11 Quantum states (output)

Let’s take stock. We learned that mea­sure­ment out­comes are rep­re­sented by the sub­spaces of a vector space. Because sub­spaces cor­re­spond one-​​to-​​one to pro­jec­tors, this is the same as saying that

  • Mea­sure­ment out­comes are rep­re­sented by the pro­jec­tors of a vector space.

We also learned that

  • The out­comes of com­pat­ible ele­men­tary tests cor­re­spond to com­muting projectors.

Finally, we decided that if the sub­spaces A and B rep­re­sent two pos­sible out­comes of a mea­sure­ment with three pos­sible out­comes, then p(AUB) = p(A) + p(B). If we fur­ther take into account that

  1. two sub­spaces cor­re­sponding to dif­ferent out­comes of the same mea­sure­ment are orthog­onal (oper­a­tionally this means that the prob­a­bility of obtaining dif­ferent out­comes in one and the same mea­sure­ment is zero, for­mally this means that every vector in one sub­space is orthog­onal to every vector in the other),
  2. pro­jec­tors are said to be orthog­onal if the cor­re­sponding sub­spaces are,
  3. the sum of two orthog­onal pro­jec­tors is another orthog­onal projector,

we arrive at the following:

  • If PA and PB are orthog­onal pro­jec­tors, then the prob­a­bility of the out­come rep­re­sented by the sum of pro­jec­tors PA + PB is the sum of the prob­a­bil­i­ties of the out­comes rep­re­sented by PA and PB, respec­tively.

These three items (or “pos­tu­lates”) allow us[1] to prove Gleason’s the­orem[2], which holds for vector spaces with at least three dimen­sions. (More recently the validity of Gleason’s the­orem has been extended to include 2-​​dimensional vector spaces.) The the­orem states that the prob­a­bility of obtaining the out­come rep­re­sented by the pro­jector P is given by

(Trace Rule)   p(P) = Tr(WP),

where W is a unique oper­ator, known as den­sity oper­ator, whose prop­er­ties will be listed presently. To obtain the trace of an oper­ator X, we apply X to the basis vec­tors a1, a2, a3,…, take the inner product of the resulting vec­tors with the same basis vec­tors, and sum over the basis vectors:

Tr(X) = <a1|Xa1> + <a2|Xa2> + <a3|Xa3> + ···

If P projects into a 1-​​dimensional sub­space con­taining the unit vector v, the trace rule reduces to

p(P) = <v|Wv>.

The World According to Quantum MechanicsThe prop­er­ties of the den­sity oper­ator (and the rea­sons why it has them) are as follows:

  • W is linear. This ensures that the third pos­tu­late is sat­is­fied: p(P1+P2) = p(P1) + p(P2), where P1 and P2 are orthogonal.
  • W is self-​​adjoint. This ensures that the prob­a­bility <v|Wv> is a real number. (What could be the meaning of a com­plex probability?)
  • W is pos­i­tive. This ensures that <v|Wv> does not come out neg­a­tive. (What could be the meaning of a neg­a­tive probability?)
  • The trace of W equals 1. This ensures that the prob­a­bil­i­ties of the pos­sible out­comes of a mea­sure­ment add up to 1. Together with the pos­i­tivity of the den­sity oper­ator, it ensures that no prob­a­bility comes out greater than 1. (What could be the meaning of a prob­a­bility greater than 1?)
  • W2 = W   or   W2 < W.

The equality of an oper­ator with its own square is char­ac­ter­istic of a pro­jector. Since the trace of W equals 1, and since the trace of a pro­jector equals the dimen­sion of the sub­space into which it projects, the equality W2 = W tells us that W projects into a 1-​​dimensional sub­space. Since we began by upgrading from a point in a phase space to a line (or 1-​​dimensional sub­space) in a vector space, we are not sur­prised by this result. If that sub­space con­tains the unit vector w, the trace rule reduces to

p(P) = <w|Pw>,

and if P projects into a 1-​​dimensional sub­space con­taining the unit vector v, it fur­ther reduces to

(Born’s Rule)   p(P) = |<v|w>|2.

If the den­sity oper­ator sat­is­fies the equality W2 = W, it is known as (or said to describe) a pure state, and the unit vector w is the so-​​called state vector. (Although W is uniquely deter­mined by w, the con­verse is not true. If w1 and w2 only differ by their phases, they deter­mine the same den­sity oper­ator and, hence, yield the same prob­a­bil­i­ties. They are there­fore phys­i­cally equivalent.)

If the den­sity oper­ator sat­is­fies the inequality W2 < W, it is known as (or said to describe) a mix­ture or mixed state. A pure state defines prob­a­bil­i­ties dis­tri­b­u­tions. It is a machine with inputs and out­puts: insert the pos­sible out­comes of the mea­sure­ment you are going to make, insert the time of the mea­sure­ment, and get the prob­a­bil­i­ties with which those out­comes are obtained. A mixed state defines a prob­a­bility dis­tri­b­u­tion over prob­a­bility dis­tri­b­u­tions. It adds a second layer of uncer­tainty to the uncer­tainty inherent in a pure state.

There are sit­u­a­tions in which this addi­tional uncer­tainty is sub­jec­tive in the same sense in which prob­a­bility dis­tri­b­u­tions over a clas­sical phase space are sub­jec­tive: the uncer­tainty arises from a lack of knowl­edge of rel­e­vant facts. But there are also sit­u­a­tions in which the addi­tional uncer­tainty is due to a lack of rel­e­vant facts. In these sit­u­a­tions it rep­re­sents an addi­tional objec­tive fuzzi­ness, over and above that asso­ci­ated with the indi­vidual algo­rithms. Later we shall come across exam­ples of this kind of uncertainty.


1. [↑] Peres, A. (1995). Quantum Theory: Con­cepts and Methods, Kluwer, p. 190.

2. [↑] Gleason, A.M. (1957). Mea­sures on the closed sub­spaces of a Hilbert space, Journal of Math­e­matics and Mechanics 6, 885–894.