I have come across to read the article talked about the quantum discord, but I stumbled upon with the abstract saying:

The amount of extra information recovered by coherent interaction is quantified and directly linked with the discord consumed during encoding. No entanglement exists at any point of this experiment. Thus we introduce and demonstrate an operational method to use discord as a physical resource.

So how can it happens? Usually quantum computation are thought to consume entanglement to performing calculation and only pure states are involved. However, when quantum discord are discussed, the mixed state description make it uneasy to understand where 'discordance' the come from.

Are there explicit (mathematical) examples that demonstrate whether it is possible to have (a) zero entanglement, non-zero discord (b) non-zero entanglement, zero discord (c) non-zero entanglement, non-zero discord ?

In classical information theory, the (Shannon) entropy is define by $ H(A) = -\sum p_i \log(p_i) $ for a given probability distribution $p_i$. The mutual information can be defined as

$$\mathcal{I}(A:B) = H(A)+H(B)-H(A,B)$$ or $$\mathcal{J}(A:B) = H(A)+H(B|A)$$

where $H(A,B)$ and $H(B|A)$ are the entropy of joint system and conditional entropy respectively. These two expressions are equivalent in the classical case, and the second one can be derived from the first one using the Bayes rule  .

Similar expressions can be defined in the quantum system, with the entropy replaced by the Von Neumann entropy:

$$H(\rho)=-Tr(\rho \ln \rho)$$

But the two expressions $\mathcal{I}(A:B)$ and $\mathcal{J}(A:B)$ are not always equivalent in quantum system and the difference is defined as the quantum discord:

$$ \mathcal{D}(A:B) = \mathcal{I}(A:B) - \mathcal{J}(A:B)$$

Therefore, it can be treated as a measure for the non-classical correlation in quantum system.

I have come across an article which talks about quantum discord (Observing the operational significance of discord consumption. M. Gu et al. Nature Physics 8, 671–675 (2012) doi:10.1038/nphys2376), but I am struggling with the abstract, which says:

The amount of extra information recovered by coherent interaction is quantified and directly linked with the discord consumed during encoding. No entanglement exists at any point of this experiment. Thus we introduce and demonstrate an operational method to use discord as a physical resource.

So how can this happen? Usually quantum computations are thought to consume entanglement to perform calculations, and only pure states are involved. However, when quantum discord is discussed, the mixed state description makes it hard to understand where 'discordance' comes from.

Are there explicit (mathematical) examples that demonstrate whether it is possible to have

• (a) zero entanglement, non-zero discord;
• (b) non-zero entanglement, zero discord; and
• (c) non-zero entanglement, non-zero discord?

In classical information theory, the (Shannon) entropy is defined by $ H(A) = -\sum p_i \log(p_i) $ for a given probability distribution $p_i$. The mutual information can be defined as

$$\mathcal{I}(A:B) = H(A)+H(B)-H(A,B)$$ or $$\mathcal{J}(A:B) = H(A)+H(B|A),$$

where $H(A,B)$ and $H(B|A)$ are the entropy of joint system and the conditional entropy, respectively. These two expressions are equivalent in the classical case, and the second one can be derived from the first one using Bayes' rule.

Similar expressions can be defined in the quantum system, with the entropy replaced by the Von Neumann entropy, $$H(\rho)=-Tr(\rho \ln \rho).$$

However, the expressions $\mathcal{I}(A:B)$ and $\mathcal{J}(A:B)$ are not always equivalent in quantum systems and the difference is called the quantum discord:

$$ \mathcal{D}(A:B) = \mathcal{I}(A:B) - \mathcal{J}(A:B).$$

Therefore, it can be treated as a measure for the non-classical correlations in a quantum system.

I have come across to read the article talked about the quantum discord, but I stumbled upon with the abstract saying:

The amount of extra information recovered by coherent interaction is quantified and directly linked with the discord consumed during encoding. No entanglement exists at any point of this experiment. Thus we introduce and demonstrate an operational method to use discord as a physical resource.

So how can it happens? Usually quantum computation are thought to consume entanglement to performing calculation and only pure states are involved. However, when quantum discord are discussed, the mixed state description make it uneasy to understand where 'discordance' the come from.

Is there a explicit (mathematical) examples that demonstrate whether it is possible to have (a) zero entanglement, non-zero discord (b) non-zero entanglement, zero discord (c) non-zero entanglement, non-zero discord ?

In classical information theory, the (Shannon) entropy is define by $ H(A) = -\sum p_i \log(p_i) $ for a given probability distribution $p_i$. The mutual information can be defined as

$$\mathcal{I}(A:B) = H(A)+H(B)-H(A,B)$$ or $$\mathcal{J}(A:B) = H(A)+H(B|A)$$

where $H(A,B)$ and $H(B|A)$ are the entropy of joint system and conditional entropy respectively. These two expressions are equivalent in the classical case, and the second one can be derived from the first one using the Bayes rule .

Similar expressions can be defined in the quantum system, with the entropy replaced by the Von Neumann entropy:

$$H(\rho)=-Tr(\rho \ln \rho)$$

But the two expressions $\mathcal{I}(A:B)$ and $\mathcal{J}(A:B)$ are not always equivalent in quantum system and the difference is defined as the quantum discord:

$$ \mathcal{D}(A:B) = \mathcal{I}(A:B) - \mathcal{J}(A:B)$$

Therefore, it can be treated as a measure for the non-classical correlation in quantum system.

I have come across to read the article talked about the quantum discord, but I stumbled upon with the abstract saying:

The amount of extra information recovered by coherent interaction is quantified and directly linked with the discord consumed during encoding. No entanglement exists at any point of this experiment. Thus we introduce and demonstrate an operational method to use discord as a physical resource.

So how can it happens? Usually quantum computation are thought to consume entanglement to performing calculation and only pure states are involved. However, when quantum discord are discussed, the mixed state description make it uneasy to understand where 'discordance' the come from.

Are there explicit (mathematical) examples that demonstrate whether it is possible to have (a) zero entanglement, non-zero discord (b) non-zero entanglement, zero discord (c) non-zero entanglement, non-zero discord ?

In classical information theory, the (Shannon) entropy is define by $ H(A) = -\sum p_i \log(p_i) $ for a given probability distribution $p_i$. The mutual information can be defined as

$$\mathcal{I}(A:B) = H(A)+H(B)-H(A,B)$$ or $$\mathcal{J}(A:B) = H(A)+H(B|A)$$

where $H(A,B)$ and $H(B|A)$ are the entropy of joint system and conditional entropy respectively. These two expressions are equivalent in the classical case, and the second one can be derived from the first one using the Bayes rule .

Similar expressions can be defined in the quantum system, with the entropy replaced by the Von Neumann entropy:

$$H(\rho)=-Tr(\rho \ln \rho)$$

But the two expressions $\mathcal{I}(A:B)$ and $\mathcal{J}(A:B)$ are not always equivalent in quantum system and the difference is defined as the quantum discord:

$$ \mathcal{D}(A:B) = \mathcal{I}(A:B) - \mathcal{J}(A:B)$$

Therefore, it can be treated as a measure for the non-classical correlation in quantum system.