QuantumPhysics.dvi

where 0≤pi≤1. In this special basis, we define the following states ofHb,

|i,b′′〉≡

∑

m

Cim|m,b′〉 (17.21)

Note that these states need to be neither orthogonal to one another, nor normalized. Hence the
pure state may be expressed as

|Ψ〉=

∑

i

|i,a〉⊗|i,b′′〉 (17.22)

We now use this expression to evaluate the density operatorρa, and find,

ρa=

∑

i,j

〈j,b′′|i,b′′〉|i,a〉〈j,a| (17.23)

But, we had already assume thatρawas actually diagonal in the basis|i,a〉, with eigenvaluespi,
so that we must have

〈j,b′′|i,b′′〉=piδij (17.24)

Thus, we conclude that the states|i,b′′〉are in fact orthogonal to one another. It now suffices to
normalize the states by

|i,b〉=

1

√

pi

|i,b′′〉 (17.25)

to recover the formula (17.18) of Schmidt’s theorem. In thisbasis now, we may compute alsoρb,
and we find,

ρb=

∑

i

pi|i,b〉〈i,b| (17.26)

Thus, the probability assignments ofρaandρb, in this basis, are identically the same. In particular,
the rank ofρacoincides with the rank ofρb, and this is referred to as theSchmidt number. It is
obviously a positive integer.

17.4 Generalized description of entangled states

Consider again, as in the preceding subsection, a full quantum system with Hilbert spaceHab, built
out of two subsystemsaandbwith respective Hilbert spacesHaandHb, so thatHab=Ha⊗Hb.
By the Schmidt purification theorem, we then have

ρa =

∑s

i=1

pi|i,a〉〈i,a|

ρb =

∑s

i=1

pi|i,b〉〈i,b|

|Ψ〉 =

∑s

i=1

√

pi|i,a〉⊗|i,b〉 (17.27)