Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
observable A is given by
interested in the continuous time evolution, but in just
its input and output relation, then the time evolution is
A =Tr[Aρ]. (18)
nothing but a unitary operator U (cid:104) (cid:105)
Thedensityoperatorcanhandleamoregeneralsituation
ψ =U ψ . (11)
out in whereaquantumstateissampledformasetofquantum
| (cid:105) | (cid:105)
states ψ withaprobabilitydistribution q . Inthis
In quantum computing, the time evolution U is some- {| k (cid:105)} { k }
case, if we measure the system in the basis i i , the
times called quantum gate. {| (cid:105)(cid:104) |}
probabilitytoobtainthemeasurementoutcomeiisgiven
by
(cid:88)
C. Qubits p = q Tr[i iρ ], (19)
i k k
| (cid:105)(cid:104) |
k
The smallest nontrivial quantum system is a two-
where ρ = ψ ψ . By using linearity of the trace
dimensionalquantumsystemC2,whichiscalledquantum k k k
| (cid:105)(cid:104) |
function, this reads
bit or qubit:
(cid:88)
p =Tr[i i q ρ ]. (20)
α0 +β 1 , (α2+ β 2 =1). (12) i | (cid:105)(cid:104) | k k
| (cid:105) | (cid:105) | | | | k
Suppose we have n qubits. The n-qubit system is de- Now we interpret that the density operator is given by
fined by a tensor product space (C2)⊗n of each two-
(cid:88)
dimensional system as follows. A basis of the system ρ= q ψ ψ . (21)
k k k
| (cid:105)(cid:104) |
is defined by a direct product of a binary state x with
k k
| (cid:105)
x 0,1 ,
k
∈{ } In this way, a density operator can represent classical
x x x , (13) mixture of quantum states by a convex mixture of den-
1 2 n
| (cid:105)⊗| (cid:105)⊗···⊗| (cid:105) sity operators, which is convenient in many cases. In
which is simply denoted by general, a positive and hermitian operator ρ being sub-
ject to Tr[ρ] = 1 can be a density operator, since it can
x x x . (14) beinterpretedasaconvexmixtureofquantumstatesvia
1 2 n
| ··· (cid:105)
spectral decomposition:
Then a state of the n-qubit system can be described as
(cid:88)
ρ= λ λ λ , (22)
(cid:88) i | i (cid:105)(cid:104) i |
ψ = α x x x . (15)
x1,x2,...,xn| 1 2 n
| (cid:105) ··· (cid:105)
x1,x2,...,xn where {|λ i and {λ i are the eigenstates and eigen-
(cid:105)} }
vectors respectively. Because of Tr[ρ] = 1, we have
The dimension of the n-qubit system is 2n, and hence (cid:80)
λ =1.
thetensorproductspaceisnothingbuta2n-dimensional i i
Fromitsdefinition,thetimeevolutionofρcanbegiven
complexvectorspaceC2n
. Thedimensionofthen-qubit
by
system increases exponentially in the number n of the
qubits. ρ(t)=e−iHtρ(0)eiHt (23)
4
or Since the Pauli operators constitute a complete basis on
the operator space, any operator A can be decomposed
ρ =Uρ U†. (24)
out in into a linear combination of P(i),
Moreover, we can define more general operations for the (cid:88)
A= aiP(i). (33)
density operators. For example, if we apply unitary op-
erators U and V with probabilities p and (1 p), respec- i
−
tively, then we have Thecoefficientai canbecalculatedbyusingtheHilbert-
ρ =pUρU†+(1 p)VρV†. (25) Schmidt inner product as follows:
out
−
ai =Tr[P(i)A]/2n, (34)
As another example, if we perform the measurement of
ρ in the basis i , and we forget about the measure- by virtue of the orthogonality
{| (cid:105)}
ment outcome, then the state is now given by a density
operator Tr[P(i)P(j)]/2n =δi,j. (35)
(cid:88) (cid:88)
Tr[i iρ]i i = i iρi i. (26) The number of the n-qubit Pauli operators P(i) is 4n,
| (cid:105)(cid:104) | | (cid:105)(cid:104) | | (cid:105)(cid:104) | | (cid:105)(cid:104) | { }
andhenceadensityoperatorρofthen-qubitsystemcan
i i
be represented as a 4n-dimensional vector
Therefore if we define a map from a density operator to
another, which we call superoperator,  r 
00...0
(cid:88) r = . . , (36)