2

I understand that when comparing two arbitrary quantum states, one may use various measures to encapsulate the difference between states such as trace distance, fidelity or relative quantum entropy. I was wondering whether there are other non-standard ways to describe the distance between states? I am guessing these may be more goal-oriented.

Edit (after an answer was posted): The question was posted out of genuine curiosity, but my goal is to find a measure which may be used to track the progress of an algorithm with a faulty oracle (such as one which randomly produces noise).

glS
  • 27,510
  • 7
  • 37
  • 125
milo
  • 39
  • 5

1 Answers1

5

One nonstandard approach to measuring distances between states is the "$\varepsilon$-smooth relative complexity distance" $\mathcal C_\varepsilon(|\psi\rangle,|\phi\rangle)$ corresponding to the minimal number of one- and two-qubit gates over all such circuits $U$ that map $|\psi\rangle$ to $|\phi\rangle$, up to some Euclidean error $\varepsilon$ and with some standard universal gate set.

For example, if $|\psi\rangle=|00\cdots0\rangle$ and $|\phi\rangle=\vert00\cdots1\rangle$, then $|\psi\rangle$ and $|\phi\rangle$ are orthogonal to one another and conventional distances between $|\psi\rangle$ and $|\phi\rangle$ are quite large but $\mathcal C_\varepsilon$ is very small, needing only a single NOT gate on the rightmost qubit. Conversely two states can be "nearly" identical in trace-norm but yet be far apart in relative complexity - think of the $|\psi\rangle=|00\cdots 0\rangle$ versus $|\phi\rangle=\sqrt{0.99}|00\cdots 0\rangle+\sqrt{0.01}|\upsilon\rangle$ for some Haar-random state $|\upsilon\rangle$. The inner product $|\langle\phi|\psi\rangle|$ is very small although the complexity $\mathcal C_\varepsilon$ exponentially large.

See here for a longish but accessible video on the metric and on some of its implications and uses.

Mark Spinelli
  • 15,378
  • 3
  • 26
  • 83