# Tag Info

### What is intuition for the trace distance between quantum states?

Short answer. The trace distance between two states more or less determines how distinguishable they are by any operational means. A trace distance of 0 means that they are indistinguishable (because ...

### Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$

Marsl is correct, and his "hint" is really more a sketch of a solution than a hint. Rather than viewing the question or its solution as just formal algebra, you can also approach his same solution ...
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### Does the trace distance have a geometric interpretation?

There is a geometric interpretation that you certainly can take seriously, but the geometry that you get is not as clean as you might have hoped. Trace distance between operator states is an example ...
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### How to prove that $\frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ hides one of $x_0$ or $x_1$?

We can bound the amount of information that can be retrieved from $|\psi\rangle$ using Holevo's bound. Alice and Bob Let us first reformulate the situation in the terms usually employed in the context ...
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### Trace distance of two classical-quantum states

Yes, since the trace norm is the sum of the absolute value of the singular values, and the singular values can be found for each of the $a$ blocks independently.
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### Prove that the trace norm is dual to the spectral norm

There are different ways to prove what you want to prove, including the solution tsgeorgios has suggested, but for the sake of gaining greater intuition I would suggest starting with the recognition ...
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### Is the trace distance between multipartite states invariant under permutations?

I'd like to add a small addition to the answer of @DaftWullie about why you would expect this operationally to be true -- without knowing permutations correspond to unitary matrices. It boils down to ...
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### Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?

Both definitions are used and authors usually make it clear which one they mean. Wikipedia also points this out under the Alternative Defintion section.
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### Relation between trace distance and inner product between pure states

A derivation of this is given in Mark Wilde's book https://arxiv.org/abs/1106.1445 equation 9.173, pages 274-275.
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### Prove that the trace norm is dual to the spectral norm

We still have $\big| \langle B, A \rangle \big| = \big|\text{Tr}(AB^{\dagger}) \big| \leq \text{Tr}|A|$ for any operator $B$ with operator norm $||B|| \leq 1$. First observe that $||B|| \leq 1$ ...
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### Trace distance bound after partial trace

Yes, the trace distance can only decrease under partial trace. One can see this via the variational characterization of the trace norm $$\|\rho\|_1 = \max_{-I \leq M \leq I} \mathrm{Tr}[M\rho]$$ ...
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