If one understands the 0-norm to be the intersection of the 1 and 2-norms, shouldn’t the trilemma really be the following (for any universe with very generic properties)?

1. Determinism, classical probabilities and quantum mechanics are all true.

2. Classical probabilities is true; the other two are false.

3. Quantum mechanics is true; the other two are false.

Thanks, that’s a very interesting and amusing way to put it! But I suppose it ultimately boils down to a matter of definition. Many people would interpret “Classical probabilities is true” to mean, not merely that all your states are probability vectors, but also that some states are nontrivial probability vectors. Likewise, they would interpret “Quantum mechanics is true” to mean that some states are nontrivial quantum superpositions.

One technical comment/correction: in case 2, the probabilistic case, you also shouldn’t rule out quantum mechanics being true—since maybe your states are “really” quantum mixed states whose density matrices just happen to be diagonal!

^{http://www.scottaaronson.com/blog/?p=1385#comment-73670}

If one understands the 0-norm to be the intersection of the 1 and 2-norms, shouldn’t the trilemma really be the following (for any universe with very generic properties)?

1. Determinism, classical probabilities and quantum mechanics are all true.

2. Classical probabilities is true; the other two are false.

3. Quantum mechanics is true; the other two are false.

Thanks, that’s a very interesting and amusing way to put it! But I suppose it ultimately boils down to a matter of definition. Many people would interpret “Classical probabilities is true” to mean, not merely that all your states are probability vectors, but also that some states are nontrivial probability vectors. Likewise, they would interpret “Quantum mechanics is true” to mean that some states are nontrivial quantum superpositions.

One technical comment/correction: in case 2, the probabilistic case, you also shouldn’t rule out quantum mechanics being true—since maybe your states are “really” quantum mixed states whose density matrices just happen to be diagonal!

^{http://www.scottaaronson.com/blog/?p=1385#comment-73670}

If one understands the 0-norm to be the intersection of the 1 and 2-norms, shouldn’t the trilemma really be the following (for any universe with very generic properties)?

1. Determinism, classical probabilities and quantum mechanics are all true.

2. Classical probabilities is true; the other two are false.

3. Quantum mechanics is true; the other two are false.

Thanks, that’s a very interesting and amusing way to put it! But I suppose it ultimately boils down to a matter of definition. Many people would interpret “Classical probabilities is true” to mean, not merely that all your states are probability vectors, but also that some states are nontrivial probability vectors. Likewise, they would interpret “Quantum mechanics is true” to mean that some states are nontrivial quantum superpositions.

One technical comment/correction: in case 2, the probabilistic case, you also shouldn’t rule out quantum mechanics being true—since maybe your states are “really” quantum mixed states whose density matrices just happen to be diagonal!

^{http://www.scottaaronson.com/blog/?p=1385 (Comment #142)}

If one understands the 0-norm to be the intersection of the 1 and 2-norms, shouldn’t the trilemma really be the following (for any universe with very generic properties)?

1. Determinism, classical probabilities and quantum mechanics are all true.

2. Classical probabilities is true; the other two are false.

3. Quantum mechanics is true; the other two are false.

Thanks, that’s a very interesting and amusing way to put it! But I suppose it ultimately boils down to a matter of definition. Many people would interpret “Classical probabilities is true” to mean, not merely that all your states are probability vectors, but also that some states are nontrivial probability vectors. Likewise, they would interpret “Quantum mechanics is true” to mean that some states are nontrivial quantum superpositions.

One technical comment/correction: in case 2, the probabilistic case, you also shouldn’t rule out quantum mechanics being true—since maybe your states are “really” quantum mixed states whose density matrices just happen to be diagonal!

^{http://www.scottaaronson.com/blog/?p=1385#comment-73670}