Vanishing Quantum Discord is Necessary and Sufficient for Completely Positive Maps

If Tr[AX] = 0 for any unitary and Hermitian matrix A, then X = 0.

“Lemma 2 If Tr[ AX ] = 0 for any unitary and Hermitian matrix A then X = 0 .”

Non-SLness can be written as Tr[ |j〉〈i| ⊗ I_B ρ_SB ] = 0, equivalently as 〈|i〉〈j| ⊗ I_B, ρ_SB〉 = 0.

“so that non-SLness can also be written as Tr[ | j 〉〈 i | ⊗ I B ρ SB ] = 0 , i.e., as 〈| i 〉〈 j | ⊗ I B , ρ SB 〉 = 0”

Tr[ |j〉〈i| ⊗ I_B ρ_SB ] = ̺_ij Tr[ φ_ij ].

“Further, note that Tr[ | j 〉〈 i | ⊗ I B ρ SB ] = ̺ ij Tr[ φ ij ]”

If ρ_SB(0) satisfies Eq. (6) with ρ(α)_B > 0, then Eq. (12) is already in the form of a CP map, with operation elements {√λ_αi E_αij}_αij.

“Now we simply note that if ρ SB (0) satisfies Eq. (6) with ρ ( α ) B > 0 (i.e., λ α j > 0 ), then Eq. (12) is already in the form of a CP map, with operation elements { √ λ α i E α ij } αij .”

If Tr[|0〉〈0| U† X U] = 0 for arbitrary U, then X = 0.

“However, U is arbitrary, so that 〈 ψ | X | ψ 〉 = 0 , ∀| ψ 〉 ( | ψ 〉 = U | 0 〉 ). This can only be true if X = 0 .”

For an SL-class initial state ρ_SB(0), the Hermitian map Φ_H: ρ_S(0) ↦ ρ_S(t) induced by the QDP (1) is completely positive iff (Φ^(α))_ij is either 0 or ρ_B^(α) for all (i,j) in the block-index set D^(α)_Φ.

“The Hermitian map Φ H : ρ S (0) ↦→ ρ S ( t ) induced by the QDP (1) is a CP map iff (Φ ( α ) ) ij = { 0 or ρ ( α ) B } ∀ ( i, j ) ∈ D ( α ) Φ :”

The QDP (1) is a linear, Hermitian map from ρ_S(0) ↦→ ρ_S(t) for any initial state ρ_SB(0).

“i.e., the QDP (1) is a linear, Hermitian map from ρ S (0) ↦→ ρ S ( t ) for any initial state ρ SB (0) .”

• cites [20] T. F. Jordan, Phys. Rev. A 71 , 034101 (2005).Not checked yet

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The matrix Φ ≡ [φ_ij] can be decomposed as Φ = ⊕_α Φ^(α), where (Φ^(α))_(i,j)∈D^(α)_Φ is either a constant φ^(α) or 0.

“Lemma 3 The matrix Φ ≡ [ φ ij ] can be decomposed as Φ = ⊕ α Φ ( α ) , where (Φ ( α ) ) ( i,j ) ∈D ( α ) Φ = φ ( α ) (a constant) or 0 .”

The total index set D_Φ for the initial state ρ_SB(0) splits into a union of disjoint index sets D(α)_Φ, so that Eqs. (5) and (6) are satisfied.

“Thus the total index set D Φ for the initial state ρ SB (0) splits into a union of disjoint index sets D ( α ) Φ , so that Eqs. (5) and (6) are satisfied”

If ρ_SB(0) is an SL-class state, then the quantum dynamical process (QDP) (1) is a linear, Hermitian map Φ_H: ρ_S(0) ↦ ρ_S(t).

“Theorem 2 If ρ SB (0) is an SL-class state then the QDP (1) is a linear, Hermitian map Φ H : ρ S (0) ↦→ ρ S ( t ) .”

The quantum dynamical process (QDP) is always a linear Hermitian map from the initial system state to the final system state if the initial system-bath state is in the special-linear (SL) class.

“We show that the QDP (1) is always a linear Hermitian map Φ H : ρ S (0) ↦→ ρ S ( t ) if ρ SB (0) in the SL-class.”

• Also stated in: § Vanishing quantum discord is necessary and sufficient for completely positive maps · p. 2 · plain

  “We now define a class of states we call 'special-linear' (SL) states for which the QDP (1) always results in a linear, Hermitian map.”

If ρ_SB(0) is an SL-class state then the QDP (1) is a linear map Φ_L : ρ_S(0) ↦→ ρ_S(t).

“Lemma 4 If ρ SB (0) is an SL-class state then the QDP (1) is a linear map Φ L : ρ S (0) ↦→ ρ S ( t ) .”

The Hermitian map Φ_H: ρ_S(0) ↦ ρ_S(t) is completely positive iff the initial system-bath state ρ_SB(0) has vanishing quantum discord (VQD), i.e., can be written as Eq. (7) with one-dimensional projectors onto the eigenvectors of ρ_S^(α).

“Theorem 3 The Hermitian map Φ H : ρ S (0) ↦→ ρ S ( t ) is a CP map iff the initial system-bath state ρ SB (0) has vanishing quantum discord (VQD), i.e., can be written as:”

The basic quantum mechanical transformation (1) is always representable as a Hermitian map between the initial and final system states.

“We have also shown that the basic quantum mechanical transformation (1) is always representable as a Hermitian map between the initial and final system states.”

Vanishing discord states are a subset of special-linear (SL) states.

“Vanishing discord states are a subset of SL states, and CP maps are a subset of Hermitian maps; we use the SL construction to prove our main result about CP maps.”

If all of φ_kk, φ_ll, φ_kl = 0, then φ_kk = φ_ll = φ_kl = φ_lk.

“Proposition 2 If all of φ kk , φ ll , φ kl = 0 then φ kk = φ ll = φ kl = φ lk .”

The quantum dynamical process (QDP) is always a linear Hermitian map from the initial system state to the final system state for arbitrary initial system-bath states.

“Here we settle both these questions by showing that the answer to the first is 'all', with the resulting map being Hermitian, and that the answer to the second is that CP maps arise exclusively from the class of separable states with vanishing quantum discord.”

• Also stated in: § Vanishing quantum discord is necessary and sufficient for completely positive maps · p. 1 · plain

  “We then argue that the restriction to the SLclass can be lifted, and that in fact the QDP (1) is always a linear Hermitian map, for arbitrary ρ SB (0) .”

• Also stated in: § Vanishing quantum discord is necessary and sufficient for completely positive maps · p. 2 · plain

  “We shall show that this is indeed the case (Theorem 2).”

The SL condition Tr[ φ_ij ] = 1 yields Tr[ |j〉〈i| ⊗ I_B ρ_SB ] = ̺_ij, which is not a constraint since ̺_ij is arbitrary.

“Note that, conversely, the SL condition Tr[ φ ij ] = 1 yields Tr[ | j 〉〈 i | ⊗ I B ρ SB ] = ̺ ij , which is not a constraint since ̺ ij is arbitrary.”

The quantum dynamical process (QDP) yields a completely positive (CP) map from the initial system state to the final system state if and only if the initial system-bath state has vanishing quantum discord, i.e., is purely classically correlated.

“Here we settle both these questions by showing that the answer to the first is 'all', with the resulting map being Hermitian, and that the answer to the second is that CP maps arise exclusively from the class of separable states with vanishing quantum discord.”

• Also stated in: § Vanishing quantum discord is necessary and sufficient for completely positive maps · p. 1 · boosted

  “We prove that the QDP yields a CP map ρ S (0) ↦→ ρ S ( t ) iff ρ SB (0) has vanishing 'quantum discord' [14], i.e., is purely classicaly correlated.”

• Also stated in: § Vanishing quantum discord is necessary and sufficient for completely positive maps · p. 2 · boosted

  “Here we show that vanishing quantum discord is not only sufficient but also necessary for the QDP to induce a CP map (Theorem 3).”

There exist precise initial state conditions for the ubiquitous class of completely positive maps.

“In particular, we have found the precise initial state conditions for the ubiquitous class of CP maps.”

The zero-measure subset of non-SL states does not spoil Theorem 2.

“Moreover, using a mapping from affine to linear maps [20], it is not hard to show that the zero-measure subset of non-SL states does not spoil Theorem 2”

• cites [20] T. F. Jordan, Phys. Rev. A 71 , 034101 (2005).Not checked yet

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If φ_kk = 0 or φ_ll = 0, then φ_kl = φ_lk = 0.

“Proposition 1 If φ kk = 0 or φ ll = 0 then φ kl = φ lk = 0 .”

The special-linear (SL) class of states is of full measure in the set of mixed bipartite states.

“we introduce a class of states we call 'special-linear' (SL), with the property of being of full measure in the set of mixed bipartite states.”

Completely positive maps arise for arbitrary joint unitaries even for certain nonfactorized initial conditions, provided the initial system-bath state is invariant under a complete set of orthogonal one-dimensional projections on the system, i.e., has vanishing quantum discord.

“A recent breakthrough due to Rodriguez et al . [12] shows that CP maps arise for arbitrary U SB even for certain nonfactorized initial conditions, namely provided the initial state ρ SB (0) is invariant under the application of a complete set of orthogonal one-dimensional projections on S , i.e., the state has vanishing quantum discord.”

• cites [12] C.A. Rodr´ ıguez et al ., J. Phys. A 41 , 205301 (2008). sourceNot checked yet

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The problem of identifying the general physical conditions under which completely positive maps are valid had remained open since it was first posed.

“For this reason it is perhaps surprising that the problem of identifying the general physical conditions under which CP maps are valid has remained open since it was first posed in a vigorous debate [5, 6].”

• cites [5] P. Pechukas, Phys. Rev. Lett. 73 , 1060 (1994). sourceNot checked yet

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• cites [6] R. Alicki, Phys. Rev. Lett. 75 , 3020 (1995). sourceNot checked yet

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Quantum subsystem dynamics is always a meaningful concept.

“This result establishes that quantum subsystem dynamics is always a meaningful concept.”

A map is Hermitian if and only if it can be written as the difference of two completely positive maps.

“It turns out that there is a tight connection between CP and Hermitian maps [9, 11]: a map is Hermitian iff it can be written as the difference of two CP maps.”

• cites [9] T.F. Jordan, A. Shaji, and E.C.G. Sudarshan, Phys. Rev. A 70 , 052110 (2004). sourceNot checked yet

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• cites [11] H. A. Carteret, D. R. Terno, and K. ˙ Zyczkowski, Phys. Rev. A 77 , 042113 (2008).Not checked yet

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Non-SL-class states lie in the hyperplane orthogonal to |i〉〈j| ⊗ I_B.

“and hence ρ SB must lie in the hyperplane orthogonal to | i 〉〈 j | ⊗ I B .”

A map Φ from n×n matrices to m×m matrices is linear if and only if it can be represented in an operator sum representation with left and right operation elements.

“Theorem 1 A map Φ : M n ↦→ M m (where M n is the space of n × n matrices) is linear iff it can be represented as”

• Also stated in: § Vanishing quantum discord is necessary and sufficient for completely positive maps · p. 1 · plain

  “where the 'left and right operation elements' { E α } and { E ′ α } are, respectively, m × n and n × m matrices.”

• cites [15] sourceNot checked yet

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• cites [16] R.D. Hill, Linear Algebra and its Applications 6 , 257 (1973). sourceNot checked yet

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For arbitrary initial system-bath states, the quantum dynamical process becomes a completely positive map if and only if the joint propagator is locally unitary, i.e., the effective system-bath interaction vanishes.

“However, it was recently shown [19] that the QDP (1) with arbitrary ρ SB (0) becomes a CP map iff a most restrictive condition is satisfied by U SB ( t ) , namely, it must be locally unitary: U SB ( t ) = U S ( t ) ⊗ U B ( t ) , i.e., the effective system-bath interaction must vanish.”

• cites [19] H. Hayashi, G. Kimura, Y. Ota, Phys. Rev. A 67 , 062109 (2003).Not checked yet

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