In this video, we will introduce the concept of pure and mixed states. In quantum mechanics, we only talk about average or expectation value for a physically observable quantity. So this means that we are implying that we make multiple measurements and take average. But multiple measurements on what? So we introduced this concept of ensemble, and by ensemble we mean a collection of identically prepared physical system. So let's take an example of the Stern-Gerlach experiment. So here is the electron source, it's an electron gun. Typically, you heat up development and produce thermally generated electron beam and it passes through this Stern-Gerlach experiment apparatus aligned along Z direction, and it will split the electrons into two beams, as you recall, one along spin up direction and the other spin down. And if we block one of them, then we get an electron beam with a well defined spin. All of them spin up along the Z direction in this particular configuration, shown here in this diagram. So this is one way to produce an ensemble, a collection of electrons all having a well defined spin state spin up along the Z direction. But what about these electron coming out of this thermal source? They have all randomly oriented spins. How do you describe a collection of electron with these randomly oriented spins? Now, if you recall the most general description of a spin state for a spin one half system or an electron, for example, is we choose a basis set. We can choose spin up along Z, spin up and down along Z direction as our basis set. And this plus and minus represents spin up and spin down along particular direction, let's call that a Z direction. And we just produce right down a superposition of these two basis there, and the c+ and c- are complex collection. This is the most general spin state in the spin 1/2 system. However, this state still has a well defined orientation determined by the coefficient. It's just that its orientation itself is random, it could be pointing any arbitrary direction depending on the values of c+ and c-. But once you fix those values of c+ and c- school options, it does have a very well defined orientation. This equation cannot describe an ensemble collection of electrons with randomly oriented spin, one electron having spin different from the other electron and so on and so forth. So how do you describe these collections of randomly oriented spins? To describe these randomly oriented spins, we introduced a fractional population. So for example if an ensemble or collection of electrons where 50% of the electrons are in spin upstate and 50% of the electrons are in spin down state. And we specify await a fractional wait for each case and say that okay, the fractional wait for spin upstate is 50%, 0.5 fractional weight for a spin down state is 0.5 also. So this w+ and w- here are different from the c+ and c- Kocian used in this superposition state. Now in these superposition state, the co option, c+ and c- are complex numbers. And therefore they define or they specify the phase relationship between the two states plus and minus, kept plus and kept minus, right? So on ensemble where all members are described by the same state, like this is called a pure ensemble. An example will be the electron beam, the collection of electrons coming out of the Stern-Gerlach experiments, after aligned along the Z direction and we block out the spin downstate and and just let spin up states pass through. And that electron beam coming out of the Stern-Gerlach experiments, is an example of pure ensemble. All of the electrons in that electron beam has the same spin state and specify can be specified by this kind of expression. In contrast, the fractional weights are always real numbers. They are not complex numbers and they do not therefore carry any phase information. And this type of mixture is sometimes called the incoherent mixture. It is akin to the optics where you basically incoherently add to light-waves, producing no interference pattern. And this ensemble that require fractional weights cannot be described by a single state kept because individual members in this ensemble are in different state. They all require different state state tet, and so the populations, or the relative fractional populations of the members particles in a particular state specified by a single state cap, all are described by these rational weights. And so this type of ensemble is called the mixed ensemble, or mixed state. So in literature, people say pure state versus mixed state, and what they really mean is really pure ensemble and mixed example, but those terminology are used interchangeably. And the fractional weights they represent fractional population, so they all have to add up to 1. Now we need to find a way to mathematically expressed this mixed example or mixed state. So let's suppose we're making a measurement on a mixed ensemble Of some observable quantity A. Now the ensemble average should be taken like this. So now, here we use a square bracket to express ensemble average. And the ensemble average will be written like this. So there is this fractional weight for different state specified by this subscript i. And that state indicated or specified by the subscript i is expressed by this state cat or state bra, alpha superscript i. And for each alpha i, we take the expectation value or average value of the observable A. And then we multiply the fractional weight and summit over all possible values of i, all possible states in the ensemble, then that's the ensemble average. Now we can then express this alpha i in terms of a superposition of the eigenkets of observable A right? Observable A because it's an observable, it's information operator and its eigenkets should form a complete or to normal set and we can express any arbitrary state cap as a superposition of those eigenkets. So if you recall that that expression looks like this. So alpha sub i is sum state. Can be expressed as a superposition of a prime. a prime are the eigenkets of the observable A. And this cooption c survey specifies the probability of finding this particular state. When you make a measurement on this particular state for observable A what is the probability of finding it in a particular eigenstate a prime? Well, that probability is given by the absolute value square of this cooption c survey prime. And what is the value of c survey prime cooption? Well, that's also easy. You just take an inner product of this state alpha, with the corresponding eigenket. And so if you use this and expressed both alpha i ket here and alpha i bra in terms of this superposition of the eigenkets of operator A. Then the equation becomes this. All right? So this inner product here, once again represents this c survey cooption. And the absolute value square is the probability of measuring a particular eigen value a prime of the operator A. So here you should notice that the probability concept enters twice when describing a mixed state. The usual probability concept here is what we are used to. For a state alpha, which is in general not an eigenstate of the operator A representing the observable that you're you are making the measurement on, right? So this alpha because it's not an eigenstate of A. We can only describe probabilistically what the results would be. The probability of measuring a particular eigen value a prime of this operator A is given by this cautions absolutely square. So that is the probability that we're used to. But then this ensemble is a mixed ensemble. The members of this ensemble are in different states. Some members are in alpha of i, other members are in alpha of j, k, l. There are many states. And the these different states has the fractional population of wi. So their the probabilistic concept enter twice or there are two different probabilities that we have to consider here. Now we choose a general basis set b. And this b or b prime here doesn't have to be the eigenstates of the observable that you're making measurements on, right? So basis said it's just a mathematical tool, you can choose whatever basis that is the most convenient for you. And so this take this definition of the ensemble leverage of a written here, in this time instead of expressing this alpha sub i as a superposition of the eigenkets of a, we use this space is right? So express alpha sub i, alpha sub i here as a superposition of b prime, right? And we can write it like this. And using the associativity, we rearrange these terms like this. So we do this summation of i first. Okay, and then we'll do the summation over b prime and b the low prime later. So if you look at this, okay? Then it is a matrix element using this basis set b prime. Okay, so this b bra here brought here and ket there, defines a matrix element in the basis set b prime of an operator defined by this guy here. So this outer product here of alpha can be considered an operator. And this here is the matrix element of this quantity. This outer product in the basis set b prime. And then you of course multiply wi to that matrix element and you summit overall i. Right, that's what this expression means. This other thing here is simply the matrix element of the operator A in the same basis set the prime that we just chose. So let us define an operator given by this outer product of alpha sum over all state i okay? With the appropriate weighting factors, fractional weights, wi representing or indicating the fractional population in state alpha sub i or alpha superscript i. So this operator is called the density operator. And if we, Write down the matrix element of this density operator, which we call the density matrix, is the quantity that we have just seen inside the parenthesis in the previous slide. Now we can write down the ensemble average as this. And so this bit here is this, which was the quantity inside the parenthesis in the last equation of the previous slide. And we multiply that too that this matrix element of A. But if you recall this here out of product of this basis set summed over all of these cats is simply an identity operator. And so this becomes identity and the resulting product simply becomes this. And what is this, well rho times A. This is a new operator and we're taking the matrix element for the same cat and some of our all all the basis cats. And that simply represents the summing over all diagonal element of the matrix, which is the trace of the matrix. So the ensemble average, of mixed state is given by the trace of the density operator, which characterises the mixed on sample that you have, right. These fractional weights and these states alpha I that individual members of the example can have. Right, this density matrix characterizes the mixed example that you have. And then the operator is of course the variable that you're making a measurement of. Or if you multiply those two and calculate the trace of that matrix, that is the ensemble average. Now recall, trace is independent of the choice of the basis set. Even as you make similarity transformations for a matrix from one to another, the trace of those matrices remain the same. So trace here is independent of the choice of your basis. You don't have to use this particular basis that B or B prime or be double prime that we're showing here, we can choose whatever is the most convenient for us. And calculate the trace of this product operator, road density operator at times A the observable. The density operator has these properties, their formation and they're also normalized, meaning that the trace of the density operator is 1. This requirement falls right out of the requirement that the fractional weights w sub I will be some doubt to equal to 1. A pure ensemble is a special case of an ensemble special case of a mixed ensemble. And pure ensemble is specified by W sub n is 1 for a particular value of N and zero for all other values. So only one of one element of the summation in the definition of the matrix density operator is non-zero and all others are zero. So your density operator has only one term in the case of pure example or pure state. When you diagonolize the matrix, and you can you can write down this operator into a matrix form by choosing an appropriate basis set. And if possible, you want to choose the basis that that diagonalise this matrix. So when you do that, then your density matrix will have zeros everywhere except one of the diagonal element. Okay, so when you have a density matrix like this, you have a pure state. In a mixed state you would have multiple diagonal elements that are non-zero. And those multiple non-zero diagonal elements will all add up to equal to 1. That's the density matrix for a mixed ensemble or mixed state. So let's take an example here, so consider a pure ensemblle where all spins are up along the Z direction. Okay, so this is the case for the result of the storm apparatus that I have shown to you earlier in this video. In that case the fractional wait for spin-up state is one and all others are zero. In this case there is spin up and down that's the only two possibilities. The only one other term that has zero fractional weight. So in this case you simply have this one term for the density operator. And if you use the spin-up and spin along the Z direction, these cats plus and minus cats as your basis vector, then this. This operator can be expressed by this matrix and only one term in the diagonal is one and everything else is zero. This is the characteristics of the density matrix. For a pure, for example, for an unpressurized electron beam with random spins from one electron to another spin state constantly change randomly. Right then, roughly, if if you just consider the spin state along the Z-direction, half of the electron will have will result in spin-up along the direction, and half of them will result in spin-down along the direction. So the fractional waits for spin up and spin down states are both one half. So your density operator can then be expressed as this. The outer product plus an outer product of minus each one has a fractional weight 0.5 multiplied to it. And if you express this operator in a matrix warm using plus and minus kets as your basis said, then you have a still diagonal matrix, but there are multiple elements being non-zero. This is the characteristic of a mixed state. And once again I want to point out that the density matrices will look different that's the operator. Okay, is the same operator, but even for the same operator, if you choose different basis set, their matrix representation will look different. The same is true for density matrices. So your your density matrix will look different if you choose a different basis set, for example, instead of this plus and minus along the direction as the basis set. We can choose spin up and spin down along X direction as our basis ket. We can do that and go through the same, you use the same definitions, the same equations for the density operator, but simply use different basis set. Spin up and spin down along X direction as a basis set, the matrix will look different.
