A basic property of this form of conditional entropy is that:
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Unlike its classical counterpart, the quantum conditional entropy can be negative.
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Unlike the classical conditional entropy, the conditional quantum entropy can be negative.
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That is, the conditional entropy of a symbol given all the previous symbols generated.
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One may also define the conditional entropy of two events and taking values and respectively, as
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The range of smoothing is provided by some fixed percentage of conditional entropy from total entropy.
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This quantity is exactly H ( Y | X ), which gives the " chain rule " of conditional entropy:
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This is equivalent to the fact that the conditional quantum entropy may be negative, while the classical conditional entropy may never be.
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The conditional entropy measures the amount of entropy remaining in one random variable when we know the value of a second random variable.
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Positive conditional entropy of a state thus means the state cannot reach even the classical limit, while the negative conditional entropy provides for additional information.
