2.11 Distances and Distinguishability Between Quantum States
31
If Bob employs the optimal measurement strategy, i.e. if Tr[P 0 (ρ − τ )] is maximized over the possible POVMs of the form {P 0 , 1 − P 0 }, one can relate the optimal
guessing probability of Bob to the trace distance between the two states prepared by
Alice [2]:
p guess (ρ, τ ) =
1
2
(1 + T (ρ, τ )) .
(2.79)
If T (ρ, τ ) = 0 the two states are the same state and the corresponding optimal
guessing probability reads p guess (ρ, τ ) = 1/2. This is expected, since Bob cannot
perform a better guess of the received state than a random guess (e.g., a coin flip),
when the two states are identical. We thus call the two states indistinguishable. In
general, Eq. 2.79 shows that the trace distance T (ρ, τ ) between two states coincides
with the distinguishing advantage of a distinguisher (in our case Bob) attempting to
distinguish them [17].
We formalize the situation where two states are “almost” indistinguishable with
the following definition.
Definition 2.14 (States indistinguishability) The states ρ and τ are
ε-indistinguishable if their trace distance is upper bounded by:
T (ρ, τ ) ≤ ε
(2.80)
i.e. if the states are indistinguishable, except for a probability at most ε/2.
We derive a closed expression for the trace distance between two pure states |ψ
and |φ:
T (ψ, φ) =
1
2
|ψψ | − |φφ | =
1
2
i
|λ i | ,
(2.81)
by computing the eigenvalues λ i of the operator O := |ψψ | − |φφ |. The eigenvalue equation of O reads:
O|λ i = |ψψ|λ i − |φφ|λ i = λ i |λ i .
(2.82)
We now take the inner product between each of the two pure states and both sides in
(2.82):
ψ|λ i − −ψ|φφ|λ i = λ i ψ|λ i
φ|ψψ|λ i − −φ|λ i = λ i φ|λ i
(2.83)
The above linear system in the variables ψ|λ i and φ|λ i can be recast as:
(1 − λ i )ψ|λ i − −ψ|φφ|λ i = 0
φ|ψψ|λ i − (1 + λ i )φ|λ i = 0
(2.84)
31
If Bob employs the optimal measurement strategy, i.e. if Tr[P 0 (ρ − τ )] is maximized over the possible POVMs of the form {P 0 , 1 − P 0 }, one can relate the optimal
guessing probability of Bob to the trace distance between the two states prepared by
Alice [2]:
p guess (ρ, τ ) =
1
2
(1 + T (ρ, τ )) .
(2.79)
If T (ρ, τ ) = 0 the two states are the same state and the corresponding optimal
guessing probability reads p guess (ρ, τ ) = 1/2. This is expected, since Bob cannot
perform a better guess of the received state than a random guess (e.g., a coin flip),
when the two states are identical. We thus call the two states indistinguishable. In
general, Eq. 2.79 shows that the trace distance T (ρ, τ ) between two states coincides
with the distinguishing advantage of a distinguisher (in our case Bob) attempting to
distinguish them [17].
We formalize the situation where two states are “almost” indistinguishable with
the following definition.
Definition 2.14 (States indistinguishability) The states ρ and τ are
ε-indistinguishable if their trace distance is upper bounded by:
T (ρ, τ ) ≤ ε
(2.80)
i.e. if the states are indistinguishable, except for a probability at most ε/2.
We derive a closed expression for the trace distance between two pure states |ψ
and |φ:
T (ψ, φ) =
1
2
|ψψ | − |φφ | =
1
2
i
|λ i | ,
(2.81)
by computing the eigenvalues λ i of the operator O := |ψψ | − |φφ |. The eigenvalue equation of O reads:
O|λ i = |ψψ|λ i − |φφ|λ i = λ i |λ i .
(2.82)
We now take the inner product between each of the two pure states and both sides in
(2.82):
ψ|λ i − −ψ|φφ|λ i = λ i ψ|λ i
φ|ψψ|λ i − −φ|λ i = λ i φ|λ i
(2.83)
The above linear system in the variables ψ|λ i and φ|λ i can be recast as:
(1 − λ i )ψ|λ i − −ψ|φφ|λ i = 0
φ|ψψ|λ i − (1 + λ i )φ|λ i = 0
(2.84)