Note some distinctive properties:1.Outcome ofa measurement is not deterministic.2. Measurement changes the system. Operation on one qubit:One can operate on a single qubit in different ways.Some examples:the Pauli-X (NOT)gate X,the Pauli-X gate Y,the Pauli-Z gate Z and the Hadamard gate H are defined as follows x=91,y=[901z=6l, Operating on a single qubit,these operators have the following effects X(al0〉+B11)=Bl0〉+a1,Y(alo〉+B11)=-i邛l0〉+ial1), Z(al0)+11)=l0)-B11), Ha10y+B1》=+e1oy+-911. 2 Multiple qubits:If we have two qubits,then a state in the 2-qubit system is lp）=aool00〉+ao1l01）+a10l10〉+a11l11, where aoo,ao1,a1o,a11 are complex numbers satisfying laool2+la0112+la1012+la1112 =1. For example,the following is an"EPR state": 1妙=100)+111) √Z And it is an example of non-product state,in the sense that it cannot be written as (a10)+ B1))(a210)+B21))for any coefficients ai,Bi. Measurement in computational basis:For any aja2E (00,01,10,11)Observe aaz with probabilityand the state becomes) Operation on two qubits:One can also operate on a two-qubit system.An important example is Controlled-NOT(CNOT),defined as follows. 00 01 CNOT 0 100 001 0 01 0Note some distinctive properties: 1. Outcome of a measurement is not deterministic. 2. Measurement changes the system. Operation on one qubit: One can operate on a single qubit in different ways. Some examples: the Pauli-X (NOT) gate 𝑋, the Pauli-X gate 𝑌, the Pauli-Z gate Z and the Hadamard gate 𝐻 are defined as follows. 𝑋 = [ 0 1 1 0 ] , 𝑌 = [ 0 −𝑖 𝑖 0 ], 𝑍 = [ 1 0 0 −1 ] , 𝐻 = 1 √2 [ 1 1 1 −1 ]. Operating on a single qubit, these operators have the following effects. 𝑋(𝛼|0〉 + 𝛽|1〉) = 𝛽|0〉 + 𝛼|1〉, 𝑌(𝛼|0〉 + 𝛽|1〉) = −𝑖𝛽|0〉 + 𝑖𝛼|1〉, 𝑍(𝛼|0〉 + 𝛽|1〉) = 𝛼|0〉 − 𝛽|1〉, 𝐻(𝛼|0〉 + 𝛽|1〉) = 𝛼 + 𝛽 √2 |0〉 + 𝛼 − 𝛽 √2 |1〉. Multiple qubits: If we have two qubits, then a state in the 2-qubit system is |𝜓〉 = 𝛼00|00〉 + 𝛼01|01〉 + 𝛼10|10〉 + 𝛼11|11〉, where α00 , 𝛼01, 𝛼10, 𝛼11 are complex numbers satisfying |𝛼00| 2 + |𝛼01| 2 + |𝛼10| 2 + |𝛼11| 2 = 1. For example, the following is an “EPR state”: |𝜓〉 = |00〉 + |11〉 √2 And it is an example of non-product state, in the sense that it cannot be written as (𝛼1 |0〉 + 𝛽1 |1〉)⊗ (𝛼2 |0〉 + 𝛽2 |1〉) for any coefficients 𝛼𝑖 ,𝛽𝑖 . Measurement in computational basis: For any 𝑎1𝑎2 ∈ {00,01,10,11}Observe 𝑎1𝑎2 with probability |𝛼𝑎1𝑎2 | 2 , and the state becomes |𝑎1𝑎2 〉. Operation on two qubits: One can also operate on a two-qubit system. An important example is Controlled-NOT (CNOT), defined as follows. CNOT = [ 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 ]