# Bra-ket notation in cryptography example

I was watching a presentation and I run into this example.

What does $$\phi$$ with the (x)t on the exponent mean?, similarly for the other symbol

This presentation seems to be about pseudorandom quantum states.

A set of quantum states $$\left\{\phi_k\right\}_k$$ indexed by some key values is said to be pseudorandom if no adversary can distinguish polynomially many copies of a state sampled by choosing a random key $$k$$ from as many copies of a Haar-random state $$|\psi\rangle$$.

In cryptography, such properties are often modeled via some games, which is the case here. The challenger will choose a bit $$b\in\{0,1\}$$ at random.

• If $$b=0$$, then they will choose a random key $$k$$ and give the adversary $$t$$ copies of $$\phi_k$$. This is what $$\phi_k^{\otimes t}$$ represents, as mentioned by @DaftWullie in his answer.
• If $$b=1$$, then they will sample a Haar-random state $$|\psi\rangle$$ and give the adversary $$t$$ copies of the same Haar-random state, which is represented by $$|\psi\rangle^{\otimes t}$$.

(By the way, if you don't know what an Haar-measure is, picking a Haar-random state means here to pick a quantum state uniformly at random among the set of all states.)

The adversary, upon being given $$t$$ copies of a state, must tell whether $$b=0$$ or $$b=1$$. If they cannot win with probability significantly larger than $$\frac12$$, which is just the probability of guessing, then the set $$\left\{\phi_k\right\}_k$$ is pseudorandom.

This is exactly what's depicted here. Note that what's potentially misleading is that both $$\varphi_k$$ and $$|\psi\rangle$$ are quantum states, even though the first one is not written using ket notation. It is usually written $$\left|\phi_k\right\rangle$$, which means that $$t$$ copies of it are usually denoted $$\left|\phi_k\right\rangle^{\otimes t}$$.

This is the tensor product symbol. Thus, something like $$\phi^{\otimes t}$$ means that you have $$t$$ copies of the state $$\phi$$.

• Could you comment also on what $\phi^{\otimes t}_k$ might mean? Commented Jul 24, 2023 at 14:00
• Without more context, not really. I assume it just means that $\phi$ depends on the parameter $k$ that was the input to the StateGen function, and you have $t$ copies of it. Commented Jul 24, 2023 at 14:27