Question about quantum state distinguishability

[physicslover123]

Homework Statement

**Problem 1: Distinguishing between two states**

Let n be a natural number. Suppose you are given one of two n-qubit states, |ψ₀⟩ or |ψ₁⟩, each equally likely (with probability 1/2). You know exactly what these two states are, but you don't know which of the two you've been handed. Also, assume the two states are not identical (two states differing only by a global phase count as identical here).

In this problem, your goal is to find a projective measurement {P, I − P} that maximizes the probability of correctly distinguishing the states. Specifically, the measurement should maximize:

(1/2) × Pr[0 | ψ₀] + (1/2) × Pr[1 | ψ₁],

where Pr[i | ψᵢ] represents the probability that the measurement outcome is "i" given the state |ψᵢ⟩.

**(i)** Carefully demonstrate that, without losing generality, the problem can be reduced to considering measurements within the two-dimensional space spanned by the states |ψ₀⟩ and |ψ₁⟩. In other words, you only need to consider measurements of the form:

{|u⟩⟨u|, I − |u⟩⟨u|},

where |u⟩ is within the span of |ψ₀⟩ and |ψ₁⟩.

(You should aim to make this proof rigorous—the reasoning involved is more subtle than it might initially appear!)

**(ii)** Determine the optimal measurement and clearly prove its optimality.

Relevant Equations

quantum mechanics eqs

I'm not entirely sure how to approach part 1. I tried the following: Call the span of the two states S. We can decompose H = S + S_perp. We consider some projector P: H -> V where V is some subspace of H. I made a new operator P_s P P_s where P_s is the orthogonal projector onto S. However, I couldn't prove idempotence, and so it wasn't a projector even though the measurement probabilities are the same. Pls help!

 

[WWGD]

Homework Statement: **Problem 1: Distinguishing between two states**

Let n be a natural number. Suppose you are given one of two n-qubit states, |ψ₀⟩ or |ψ₁⟩, each equally likely (with probability 1/2). You know exactly what these two states are, but you don't know which of the two you've been handed. Also, assume the two states are not identical (two states differing only by a global phase count as identical here).

In this problem, your goal is to find a projective measurement {P, I − P} that maximizes the probability of correctly distinguishing the states. Specifically, the measurement should maximize:

(1/2) × Pr[0 | ψ₀] + (1/2) × Pr[1 | ψ₁],

where Pr[i | ψᵢ] represents the probability that the measurement outcome is "i" given the state |ψᵢ⟩.

**(i)** Carefully demonstrate that, without losing generality, the problem can be reduced to considering measurements within the two-dimensional space spanned by the states |ψ₀⟩ and |ψ₁⟩. In other words, you only need to consider measurements of the form:

{|u⟩⟨u|, I − |u⟩⟨u|},

where |u⟩ is within the span of |ψ₀⟩ and |ψ₁⟩.

(You should aim to make this proof rigorous—the reasoning involved is more subtle than it might initially appear!)

**(ii)** Determine the optimal measurement and clearly prove its optimality.
Relevant Equations: quantum mechanics eqs

I'm not entirely sure how to approach part 1. I tried the following: Call the span of the two states S. We can decompose H = S + S_perp. We consider some projector P: H -> V where V is some subspace of H. I made a new operator P_s P P_s where P_s is the orthogonal projector onto S. However, I couldn't prove idempotence, and so it wasn't a projector even though the measurement probabilities are the same. Pls help!

It seems at first sight your operator O:=PsPPs is idempotent. Have you computed (PsPPs)(PsPPs)?

 

[physicslover123]

Yes I have, and it is not idempotent in general because we don't necessarily know that P(|x>) \in S for |x>\in S, in other words the projector must be invariant on the subspace S, but that may not be true in general

 

[WWGD]

By the way, Welcome to PF!. But, according to the definition of Projection I'm aware of, a Projection operator into S maps x into S and, by definition too, if P is a projection operator, $P(Ps)=P^2(S)=Ps \in S$. Is that what you meant? Btw: Could you please use ## to wrap your Latex code? Maybe someone with a more "Physichy" perspective, maybe @hutchphd can elaborate, expand here?

 

[physicslover123]

So the issue is that $P: \mathcal{H} \rightarrow V$ is any projector from the whole space to a subspace $V$. This subspace may not necessarily be $S$ in general. If that's the case, then a vector in $S$ can acquire a component orthogonal to $S$ under the projection. The way I'm reading the question, given an arbitrary projective measurement which is associated with some projector, we want to find a new projector onto $S$ that keeps the measurement probabilities the same.