Entangled, mixed state with conditional entropy zero

In summary, "Entangled, mixed state with conditional entropy zero" refers to a quantum state that exhibits entanglement while having a conditional entropy of zero. This indicates that when one part of the system is measured, it provides complete information about the other part, despite the overall mixedness of the state. Such states challenge classical intuitions about information and entropy, highlighting unique properties of quantum systems and their correlations.
  • #1
greypilgrim
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Hi.
The classical (Shannon) conditional entropy is never negative. It can be written as ##H(Y|X)=H(X,Y)-H(X)## which allows for a quantum generalization using von Neumann entropy. In the case of entangled states, it can become negative.

I guess it should be possible to construct an entangled, mixed (bipartite) state where ##H(Y|X)## is exactly zero (though I don't know how to exactly do that). Does this have a specific meaning?
 
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  • #2
Conditional entropy in classical probability is always positive because the entropy of the total system is always greater than or equal to the entropy of its parts. This is no longer true in QM and it can be negative because you can have maximum knowledge of the whole system (it's in a pure state), but less than optimal knowledge of the parts (subsystems are in a mixed state). It vanishing is just the special case where the entropies happen to be equal, it has no additional special meaning.
 
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FAQ: Entangled, mixed state with conditional entropy zero

What is an entangled, mixed state with conditional entropy zero?

An entangled, mixed state with conditional entropy zero is a quantum state involving two or more subsystems where the total system is in a mixed state, but the conditional entropy of one subsystem given the other is zero. This implies that despite the mixed nature of the overall state, there is a perfect correlation between the subsystems.

How can a mixed state have conditional entropy zero?

Conditional entropy zero in a mixed state can occur when the mixed state is a classical mixture of pure entangled states. In such cases, knowledge of one subsystem allows for perfect prediction of the other, leading to zero conditional entropy, despite the overall state being mixed.

What is the significance of conditional entropy being zero in quantum information theory?

Conditional entropy being zero is significant because it indicates that the subsystems are maximally entangled in a way that perfect knowledge of one subsystem implies perfect knowledge of the other. This is a key resource for quantum information protocols such as quantum teleportation and superdense coding.

How is an entangled, mixed state with conditional entropy zero different from a pure entangled state?

An entangled, mixed state with conditional entropy zero differs from a pure entangled state in that the former is a statistical mixture of different pure states, while the latter is a single quantum state with no classical uncertainty. Despite this difference, the conditional entropy zero property ensures strong correlations similar to those in pure entangled states.

Can you provide an example of an entangled, mixed state with conditional entropy zero?

An example of an entangled, mixed state with conditional entropy zero is a classical mixture of Bell states. For instance, if a system is in a 50-50 mixture of the Bell states |Φ+⟩ and |Φ-⟩, the overall state is mixed, but measuring one qubit perfectly determines the state of the other, resulting in zero conditional entropy.

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