∑

    x∈Xr /C⊥G,3PX

    r1(sx ) ≤ 1 − (1 − p)6(1 + 6p) (A.15)

    1 −∑

    y∈Xr /C⊥G,2PX

    r2(t y) ≤ 1 − (1 − p)6(1 + 6p) (A.16)

    http://dx.doi.org/10.1007/978-3-319-45241-8_5http://dx.doi.org/10.1007/978-3-319-45241-8_5http://dx.doi.org/10.1007/978-3-319-45241-8_5

  • Appendix A: Solutions of Exercises 215

    because CG,1 = C⊥G,3 and CG,2 = C⊥G,4. Combining both evaluations, we obtain thedesired statement.

    Exercise 6.1(1, 1, 2, 2, 2, 2), (1, 2, 1, 2, 2, 2), (1, 2, 2, 1, 2, 2), (1, 2, 2, 2, 1, 2), (1, 2, 2, 2, 2, 1),(2, 1, 1, 2, 2, 2), (2, 1, 2, 1, 2, 2), (2, 1, 2, 2, 1, 2), (2, 1, 2, 2, 2, 1), (2, 2, 1, 1, 2, 2),(2, 2, 1, 2, 1, 2), (2, 2, 1, 2, 2, 1), (2, 2, 2, 1, 1, 2), (2, 2, 2, 1, 2, 1), (2, 2, 2, 2, 1, 1).

    Exercise 6.2When n = 1, the random variable n has covariance Ci, j . For general n, the randomvariable nn is the sample mean of the above random variable with n independenttrials. So, we can apply law of large number and the central limit theorem. So, weobtain Lemma 6.3.

    Exercise 6.3Since ∂H( p)

    ∂ pi= log pi − 1, we have

    ∑i, j

    ∂H( p)∂ pi

    Ci, j∂H( p)∂ p j

    =∑i ′

    (∂H( p)∂ pi ′

    )2 pi ′ −∑i, j

    ∂H( p)∂ pi

    pi p j∂H( p)∂ p j

    =∑i ′

    (log pi ′ − 1)2 pi ′ −∑i, j

    (log pi − 1)pi p j (log p j − 1)

    = (∑i ′

    pi ′ log p2i ′) − H( p) + 1 − (H( p) − 1)2

    = (∑i ′

    pi ′ log p2i ′) − H( p)2 = V ( p).

    Exercise 6.4The relation (6.6) and Lemma 6.6 guarantee that the random variables

    √n(H( nn ) −

    H( p)) asymptotically obeys the normal distribution with average 0 and variance∑i, j

    ∂H( p)∂ pi

    Ci, j∂H( p)∂ p j

    , which equals V ( p) as shown in (6.14). Further, as discussed

    in the end of Subsection 6.2.1, the differences√n(H( nn ) − H( p)) −

    √n( 1n log

    n!n! −

    H( p)) and√n(H( nn ) − H( p)) −

    √n( 1n log dim Vn − H( p)) converges to zero in

    probability. So, the remaining random variables asymptotically obey the normaldistribution with average 0 and variance V ( p).

    http://dx.doi.org/10.1007/978-3-319-45241-8_6http://dx.doi.org/10.1007/978-3-319-45241-8_6http://dx.doi.org/10.1007/978-3-319-45241-8_6http://dx.doi.org/10.1007/978-3-319-45241-8_6http://dx.doi.org/10.1007/978-3-319-45241-8_6

  • 216 Appendix A: Solutions of Exercises

    Exercise 6.5Recall (2.74) and (4.59) in [44] as

    dim U n+k2 ,

    n−k2

    (Sn) = k + 1n + 1

    (n + 1

    n+k2 + 1

    )

    χ n+k2 ,

    n−k2

    (ρ) = pn+k2 +1(1 − p) n−k2 − p n−k2 (1 − p) n+k2 +1

    (2p − 1) .

    Thus, we obtain

    Q[ρ⊗n](n + k

    2,n − k2

    )= dim U n+k

    2 ,n−k2

    (Sn) · χ n+k2 ,

    n−k2

    (ρ)

    = k + 1(2p − 1)(n + 1)

    (B

    (n + 1, p, n + k

    2+ 1

    )− B

    (n + 1, p, n − k

    2+ 1

    )).

    Exercise 6.6Assume that f satisfies the axiom of the distance as well as the additive conditionf (ρ⊗n,σ⊗n) = n f (ρ,σ). Any state ρU,n satisfies

    n f (ρ,σ) = f (ρ⊗n,σ⊗n) ≤ f (ρ⊗n, ρU,n) + f (σ⊗n, ρU,n).

    Hence, at least one of f (ρ⊗n, ρU,n) and f (σ⊗n, ρU,n) increases linearly. So, no stateis universal approximation with respect to f .

    Exercise 6.7

    Lemma 6.4 guarantees that the limit of the desired probability is 1√2π

    ∫ R2√V ( p)∞ e−

    x2

    2 dx .

    Exercise 6.8

    Corollary 6.1 guarantees that the limit of the desired probability is 1√2π

    ∫ R2√V ( p)∞ e−

    x2

    2 dx .

    Exercise 6.9The relation (6.59) shows that limn→∞ 1√n (log Tr Q[H( p)+ R2√n , n]− H( p)) ≤ R2.Conversely, we have

    limn→∞

    1√n(log Tr Q[H( p) + R2√

    n, n] − H( p))

    ≥ limn→∞

    1√n(log dimWn − H( p))

    ≥ limn→∞

    1√n(log dim Vn − H( p)) = R2,

    http://dx.doi.org/10.1007/978-3-319-45241-8_2http://dx.doi.org/10.1007/978-3-319-45241-8_4http://dx.doi.org/10.1007/978-3-319-45241-8_6http://dx.doi.org/10.1007/978-3-319-45241-8_6http://dx.doi.org/10.1007/978-3-319-45241-8_6

  • Appendix A: Solutions of Exercises 217

    where n is chosen to satisfy log H( nn ) = nH( p) +√nR2. The final equation fol-

    lows from (6.20) and (6.21). So, we obtain limn→∞ 1√n (log Tr Q[H( p) + R2√n , n] −H( p)) = R2.

    Corollary 6.1 with respect to√n(H( nn ) − H( p)) guarantees that

    limn→∞Tr ρ

    ⊗nP Q[H( p) +

    R2√n, n] = 1√

    ∫ R2√V (ρP )

    ∞e−

    x2

    2 dx . (A.17)

    Exercise 6.10φ2 is not injective because the map φ2 maps ((1, 1), (1, 2), (2, 1), (2, 2)) �→((0, 0), (0, 0, 0), (0, 0, 0), (0, 0, 0, 0)).

    Exercise 6.11When we replace (0, 1) by (1), φ1 is modified as (1, 2) �→ ((0), (1)). In this modi-fication, φn1 is invertible. So, the original φ

    n is also invertible. Hence, φ1 is uniquelydecodable.

    Exercise 6.12When φ2 is defined as (1, 2) �→ ((0), (1, 0)), φ1 · φ2 maps ((1, 1), (1, 2), (2, 1),(2, 2)) �→ ((0, 0), (0, 1, 0), (0, 1, 0), (0, 1, 1, 0)). So, φ1 · φ2 is not invertible.

    Exercise 6.13For any i1 and i2 �= j2, we have φ1(i1)φ2(i2) �= φ1(i1)φ2( j2). Also, for any i1 �= j1and i2, we have φ1(i1)φ2(i2) �= φ1( j1)φ2(i2). Similarly, for any i1 �= j1 and i2 �= j2,we have φ1(i1)φ2(i2) �= φ1( j1)φ2( j2). So, φ1 · φ2 is a prefix code.

    Exercise 6.14(1, 2, 3, 4) �→ ((1, 0), (1, 1, 0), (1, 1, 1), (0)).

    Exercise 6.15We list the sequences by grouping 6 sequences together as follows.

    ((1, 1), (1, 2), (2, 1), (2, 2), (2, 2)), ((1, 1), (1, 2), (2, 2), (2, 1), (2, 2)),((1, 1), (1, 2), (2, 2), (2, 2), (2, 1)), ((1, 2), (1, 1), (2, 1), (2, 2), (2, 2)),((1, 2), (1, 1), (2, 2), (2, 1), (2, 2)), ((1, 2), (1, 1), (2, 2), (2, 2), (2, 1)),

    ((1, 1), (2, 1), (1, 2), (2, 2), (2, 2)), ((1, 1), (2, 2), (1, 2), (2, 1), (2, 2)),((1, 1), (2, 2), (1, 2), (2, 2), (2, 1)), ((1, 2), (2, 1), (1, 1), (2, 2), (2, 2)),((1, 2), (2, 2), (1, 1), (2, 1), (2, 2)), ((1, 2), (2, 2), (1, 1), (2, 2), (2, 1)),

    ((1, 1), (2, 1), (2, 2), (1, 2), (2, 2)), ((1, 1), (2, 2), (2, 1), (1, 2), (2, 2)),((1, 1), (2, 2), (2, 2), (1, 2), (2, 1)), ((1, 2), (2, 1), (2, 2), (1, 1), (2, 2)),((1, 2), (2, 2), (2, 1), (1, 1), (2, 2)), ((1, 2), (2, 2), (2, 2), (1, 1), (2, 1)),

    ((1, 1), (2, 1), (2, 2), (2, 2), (1, 2)), ((1, 1), (2, 2), (2, 1), (2, 2), (1, 2)),((1, 1), (2, 2), (2, 2), (2, 1), (1, 2)), ((1, 2), (2, 1), (2, 2), (2, 2), (1, 1)),

    http://dx.doi.org/10.1007/978-3-319-45241-8_6http://dx.doi.org/10.1007/978-3-319-45241-8_6http://dx.doi.org/10.1007/978-3-319-45241-8_6

  • 218 Appendix A: Solutions of Exercises

    ((1, 2), (2, 2), (2, 1), (2, 2), (1, 1)), ((1, 2), (2, 2), (2, 2), (2, 1), (1, 1)),

    ((2, 1), (1, 1), (1, 2), (2, 2), (2, 2)), ((2, 2), (1, 1), (1, 2), (2, 1), (2, 2)),((2, 2), (1, 1), (1, 2), (2, 2), (2, 1)), ((2, 1), (1, 2), (1, 1), (2, 2), (2, 2)),((2, 2), (1, 2), (1, 1), (2, 1), (2, 2)), ((2, 2), (1, 2), (1, 1), (2, 2), (2, 1)),

    ((2, 1), (1, 1), (2, 2), (1, 2), (2, 2)), ((2, 2), (1, 1), (2, 1), (1, 2), (2, 2)),((2, 2), (1, 1), (2, 2), (1, 2), (2, 1)), ((2, 1), (1, 2), (2, 2), (1, 1), (2, 2)),((2, 2), (1, 2), (2, 1), (1, 1), (2, 2)), ((2, 2), (1, 2), (2, 2), (1, 1), (2, 1)),

    ((2, 1), (1, 1), (2, 2), (2, 2), (1, 2)), ((2, 2), (1, 1), (2, 1), (2, 2), (1, 2)),((2, 2), (1, 1), (2, 2), (2, 1), (1, 2)), ((2, 1), (1, 2), (2, 2), (2, 2), (1, 1)),((2, 2), (1, 2), (2, 1), (2, 2), (1, 1)), ((2, 2), (1, 2), (2, 2), (2, 1), (1, 1)),

    ((2, 1), (2, 2), (1, 1), (1, 2), (2, 2)), ((2, 2), (2, 1), (1, 1), (1, 2), (2, 2)),((2, 2), (2, 2), (1, 1), (1, 2), (2, 1)), ((2, 1), (2, 2), (1, 2), (1, 1), (2, 2)),((2, 2), (2, 1), (1, 2), (1, 1), (2, 2)), ((2, 2), (2, 2), (1, 2), (1, 1), (2, 1)),

    ((2, 1), (2, 2), (1, 1), (2, 2), (1, 2)), ((2, 2), (2, 1), (1, 1), (2, 2), (1, 2)),((2, 2), (2, 2), (1, 1), (2, 1), (1, 2)), ((2, 1), (2, 2), (1, 2), (2, 2), (1, 1)),((2, 2), (2, 1), (1, 2), (2, 2), (1, 1)), ((2, 2), (2, 2), (1, 2), (2, 1), (1, 1)),

    ((2, 1), (2, 2), (2, 2), (1, 1), (1, 2)), ((2, 2), (2, 1), (2, 2), (1, 1), (1, 2)),((2, 2), (2, 2), (2, 1), (1, 1), (1, 2)), ((2, 1), (2, 2), (2, 2), (1, 2), (1, 1)),((2, 2), (2, 1), (2, 2), (1, 2), (1, 1)), ((2, 2), (2, 2), (2, 1), (1, 2), (1, 1)).

  • References

    1. E. Bagan,M. Baig, R.Munoz-Tapia, Quantum reverse-engineering and reference frame align-ment without non-local correlations. Phys. Rev. A 70, 030301 (2004)

    2. E. Bagan, M.A. Ballester, R.D. Gill, A. Monras, R. Munoz-Tapia, Optimal full estimation ofqubit mixed states. Phys. Rev. A 73, 032301 (2006)

    3. M. Bellare, S. Tessaro, A. Vardy, Semantic security for the wiretap channel, in Proceedingsof the 32nd Annual Cryptology Conference, vol. 7417 (2012), pp. 294–311

    4. C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Concentrating partial entanglementby local operations. Phys. Rev. A 53, 2046 (1996)

    5. C.H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing, inProceedings of IEEE International Conference on Computers, Systems and Signal Processing(Bangalore, India, 1984), pp. 175–179

    6. C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Teleporting anunknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev.Lett. 70, 1895 (1993)

    7. C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Mixed-state entanglement andquantum error correction. Phys. Rev. A 54, 3824–3851 (1996)

    8. C.H. Bennett, S.J. Wiesner, Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881 (1992)

    9. R. Bhatia,Matrix Analysis (Springer, New York, 1996)10. I. Bjelakovic, J.-D. Deuschel, T. Kruger, R. Seiler, R. Siegmund-Schultze, A. Szkola, A

    quantum version of Sanov’s theorem. Commun. Math. Phys. 260(3), 659–671 (2005)11. N.A. Bogomolov,Minimaxmeasurements in a general statistical decision theory. Teor. Veroy-

    atnost. i Primenen. 26, 798–807 (1981) (English translation: Theory Probab. Appl. 26, 4,787–795 (1981))

    12. D.Bouwmeester,A.Ekert,A.Zeilinger (eds.),ThePhysics ofQuantum Information:QuantumCryptography (Quantum Teleportation, Quantum Computation, Springer, 2000)

    13. V. Bužek, R. Derka, S. Massar, Optimal quantum clocks. Phys. Rev. Lett. 82, 2207 (1999)14. A.R. Calderbank, E.M.Rains, P.W. Shor, N.J.A. Sloane, Quantum error correction and orthog-

    onal geometry. Phys. Rev. Lett. 78, 405–408 (1997)15. A.R. Calderbank, E.M. Rains, P.W. Shor, N.J.A. Sloane, Quantum error correction via codes

    over GF(4). IEEE Trans. Inform. Theor. 44, 1369–1387 (1998)16. A.R. Calderbank, P.W. Shor, Good quantum error correcting codes exist. Phys. Rev. A 54,

    1098–1105 (1996)17. J.L. Carter, M.N. Wegman, Universal classes of hash functions. J. Comput. Syst. Sci. 18,

    143–154 (1979)

    © Springer International Publishing AG 2017M. Hayashi, A Group Theoretic Approach to Quantum Information,DOI 10.1007/978-3-319-45241-8

    219

  • 220 References

    18. N.J. Cerf, A. Ipe, X. Rottenberg, Cloning of continuous quantum variables. Phys. Rev. Lett.85, 1754–1757 (2000)

    19. M.-D. Choi, Completely positive linear maps on complex matrices. Linear Algebra Appl.285–290 (1975)

    20. G. Chiribella, G.M. D’Ariano, M.F. Sacchi, Optimal estimation of group transformationsusing entanglement. Phys. Rev. A 72, 042338 (2005)

    21. G. Chiribella, G.M. D’Ariano, P. Perinotti, M.F. Sacchi, Efficient use of quantum resourcesfor the transmission of a reference frame. Phys. Rev. Lett. 93, 180503 (2004)

    22. M. Christandl, The Structure of Bipartite Quantum States Insights from Group Theory andCryptography, PhD thesis, University of Cambridge, 2006

    23. L. Collatz, U. Sinogowitz, Spektren endlicher Grafen. Abh. Math. Sem. Univ. Hamburg 21,63–77 (1957)

    24. H. Cramer,Mathematical Methods of Statistics (Princeton University Press, 1946)25. I. Csiszár, J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems

    (Academic Press, 1981). (Cambridge University Press, 2nd edn., 2011)26. M.J. Donald, M. Horodecki, Continuity of relative entropy of entanglement. Phys. Lett. A

    264, 257–260 (1999)27. M. Donald, M. Horodecki, O. Rudolph, The uniqueness theorem for entanglement measures.

    J. Math. Phys. 43, 4252–4272 (2002)28. R.G. Gallager, Information Theory and Reliable Communication (Wiley, 1968)29. N. Gisin, S. Massar, Optimal quantum cloning machines. Phys. Rev. Lett. 79, 2153 (1997)30. D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming

    bound. Phys. Rev. A 54, 1862–1868 (1996)31. D.Gottesman, Theory of fault-tolerant quantum computation. Phys. Rev. A 57, 127–37 (1998)32. D. Gottesman, Fault-tolerant quantum computation with higher-dimensional systems. Chaos,

    Solitons Fractals 10, 1749–1758 (1999)33. M. Grassl, M. Rotteler, T. Beth, Efficient quantum circuits for non-qubit quantum error-

    correcting codes. Int. J. Found. Comput. Sci. (IJFCS) 14(5), 757–775 (2003)34. M. Hamada, Teleportation and entanglement distillation in the presence of correlation among

    bipartite mixed states. Phys. Rev. A 68, 012301 (2003)35. M. Hamada, Notes on the fidelity of symplectic quantum error-correcting codes. Int. J. Quan-

    tum Inf. 1(4), 443–463 (2003)36. M. Hamada, Reliability of Calderbank-Shor-Steane codes and security of quantum key dis-

    tribution. J. Phys. A: Math. Gen. 37(34), 8303–8328 (2004)37. T.S. Han, K. Kobayashi, Mathematics of Information and Coding (American Mathematical

    Society, 2001)38. A.W. Harrow, M.A. Nielsen, How robust is a quantum gate in the presence of noise? Phys.

    Rev. A 68, 012308 (2003)39. T. Hashimoto, A. Hayashi, M. Hayashi, M. Horibe, Unitary-process discrimination with error

    margin. Phys. Rev. A 81, 062327 (2010)40. M.B. Hastings, A counterexample to additivity of minimum output entropy. Nat. Phys. 5, 255

    (2009)41. A. Hayashi, T. Hashimoto, M. Horibe, Extended quantum color coding. Phys. Rev. A 71,

    012326 (2005)42. M. Hayashi, Quantum Information: An Introduction (Springer, 2006) (Originally published

    in Japanese in 2004)43. M. Hayashi, S. Ishizaka, A. Kawachi, G. Kimura, T. Ogawa, Introduction to Quantum Infor-

    mation Science, Graduate Texts in Physics (Springer, 2015) (Originally published in Japanesein 2012)

    44. M. Hayashi, Group Representations for Quantum Theory, (Springer, 2016) (Originally pub-lished in Japanese in 2014)

    45. M. Hayashi, Quantum Information: An Introduction (Springer, 2006)46. M. Hayashi, Asymptotic estimation theory for a finite dimensional pure state model. J. Phys.

    A: Math. Gen. 31, 4633–4655 (1998). (It is also appeared as Chapter 23 of Asymptotic Theoryof Quantum Statistical Inference, M. Hayashi eds.)

  • References 221

    47. M. Hayashi, Optimal sequence of POVMs in the sense of Stein’s lemma in quantum hypoth-esis. J. Phys. A: Math. Gen. 35, 10759–10773 (2002)

    48. M. Hayashi, Exponents of quantum fixed-length pure state source coding. Phys. Rev. A 66,032321 (2002)

    49. M. Hayashi, K. Matsumoto, Asymptotic performance of optimal state estimation in quantumtwo level system. arXiv:quant-ph/0411073

    50. M. Hayashi, K. Matsumoto, Quantum universal variable-length source coding. Phys. Rev. A66, 022311 (2002)

    51. M. Hayashi, K. Matsumoto, Simple construction of quantum universal variable-length sourcecoding. Quant. Inf. Comput. 2, Special Issue, 519–529 (2002). arXiv:quant-ph/0209124

    52. M. Hayashi, Parallel treatment of estimation of SU(2) and phase estimation. Phys. Lett. A354(3), 183–189 (2006)

    53. M. Hayashi, Practical evaluation of security for quantum key distribution. Phys. Rev. A 74,022307 (2006)

    54. M. Hayashi, Universal approximation of multi-copy states and universal quantum losslessdata compression. Commun. Math. Phys. 293(1), 171–183 (2010)

    55. M. Hayashi, Universal coding for classical-quantum channel. Commun. Math. Phys. 289(3),1087–1098 (2009)

    56. M. Hayashi, Upper bounds of eavesdropper’s performances in finite-length code with thedecoy method. Phys. Rev. A 76, 012329 (2007)

    57. M.Hayashi, General theory for decoy-state quantum key distributionwith an arbitrary numberof intensities. New J. Phys. 9, 284 (2007)

    58. M.Hayashi, D.Markham,M.Murao,M.Owari, S. Virmani, Bounds onmultipartite entangledorthogonal state discrimination using local operations and classical communication. Phys.Rev. Lett. 96, 040501 (2006)

    59. M. Hayashi, D. Markham, M. Murao, M. Owari, S. Virmani, The geometric measure ofentanglement for a symmetric pure state with positive amplitudes. J. Math. Phys. 50, 122104(2009)

    60. M. Hayashi, Exponential decreasing rate of leaked information in universal random privacyamplification. IEEE Trans. Inf. Theor. 57, 3989–4001 (2011)

    61. M. Hayashi, T. Tsurumaru, Concise and tight security analysis of the Bennett-Brassard 1984protocol with finite key lengths. New J. Phys. 14, 093014 (2012)

    62. M. Hayashi, Fourier Analytic Approach to Quantum Estimation of Group Action. Commun.Math. Phys. 347(1), 3–82 (2009)

    63. M. Hayashi, Universal channel coding for general output alphabet (2016).arXiv:1502.02218v2

    64. P.M. Hayden, M. Horodecki, B.M. Terhal, The asymptotic entanglement cost of preparing aquantum state. J. Phys. A: Math. Gen. 34, 6891–6898 (2001)

    65. M. Hein, J. Eisert, H.J. Briegel, Multi-party entanglement in graph states. Phys. Rev. A 69,062311 (2004)

    66. M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, H.-J. Briegel, Entanglementin graph states and its applications, in Proceedings of the International School of Physics“Enrico Fermi”, vol. 162 ed. by G. Casati, D.L. Shepelyansky, P. Zoller, G. Benenti (IOPPress, 2006), pp. 115–218

    67. F. Hiai, Matrix analysis: matrix monotone functions, matrix means, and majorization. Inter-discip. Inf. Sci. 16, 139–248 (2010)

    68. F. Hiai, D. Petz, The proper formula for relative entropy and its asymptotics in quantumprobability. Commun. Math. Phys. 143, 99–114 (1991)

    69. T. Hiroshima, Majorization criterion for distillability of a bipartite quantum state. Phys. Rev.Lett. 91, 057902 (2003)

    70. A.S. Holevo, Covariant measurements and uncertainty relations. Rep. Math. Phys. 16, 385–400 (1979)

    71. A.S. Holevo, The capacity of the quantum channel with general signal states. IEEE Trans.Inf. Theor. 44, 269 (1998)

    http://arxiv.org/abs/quant-ph/0411073http://arxiv.org/abs/quant-ph/0209124http://arxiv.org/abs/1502.02218v2

  • 222 References

    72. A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Ams-terdam, 1982) Originally published in Russian (1980) (2nd Edition, Springer 2012)

    73. P. Horodecki, Separability criterion and inseparable mixed states with positive partial trans-position. Phys. Lett. A 232, 333 (1997)

    74. M. Horodecki, P. Horodecki, Reduction criterion of separability and limits for a class ofdistillation protocols. Phys. Rev. A 59, 4206 (1999)

    75. M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed states: necessary and suf-ficient conditions. Phys. Lett. A 223, 1–8 (1996)

    76. R. Howe, E.C. Tan, Non-Abelian Harmonic Analysis: Applications of SL(2, R) (Springer,1992)

    77. R.Hübener,M.Kleinmann, T.C.Wei, C.González-Guillén,O.Gühne, The geometricmeasureof entanglement for symmetric states. Phys. Rev. A 80, 032324 (2009)

    78. A. Jamiolkowski, Linear transformations which preserve trace and positive semidefinitenessof operators. Rep. Math. Phys. 3, 275 (1972)

    79. G.A. Jones, J.M. Jones, Information and Coding Theory (Springer, 2000)80. R. Jozsa, M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 81, 1714 (1998)81. A. Klappenecker,M. Rötteler, Beyond stabilizer codes II: Clifford codes. IEEE Trans. Inform.

    Theor. 48(8), 2396–2399 (2002)82. A. Klappenecker, P.K. Sarvepalli, Clifford code constructions of operator quantum error-

    correcting codes. IEEE Trans. Inform. Theor. 54(12), 5760–5765 (2008)83. A. Klappenecker, P.K. Sarvepalli, Encoding subsystem codes. Int. J. Adv. Secur. 2(2 & 3),

    142–155 (2009)84. E. Knill, Group representations, error bases and quantum codes (1996).

    arXiv:quant-ph/960804985. M. Koashi, N. Imoto, Compressibility of mixed-state signals. Phys. Rev. Lett. 87, 017902

    (2001)86. J. Von Korff, J. Kempe, Quantum advantage in transmitting a permutation. Phys. Rev. Lett.

    93(26), 260502 (2004)87. M. Keyl, R.F.Werner, Estimating the spectrum of a density operator. Phys. Rev. A 64, 052311

    (2001)88. K. Kraus, States, Effects and Operations (Springer, 1983)89. A. Luis, J. Perina, Optimum phase-shift estimation and the quantum description of the phase

    difference. Phys. Rev. A 54, 4564 (1996)90. D. Markham, A. Miyake, S. Virmani, Entanglement and local information access for graph

    states. New J. Phys. 9, 194 (2007)91. A.W.Marshall, I. Olkin, Inequalities: Theory of Majorization and Its Applications (Academic

    Press, 1979)92. R.Matsumoto, Conversion of a general quantum stabilizer code to an entanglement distillation

    protocol. J. Phys. Math. Gen. 36(29), 8113–8127 (2003)93. K. Matsumoto, M. Hayashi, Universal distortion-free entanglement concentration. Phys. Rev.

    A 75, 062338 (2007)94. K. Matsumoto, T. Shimono, A. Winter, Remarks on additivity of the Holevo channel capacity

    and of the entanglement of formation. Comm. Math. Phys. 246(3), 427–442 (2004)95. D. Mayers, Quantum key distribution and string oblivious transfer in noisy channels, in

    Advances in Cryptography Proceedings of Crypto’96 (1996), pp. 343–35796. D. Mayers, Unconditional security in quantum cryptography. J. Assoc. Comp. Mach. 48,

    351–406 (2001)97. N.D. Mermin, Quantum Computer Science: An Introduction (Cambridge University Press,

    2007)98. T. Miyadera, Information-disturbance theorem for mutually unbiased observables. Phys. Rev.

    A 73, 042317 (2006)99. M.A. Naı̌mark, Comptes rendus (Doklady) de l’Acadenie des science de l’URSS, 41. 9, 359

    (1943)

    http://arxiv.org/abs/quant-ph/9608049

  • References 223

    100. M.A. Nielsen, Conditions for a class of entanglement transformations. Phys. Rev. Lett. 83,436 (1999)

    101. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (CambridgeUniversity Press, Cambridge, 2000)

    102. M. Ohya, D. Petz, Quantum Entropy and Its Use (Springer, New York, 1993)103. M. Owari, M.B. Plenio, E.S. Polzik, A. Serafini, M.M. Wolf, Squeezing the limit: quantum

    benchmarks for the teleportation and storage of squeezed states. New J. Phys. 10, 113014(2008)

    104. M. Ozawa, On the noncommutative theory of statistical decisions. Res. Rep. Inf. Sci. A-74(1980)

    105. M. Ozawa, Quantum state reduction and the quantum Bayes principle, in Quantum Commu-nication, Computing, and Measurement, ed. by O. Hirota, A.S. Holevo, C.M. Caves (Plenum,New York, 1997), pp. 233–241

    106. M. Ozawa, An Operational approach to quantum state reduction. Ann. Phys. 259, 121–137(1997)

    107. M. Ozawa, Quantum state reduction: an operational approach. Fortschr. Phys. 46, 615–625(1998)

    108. S. Popescu, Bell’s inequalities versus teleportation: what is nonlocality? Phys. Rev. Lett. 72,797–799 (1994)

    109. M. Reed, B. Simon,Methods of Modern Mathematical Physics I: Functional Analysis (Aca-demic Press, 1980)

    110. R. Renner, Security of quantum key distribution, PhD thesis, ETH Zurich, 2005.arXiv:quant-ph/0512258

    111. B. Schumacher, Sending entanglement through noisy quantum channels. Phys. Rev. A 54,2614–2628 (1996)

    112. B. Schumacher, M.A. Nielsen, Quantum data processing and error correction. Phys. Rev. A54, 2629 (1996)

    113. B. Schumacher, M.D. Westmoreland, Sending classical information via noisy quantum chan-nels. Phys. Rev. A 56, 131 (1997)

    114. C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379– 423and 623–656 (1948)

    115. N. Sharma, N.A. Warsi, Fundamental bound on the reliability of quantum information trans-mission. Phys. Rev. Lett. 110(8), 080501 (2013)

    116. A. Shimony, Ann. N.Y. Acad. Sci. 755, 675 (1995)117. P. Shor, J. Preskill, Simple proof of security of the BB84 quantumkey distribution protocol.

    Phys. Rev. Lett. 85, 441–444 (2000)118. A.M. Steane, Multiple particle interference and quantum error correction. Proc. Roy. Soc.

    Lond. A 452, 2551–2577 (1996)119. W.F. Stinespring, Positive functions on C∗ algebras. Proc. Am. Math. Soc. 6, 211 (1955)120. D.R. Stinson, Universal hash families and the leftover hash lemma, and applications to cryp-

    tography and computing. J. Combin. Math. Combin. Comput. 42, 3–31 (2002)121. A. Streltsov, H. Kampermann, D. Bruß, Linking a distance measure of entanglement to its

    convex roof. New J. Phys. 12, 123004 (2010)122. T. Tsurumaru,M. Hayashi, Dual universality of hash functions and its applications to classical

    and quantum cryptography. IEEE Trans. Inf. Theor. 59(7), 4700–4717 (2013)123. V. Vedral, M.B. Plenio, Phys. Rev. A 57, 1619 (1998)124. G. Vidal, R. Tarrach, Robustness of entanglement. Phys. Rev. A 59, 141 (1999)125. X.B.Wang, T.Hiroshima,A. Tomita,M.Hayashi,Quantum InformationwithGaussian States.

    Phys. Rep.: Rev. Sect. Phys. Lett. 448, 1–111 (2007)126. S. Watanabe, T. Matsumoto, T. Uyematsu, Noise tolerance of the BB84 protocol with random

    privacy amplification. Int. J. Quantum Inf. 4(6), 935–946 (2006)127. T.-C. Wei, P.M. Goldbart, Phys. Rev. A 68, 042307 (2003)128. R.F. Werner, Optimal cloning of pure states. Phys. Rev. A 58, 1827 (1998)129. M.M. Wilde, Quantum Information Theory (Cambridge University Press, 2013)

    http://arxiv.org/abs/quant-ph/0512258

  • 224 References

    130. H.M. Wiseman, R.B. Killip, Adaptive single-shot phase measurements: a semiclassicalapproach. Phys. Rev. A 56, 944–957 (1997)

    131. A.D. Wyner, The wire-tap channel. Bell Syst. Tech. J. 54, 1355–1387 (1975)132. H. Zhu, L. Chen, M. Hayashi, Additivity and non-additivity of multipartite entanglement

    measures. New J. Phys. 12, 083002 (2010)

  • Index

    AAdditivity, 22, 179Antisymetric state, 60Approximate state cloning, 115Approximation of quantum state, 179Averaged channel, 112Averaged codeword length, 192, 194Averaged density matrix, 188

    BBasis verification, 159Bayes method, 115Bayesian method, 71, 72, 93Bayesian prior distribution, 71BB84 protocol, 158

    twirling-type modified, 161Bell diagonal state, 46Binary entropy, 29Bipartite system, 8, 46Bit error, 136Bound entangled state, 51Bures distance, 23

    CChoi-Jamiolkowski representation, 10–12,

    111–114Classical-quantum channel, 197Clifford code, 143Clifford group, 100, 101Code pair, 131Code space, 122Codeword, 192Coherent information, 28, 139Coherent state, 70, 76, 78, 79, 92, 117Coherent vector, 86

    Compact, 72Complete graph, 49Completely mixed state, 2, 28, 31Completely orthonormal system, 1Completely positivity (CP), 10Completeness, 1Complex conjugate, 5Complex conjugate representation, 58Computational basis, 159Concave function, 14Concavity, 20, 21, 26, 31, 152, 156, 157, 161Conjugate depolarizing channel, 113CONS, 1Contracting covariant channel, 114Control NOT (C-NOT), 49, 65, 66Convex combination, 4, 14Convex function, 14, 15Convex hull, 14Convex roof, 14, 67Convex set, 14Convexity, 20, 21, 24, 31, 115, 125Covariance condition, 114Covariant, 69, 71, 73, 74, 111CP map, 10CSS code, 135

    D1D Cluster state, 49, 652D Cluster state, 50Decoder, 132Decoding, 122Density matrix, 2Density operator, 2Depolarizing channel, 113Design

    S-, 104

    © Springer International Publishing AG 2017M. Hayashi, A Group Theoretic Approach to Quantum Information,DOI 10.1007/978-3-319-45241-8

    225

  • 226 Index

    Dicke state, 59Discrete Heisenberg group, 44, 140, 145Discrete Heisenberg representation, 43, 45,

    46, 60, 98, 112, 133, 139, 152, 158Discrete symplectic group, 46, 99Double universal2, 127Dual computational basis, 43, 45, 159

    EEmpirical distribution, 164Encoder, 132Encoding, 122Entangled, 4Entanglement, 5Entanglement-breaking channel, 12Entanglement concentration, 185Entanglement cost, 63Entanglement distillation, 62, 146, 184Entanglement fidelity, 28, 132–136, 138,

    143, 145, 149, 188Entanglement formation, 67Entanglement monotone, 51Entanglement relative entropy, 52Entanglement swapping, 42Entropy, 163Environment system, 11, 12, 68, 149–154,

    161Equivalent

    entangled state, 46Error correction, 159, 161Error estimation, 159Error-free variable-length code, 192Error function, 71Error probability, 76Estimate, 71, 73, 94, 95, 97, 165, 169–171,

    178Extended covariant state family, 116Extending covariant channel, 114Extremal point, 14

    FFidelity, 23, 24, 28, 76, 79, 85, 89, 91, 147,

    149, 178, 190–192Fixed-length data compression, 188Fundamental representation, 57

    GGenerated by matrix

    covariant POVM, 74Generating group, 143Generation of approximating state, 80

    Geometric measure, 52GHZ state, 50, 66Global logarithmic robustness, 53Graph, 48Graph state, 48, 65Group invariance, 71

    HHamming code, 123Heisenberg representation, 108Hilbert space, 1Hölder inequality, 17, 18

    matrix, 19Holevo capcity, 68

    IIndependent and identical trial, 163Indistilable condition, 50, 51Inertia group, 141Information leakage, 12Instrument, 13Irreducible decomposition, 74, 75, 88, 89,

    99, 102, 111, 113, 141, 146, 170

    JJeffreys prior distribution, 181Jensen inequality, 16, 191Joint convexity, 22, 24, 31, 52, 155–157

    KKraft inequality, 193Kraus representation, 11–13, 28, 40, 132,

    133

    LLarge deviation, 165Leaked information, 149, 152Local complementation, 48, 50, 65Local unitary, 46Locality condition, 40LOCC, 39Logarithmic inverse fidelity, 23Logarithmic inverse pseudo fidelity, 24, 52Logarithmic inverse square fidelity, 23, 24,

    30, 52Logarithmic robustness, 53

    MMajorization, 15

  • Index 227

    Majorization condition, 51Many-body system, 8Matrix concave, 182Matrix concave function, 18Matrix concavity, 195Matrix convex function, 18Matrix Hölder inequality, 19, 20Matrix monotone, 18Matrix reverse Hölder inequality, 19, 20Max-entropy, 21, 31Maximally entangled state, 7, 10, 41, 57, 62Maximum likelihood decoder, 131, 134Maximum likelihood decoding, 122McMillian inequality, 193Merit fuction, 71Metaplectic representation, 46, 99Min-entropy, 21, 31Mini-max, 181, 184Mini-max method, 71, 72, 93, 115Mixed state, 2Modified BB84 protocol, 158Multipartite system, 8Multi-party system, 8Mutual information, 25

    NNon-demolition measurement, 13Norm, 19n-positivity, 10

    OOne-way LOCC, 40One-way LOCC POVM, 40Optimal distinguishing probability, 98–101Origin, 70Orthogonal subgroup, 135

    PPartial trace, 5Partial transpose, 50Pauli channel, 113, 134, 138Permutation group, 60, 101Phase error, 136Pinching, 13Positive operator-valued measure (POVM),

    2Positive partial transpose (PPT), 50Positive semi definite, 2Positivity, 9PPT condition, 50PPT monotone, 61

    PPT quantum channel, 61PPT state, 50Prefix, 193Prefix code, 193Privacy amplification, 131, 155, 160Projection postulate, 13Projecton valued measure (PVM), 2Pseudo fidelity, 30Pseudo square distance, 22Pseuod fidelity, 23Public channel, 161Pure state, 2Purification, 6, 11, 12, 28, 67Puseudo fidelity, 82, 85, 87

    QQuantum channel, 9Quantum code, 132Quantum error correction, 132Quantum Hunt-Stein theorem, 72Quantum key distribution, 158Quantum prefix code, 195Quantum system, 1Quantum teleportation, 41Quantum transmission information, 26, 197

    RReduced density matrix, 5Reduction condition, 51Redundancy, 196Reference system, 6, 11Relative entropy, 23, 24, 30, 31, 52, 61, 163Relative max-entropy, 23, 24, 30, 31Relative min-entropy, 23, 24, 30, 31Relative Rényi entropy, 23, 24, 30, 31, 52Rényi entropy, 20, 21, 31Representation

    discrete Heisenberg, 43, 45, 46, 60, 98,112, 133, 139, 152, 158

    fundamental, 57Heisenberg, 108metaplectic, 46, 99tensor product, 60

    Reverse Hölder inequality, 17, 18matrix, 19

    Ring state, 49, 65

    SSatabilizer state, 64Schmidt basis, 6Schmidt coefficient, 6, 46

  • 228 Index

    Schmidt decomposition, 6, 185Schmidt rank, 6, 46Schur concave function, 15, 61Schur convex function, 15Schur duality, 163, 167Schur function, 61Secure random number, 158Self convex roof, 14Self orthogonal subgroup, 133Semi direct product, 145Separable, 40Separable condition, 50Separable POVM, 40Separable state, 4Shannon entropy, 21Stabilizer, 134Stabilizer code, 134Stabilizer state, 46, 48, 60, 64, 65Star graph, 49State family, 69State vector, 1Stinespring representation, 11, 12, 28, 30, 68Stirling formula, 164Strictly self-orthogonal subgroup, 46, 60, 64SU(1, 1), 70, 86su(1, 1), 76–78SU(2), 70, 89, 91su(2), 76–78, 85, 87–89, 91, 118SU(r), 167, 169, 182su(r), 85, 119Subsystem code, 143Symmetric function, 15System, 1

    TTensor product representation, 60Tensor product space, 4, 7, 170, 186, 196Tensor product state, 4, 8, 50, 56, 169, 173,

    196Tensor product system, 169

    Three-digit code, 122Torsion condition, 135TP-CP map, 10Trace norm distance, 23, 24, 30, 31, 157Trace-preserving, 9Transitive, 69Transmission information, 150Transposed complex conjugate matrix, 5Transposed matrix, 5Twirling, 112, 113, 148, 149, 152, 161Twirling type-modified BB84 protocol, 161Two-body system, 8Two-party system, 8Two-way LOCC, 40Two-way LOCC POVM, 40Type, 164Typical state reduction, 13

    Uu(r), 85, 119Unimodular, 74Uniquely decodable, 193, 195Unital, 12Universal, 163, 171, 179, 186, 192, 194, 197,

    198Universal approximation, 179

    VVandermonde determinant, 172, 178Varentropy, 21Variable-length data compression, 190, 194Vector state, 2von Neumann entropy, 20, 21, 23, 26, 31,

    156

    YYoung inequality, 17

    Appendix A Solutions of ExercisesAppendix ReferencesIndex