Calculating Subsystem B Probability in a Two-System Setup

[nashed]

This is a rather basic question but I seem to be stuck, given a system made up of two subsystems how do I compute the probability of subsystem B to be in some state (for example I've got |p>=|a>|b> if I measure the second part what's the probability of it being found in state |c>, how am I supposed to deal with |a>?)

 

[blue_leaf77]

for example I've got |p>=|a>|b> if I measure the second part what's the probability of it being found in state |c>, how am I supposed to deal with |a>?

The blunt answer, disregarding of whether or not the system is of fermionic or bosonic, is to sum all probabilities of the outcome in which the second component is ##|c\rangle## from your measurement.
A simple example is finding the probability of a particle to be found in a region ##c<z<c+\Delta z## where ##\Delta z## is very small, if the probability associated with this particle is denoted as ##\rho(x,y,z)##, then you would want to calculate ##\Delta z \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \rho(x,z,c) \ dx\ dy##.

 

[nashed]

just to make sure, if I've got a system of 2 qubits (let's say ## |\psi \rangle = \frac 1 \sqrt(2) (|00\rangle + |11\rangle ) ## ) and I want to check what's the probability of measuring the second qubit to be a ## |0\rangle ## I"ll be checking $$ \langle \psi | \sum_i |i\rangle\langle i| \otimes |0\rangle\langle 0| |\psi \rangle $$
and then I"ll have $$ \sum_i |i\rangle\langle i| = I $$ so I get $$ \langle \psi | I \otimes |0\rangle\langle 0| |\psi \rangle $$

 

[blue_leaf77]

just to make sure, if I've got a system of 2 qubits (let's say ## |\psi \rangle = \frac 1 \sqrt(2) (|00\rangle + |11\rangle ) ## ) and I want to check what's the probability of measuring the second qubit to be a ## |0\rangle ## I"ll be checking $$ \langle \psi | \sum_i |i\rangle\langle i| \otimes |0\rangle\langle 0| |\psi \rangle $$
and then I"ll have $$ \sum_i |i\rangle\langle i| = I $$ so I get $$ \langle \psi | I \otimes |0\rangle\langle 0| |\psi \rangle $$

That doesn't look right to me. ##|\psi\rangle## is a composite state while either ##|0\rangle## or ##|1\rangle## is just one component of the whole system, the scalar inner product between them is undefined. You will need to project ##|\psi\rangle## to the subspace where the second bit iz ##|0\rangle## and calculate the norm of resulting state.

 

[bhobba]

This is a rather basic question but I seem to be stuck, given a system made up of two subsystems how do I compute the probability of subsystem B to be in some state (for example I've got |p>=|a>|b> if I measure the second part what's the probability of it being found in state |c>, how am I supposed to deal with |a>?)

From the state its easily seen they are separate systems. Measuring one has no effect on the other. Now if they were entangled that is another matter, but the subject for another thread to explain what entanglement is and what's going on. The short answer is if entangled they are in a mixed state when observing just one system - but like I said requires its own thread. I have posted the math of it before and will need to dig it up which can take a while.

Thanks
Bill

 

[nashed]

I actually know what entanglement is, my main problem is that I'm used to working with operators on the whole system so when I was told that I need to calculate a probability on a subsystem I was taken aback.

My thought process is that I should sum over all possible states for the first part of the system, only I have no idea how to do that.

 

[bhobba]

My thought process is that I should sum over all possible states for the first part of the system, only I have no idea how to do that.

Simply apply it to the state of the sub-system - forget the other system.

Technically it lies in the combined vector space of the two systems but operators on one system commutes with state of the other - it's the identity operator.

Thanks
Bill