18.435/2.111 POVM LectureWe have already seen how von Neumann, or projective, measurements work. Aswe have mentioned, these are not the only kind of measurements. The most generalkind of measurements are called POVM’s. POVM stands for positive operator valuedmeasure. We will not be talking abo ut the most general kind of POVM (in which thereis a continuous family of outcomes, and sums are replaced by integrals; this is whythey are called measures). We will only deal with the case where there are a finitenumber of possible outcomes, over a finite dimensional quantum space. In this case,the elements are Hermitian matrices (finite dimensional positive operators).Before we can talk about POVM’s, we should probably review the case of projectivemeasurements where some of the projections are on subspaces of dimension higher than1. We use such measurements implicitly, when we measure some qubits of a quantumcomputer, and leave other qubits untouched, but it has been a while since we gavea formal mathematical definition (if we did), so I will review this now. Suppose wehave a set of projectors onto subspaces Π1, Π2, . . ., Πk, with the property that thesesubap ces are orthogonal, soΠiΠj= 0 if i 6= jand these subspaces span the entire space; that is,kXi=1= I.Then there is a projective measurement associated with these subspaces which takes aquantum state | ψi to the stateΠi| ψihψ | Πi| ψi1/2with probabilityhψ | Πi| ψi .That is, it projects the state | ψi onto the i’th subspace with probability propo rt io na l tothe square of the length of this projection. It should be clear that the case where eachsubspace has dimensio n 1 corresponds t o measuring with respect to an orthonormalbasis, the best known case of quantum measurements.To motivate POVM’s let us consider an example. Suppose we have a qubit, so itsstate is a unit vector in the space with basis | 0i and | 1 i. We can embed this space in alarge space by simply adding a number of extra basis vectors. In coordinate notation,this corresponds to adding extra 0’s to the coordinates of each state vector.Let’s add the basis vector | 2i to a qubit. Now, there are orthono rmal basis ofthis 3-dimensional space where none of the basis vectors lie in the subspace containingthe original qubit. What happens when we choose one of these bases for a projective1measurement? What effect does this measurement have on the o riginal qubit? Let’slook at the example given by the orthonormal basis:√2√3| 0i +1√3| 2i−1√6| 0i +1√2| 1i +1√3| 2i−1√6| 0i −1√2| 1i +1√3| 2i .When we take our vector | ψi and measure it using the above basis, what happens?The probability of the first outcome is(√2√3h0 | +1√3h2 |) | ψi2=√2√3h0|ψi2since h2 | is orthogonal to | ψi. If we define the unnormalized quantum states| e1i =√2√3| 0i| e2i = −1√6| 0i +1√2| 1i| e3i = −1√6| 0i −1√2| 1iwe similarly see that the probability of outcome i is|hei|ψi|2.Now, suppose we have a number of these unnormalized vectors | eii, and we askwhen can t he above rule for choosing probabilities of outcomes possibly form a mea-surement. A necessary condition is that the probabilities add to 1; that is,kXi=1|hei|vi|2= 1for a ll unit vectors | vi in our quantum state space. This condition is equivalent tokXi=1hv|eiihei|vi = 1and moving the sum inside the hv | · | vi we havehv | kXi=1| eiihei|!| vi = 1for all unit vectors | vi. However, any Hermitian matrix M satisfying hv | M | vi = 1for all unit | vi must be the identity matrix (one can easily prove all its eigenvalues are1). Thus, we have the necessary conditionkXi=1| eiihei| = I.2It turns out that this is necessary and sufficient for a collection of unnormalized vectors| eii to be the special kind of POVM all of whose elements are rank 1. We next showthat if we have a collection of such | eii, we can achieve the above outcome probabilitiesby using a projective measurement in a higher dimensional space.Suppose that we have k unnormalized quantum states | eii in n dimensions suchthatkXi=1| eiihei| = I.Let us consider the k × n matrix M obtained by putting these vectors in the columnsof a matrix. The ent ry Mi,jis the i’th coordinate of | eji, or h i| eji. The matrix thuslooks likeh1|e1i h1|e2i . . . h1|ekih2|e1i h2|e2i . . . h2|eki· · · · · · · · · · · ·hn|e1i hn|e2i . . . h n| ekiNow, I’d like to claim that all of the rows fo this matrix are orthonormal. Let usconsider t he inner product of row i and row i′. We have that this iskXj=1hi|eJiejiHowever, we can move the sum into the middle, where we obta in the identity matrix,becaus of the condition on the | eji. We thus have that the inner product of rows i andi′ishi′|kXj=1| ejihej| | i′i= hi′|ii = δi′,iWe now have a set of n orthonormal rows in a k-dimensio nal space. By using Gram-Schmidt, we can extend these to a set of k orthonormal rows. Since any square matrixwhose rows are orthonormal is unitary, and thus has orthonomral columns, the columnsof this new k × k matrix correspond to a projective measurement. If this measurementis r estricted to act on the n-dimensional subspace given by the first n basis vectors,this becomes the POVM given by the | eji that we started with.Thus, we have discovered that if we start with any projective measurement withrank 1 projectors on a large space, and restrict to a smaller space, it can be extressedas a POVM given by a set of unnormalized vectors | eii with the conditionkXi=1| eiihei| = I.Conversely, any POVM with rank 1 elements | eii satisfying this condition can beexpressed as a von Neumann measurement in a higher dimensional space, of which theoriginal space is a subspace.3Now, we can deal with POVM’s with elements greater than ra nk 1. suppose youhave a set of matrices EiwithPki=1Ei= I Then, if we let the probability of outcome iwhen the measurement is applied to | ψi be pi= hψ | Ei| ψi, then essentially the samecalculation as above:Xipi=Xihψ | Ei| ψi=XiTr | ψihψ | Ei= Tr | psiihpsi |XiEi= Tr | psiihpsi |= 1shows that the sum o f the probabilities is equal to 1. Thus, a set of matrices Eiispotentially a POVM measurement. Can this measurement be realized? We will showthat it is always possible t o realize such a POVM as a projective measurement in higherdimensions.How can we do this? What we will do is refine the measurement corresponding tothe set {Ei} to a