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Questions about quantum mechanics reducing the complexity of classical models 
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#1
May612, 04:05 AM

P: 159

I have some questions about this paper: http://arxiv.org/abs/1102.1994v2
The author computes the entropy of the classical simulator using the Shannon entropy, then computes the entropy of the quantum simulator using von Neumann entropy and gets a smaller number, thus concluding that quantum simulators require smaller input. Firstly, are these two measures directly comparable? For example, the von Neumann entropy of a pure state is 0, even if it's maximally entangled. The corresponding classical entropy would be nonzero (and maximized). Also, it doesn't seem like the quantum simulator is using any less internal memory. It still seems to require log(S) qubits (S = space of causal states), which is the same internal memory as the classical model. 


#2
May712, 02:07 PM

P: 729

Btw: I haven't read the paper. 


#3
May712, 05:20 PM

P: 159

But the thing is: it's obvious that the information of classical and quantum states is not directly comparable. For example, consider the case of 2 bits: {00,01,10,11}. Let's say we have a classical distribution over these bits P(00)=P(01)=0, P(10)=P(11)=1/2. The entropy of this distribution is 1. The corresponding quantum state would be (10> + 11>)/√2. By 'corresponding' I mean performing a measurement on this quantum state would give us the same results as sampling from the classical distribution. However, the entropy of the quantum state is 0. However, this is not the main focus of my question. My main issue is that the internal memory of the simulator  the thing that is ostensibly more important when considering which model is 'simpler'  is the same in both models (actually, it's increased in the quantum case because it has to be reversible). 


#4
May712, 07:17 PM

P: 729

Questions about quantum mechanics reducing the complexity of classical models
The full state (qubits + measurement device) gets entangled but remains pure. Only the qubit state alone is mixed. This is a very remarkable property of QM: the entropy of a subsystem can be greater than the entropy of the whole system. I don't have time to read the paper and contribute to more specific issues, sorry. 


#5
May712, 07:41 PM

P: 159

Everything you said is correct, but I don't see how it relates to my question.



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