Product States (pt. 1)CSE 490Q: Quantum Computation

Quantum Computing

• States are vectors of complex numberswhose magnitudes squared sum to 1

• Operations are…• unitary matrices

• do not change length• measurements

• force into a classical state• prob = magnitude squared

Product States

• Consider a state of the form(p|0> + q|1>) ⊗ (s|0> + t|1>)= ps|00> + pt|01> + qs|10> + qt|11>

• Is every 2-qubit state a|00> + b|01> + c|10> + d|11>of this form?

• Consider 0.707|00> + 0.707|11>• since |01> is missing, we have pt=0• if p = 0, then ps = 0 not 0.707• if t = 0, then qt = 0 not 0.707

Product States

• Not every state is of the form (p|0> + q|1>) ⊗ (s|0> + t|1>)

• Those that are are called product states• can describe the two qubits individually

• All other states are called entangled states• cannot describe the two qubits individually• accurate description requires describing them together

Entangled States

• The properties of entangled states like 0.707|00> + 0.707|11> were studied by Einstein, Podolski, and Rosen in 1935• we those states “EPR pairs”• one of the four Bell states. others are

• 0.707|00> - 0.707|11>• 0.707|01> + 0.707|10>• 0.707|01> - 0.707|10>

• these are “maximally entangled” states

• Einstein et al. were trying to prove quantum mechanics was wrongbut history remembers them for teaching us about entanglement

Product Operations

• If U and V are unitaries, then U ⊗ V is a 2-qubit unitary

• Are all 2-qubit unitaries of this form?• No!

Product Operations

• Consider CNOT: flip the second bit iff the first bit is 1

• CNOT acts as identify on the first qubit,so it would have to be I ⊗ V for some V

• CNOT|10> = |11>, so (I ⊗ V)|10> = |11>,which means V|0> = |1>

• But then (I ⊗ V)|00> = |01> while CNOT|00> = |00>

Product States

• When a new qubit is added on, it forms a product state• we go from a|0> + b|1> to (a|0> + b|1>) ⊗ |0>

• It is okay to “throw away” a qubit if it is not entangled with the other• we are left with just the other qubit

• If they are entangled, we could measure one of the bits• the result would now be a product state |0> ⊗ |x> or |1> ⊗ |y>

• What happens if we don’t measure it?

Marginals

• Consider probabilistic state p|00> + q|01> + r|10> + s|11>where p + q + r + s = 1

• If we “throw away” the second qubit,this becomes p|0> + q|0> + r|1> + s|1> = (p+q)|0> + (r+s)|1>

• This is still a valid state since (p+q) + (r+s) = 1

• (This is a “marginal” distribution.)

Marginals

• This does not work on quantum states.

• Consider the state 0.707|00> + 0.707|01>

• Throwing away the second qubit gives 0.707|0> + 0.707|0> = 1.414|0>

that is not a valid state!

Throwing Away A Bit

• This is not purely theoretical

• Consider the following scenario• Adrian and Bianca create an EPR pair• Each takes hold of one of the two qubits• Adrian takes his qubit and climbs into a space ship• He flies to the nearest black hole and throws it in

Throwing Away A Bit

• Ways of throwing away a bit• toss it into a black hole• toss it into the Sun• toss it into molten lava

• As a wise man once, said “man, it’s gone!”

Throwing Away A Bit

• Nature must do something in this case• So what does it do?

• We can do an experiment to find out

• Nature measures the bit we threw away• turning it into a product state

Product States

• When a new qubit is added on, it forms a product state• we go from a|0> + b|1> to (a|0> + b|1>) ⊗ |0>

• We can throw away a qubit• if it is not entangled, it just disappears• if it is are entangled, the thrown away bit is measured

(and forgotten)• left with |x> from |0> ⊗ |x> or |y> from |1> ⊗ |y>

Nature Has No Other Choice

• Consider the following scenario• Adrian and Bianca create an EPR pair• Adrian takes his bit and climbs into a space ship• He says, “I’m flying to the black hole”• When he gets to the black hole,

he wonders, ”which half did I get?”• So he decides to measure it

• Bianca’s half cannot behave differently when he measures itvs when he throws it in the black hole• that would allow faster-than-light communication

Nature Has No Other Choice

• Adrian could measure his qubit when he gets to Mars• Or when he gets in his spaceship• Or just after he leaves the room

• To prevent Bianca from knowing whether Adrian has measured it yet,Nature must make her half always behave like he measured it immediately

• Consequence: part of an entangled state behaves as ifthe other part has been measured

Is It Really Changed?

• Consider this possibility• Adrian walks out the door• Bianca’s qubit acts like the Adrian’s was measured• Adrian changes his mind and comes back• now they have both parts again

• And nothing happened to the two parts!

• Consequence: Nature cannot “do” anything to Bianca’s qubitsince Adrian could come back at any time

What Is Measurement?

• Don’t think of measurement as something happening to the state

• A part of an entangled-state behaves like a probabilistic state• (a probability distribution over states)• the state has not changed… you’re just seeing part of it

• Measurement occurs when your state becomes entangledwith the environment (the measuring device)• building a QC requires shielding it from the environment