2.3
Summary of Chapter 2
In this chapter, I have discussed the notion of quantum correlations, starting by describing quantum entanglement, before introducing quantum correlations beyond entanglement with a focus on quantum discord. Quantum entanglement is known to have numerous applications in quantum information protocols, and recently, quantum discord has been shown to be important in many protocols involving mixed states. The relationship between entanglement and discord, particularly in mixed states, is of interest as the search for implementable quantum protocols continues. The Koashi-Winter relation provides a useful way to relate entanglement to discord and suggests the two are closely connected.
In the next two chapters I aim to advance understanding of discord and entanglement by studying their behaviour in multipartite mixed-state systems. In this way, I hope to shine a light on some interesting features of quantum correlations in mixed states. This is important because all realistic protocols run in noisy environments resulting in mixed states. By understanding all possible forms of correlation present, we can maximise our ability to utilise them effectively.
3
Discord Increase Under Local Loss
Entangled states have no local description and therefore an interaction between states is required to create entanglement. It is impossible to create entanglement using local oper- ations and classical communications [118]. It therefore follows that it is also impossible to increase entanglement using local operations. This is the reason that entanglement distri- bution requires transmission of a quantum state between different parties. Entanglement is invariant under local unitary operations [118], however nonunitary local operations can reduce entanglement by degrading the correlations, for example by dissipation.
Whereas entanglement can be seen as a consequence of the superposition principle, discord [158, 105] is related to the non-commutativity of observables and, in mixed states, can be created by local operations and classical communication. For example, discord can be created by correlated modulation of two quantum states as described in Section 1.5.3. Similarly to entanglement, quantum discord is invariant under local unitary operations [154], but it can change under local nonunitary operations. However, unlike entanglement, discord can increase under local operations [197, 34]. This can only occur under local operations on the measured part of the state; discord can never increase if the operation is on the unmeasured system. In addition, discord can even be created from a classically correlated state by local operations or even loss [42, 120]. This is surprising, as loss is normally considered to decrease the quality of a quantum state, but if loss can increase discord, perhaps loss can sometimes be made useful.
3.1
Discord increase with discrete variables
Streltsov et al. [197] described the conditions by which discord in a discrete variable system can increase under the action of a noisy channel, where a noisy channel is one that can be described by a completely positive trace preserving map. Consider the classically correlated state of two qubits
ρcc = 1 2|0 Aih0A| ⊗ |0Bih0B|+1 2|1 Aih1A| ⊗ |1Bih1B|. (3.1)
From this state, a quantum correlated state can be created by applying a local noisy channel to mode A. A local measurement on modeA, followed by a replacement, leads to the state ρ= 1 2|0 Aih0A| ⊗ |0Bih0B|+1 2|+ Aih+A| ⊗ |1Bih1B|, (3.2) where |+Ai = √1
2(|0i +|1i). After this operation, the states that make up mode A
form a non-orthogonal basis, which means there is now discord present in the state if the measurement is performed on mode A. The quantum channel needed to implement this change is the completely positive trace-preserving map
ρ= ΛA(ρcc) =E1ρE1†+E2ρE2† (3.3)
with local Kraus operators E1 =|0Aih0A|and E2 = |+Aih1A|. The state in Eq. (3.2) is
called quantum-classical because it has zero right discord, while the left discord is nonzero. This operation shows the ease with which discord can be created using local operations on a discrete variable state.
Streltsovet al. [197] went further by showing which channels could possibly cause an increase in discord. For qubits, they showed that a channel can cause discord increase only if it is neither semi-classical nor unital. A unital channel is one that maps the maximally mixed state 121onto itself, i.e., Λ(121) = 121. A semi-classical channel is one that maps all input states into a state that is diagonal in the same basis
Λsc(ρ) =
X
k
p(k)|kihk|. (3.4)
This is not to say that a channel that is neither semi-classical nor unital will always cause an increase in discord. It just means that there is at least one state that will undergo an increase in discord when passed through this channel. It can be easily seen that the channel in Eq. (3.3) is non-unital by applying it to the maximally mixed state ρ = 121. For larger finite-dimensional systems it has been shown that the only operations that can never create discord are local commutativity-preserving operations [119].
An example of a situation where dissipative loss results in the emergence of discord was shown by Ciccarello and Giovannetti [42]. Consider the state
ρ0 = 1 2|0 Aih0A| ⊗ |+Bih+B|+1 2|1 Aih1A| ⊗ |−Bih−B|, (3.5) where|±Ai= √1
2(|0i ± |1i). Clearly this is a classically correlated state since it is diagonal
in two local orthonormal bases. Now modeB is subject to a dissipative Markovian bath, while mode A remains untouched. This channel is described by the quantum map
Λ(ρ0) =E0ρE0†+E1ρE1†, (3.6)
where E0 = |0Bih0B|+
√
1−p|1Bih1B| and E1 =
√
p|0Bih1B|. If |1i is the state with one photon and |0i is the state with zero photons, then this can be thought of as a lossy channel with probability p that the photon will be lost. Application of this channel to
ρ0 results in a state with non-zero discord. This can be seen because the channel acts
differently on the two parts of mode B in ρ0. This results in a state that is no longer