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42) are, by definition, members of Rd+ . 39) we obtain + p ˜ ↓ (Q + R) λ + p ˜ ↓ (Q) + λ ˜ ↓ (R) λ ≺w . 44) i=1 where we have removed the ordering of the vector on the left hand side. The term inside square brackets on the right hand side of Eq. 44) can be bounded from + + ˜ ↓ (Q) + λ ˜ ↓ (R) , yielding above by λ i i d ˜ + (Q + R) λ i p i=1 d ≤ ˜ ↓ (Q) λ i + ˜ ↓ (R) + λ i p + . 46) i=1 which shows that the pth root of the quantity on the right hand side of Eq. 45) satisfies the triangle inequality, we obtain 1/p d ˜ + (Q λ i + R) i=1 p 1/p d ≤ ˜ + (Q) λ i p 1/p d ˜ + (R) λ i + i=1 p .

Finally, due to the normalization condition i Ki† Ki ≤ IB [2], it follows that ||IA ⊗ Ki || ≤ 1 and IA ⊗ Ki† ≤ 1. Combined with Eq. 2. 2) and are there- fore entanglement monotones. A similar argument can be given for the monotonicity of the functions NpTA (ρ). Note that the negativity given by Eq. 33) is one member of this new family A of entanglement monotones since N (ρ) = MT 1 (ρ). Further, when we restrict our attention to the two-qubit case (d = 4) the partial transpose of ρ has at most one A negative eigenvalue, implying that MT 2 (ρ) also reduces to the negativity in this situation.

From this point of view, we are justified in asking information-theoretic questions about these states. 3. Quantum Information Theory approach. This theorem establishes the qubit as a resource for performing quantum communication by quantifying the number of qubits, transmitted from sender to receiver, that are asymptotically necessary and sufficient to faithfully transmit unknown pure quantum states randomly selected from an arbitrary, but known, source ensemble. Schumacher’s result generalizes Shannon’s noiseless channel coding theorem [25], which quantifies the minimum number of bits (in an asymptotic sense) required to reliably encode the output of a given classical information source, to the quantum case.

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