nLab
tensor product of Banach spaces

Context

Functional analysis

Monoidal categories

Tensor products of Banach spaces

Idea

There are various norms that may be placed on the tensor product of the underlying vector spaces of two Banach spaces; the result is not usually complete, but of course we may take its completion. One of these, the projective tensor product, makes Ban (the category of Banach spaces and short linear maps) into a closed monoidal category, but any of them makes BanBan into a symmetric monoidal category. If we start with Hilbert spaces, then a different choice of norm is needed to make the result into a Hilbert space; then Hilb also becomes a closed symmetric monoidal category.

Definitions

Let VV and WW be Banach spaces, and let VWV \otimes W be their tensor product as vector spaces. To define a tensor product of VV and WW as Banach spaces, we will place a norm on VWV \otimes W, making a normed vector space; the only difference in the following definitions is which norm to use. Then we take the completion V^WV {\displaystyle\hat{\otimes}} W, which is a Banach space.

Definition (projective tensor product)

Every element of VWV \otimes W may be written (in many different ways) as a formal linear combination of formal tensor products of elements of VV and WW (suppressing the symbol \otimes):

iα iv iw i. \sum_i \alpha_i v_i w_i .

Let the projective cross norm x π{\|x\|_\pi} of an element xx of VWV \otimes W be

x πinf{ i|α i|v i Vw i W|x= iα iv iw i}. {\|x\|_\pi} \coloneqq \inf \{ \sum_i {|\alpha_i|} {\|v_i\|_V} {\|w_i\|_W} \;|\; x = \sum_i \alpha_i v_i w_i \} .

Then the projective tensor product V^ πWV {\displaystyle\hat{\otimes}_\pi} W of VV and WW is the completion of VWV \otimes W under the projective cross norm.

Definition (injective tensor product)

If λ\lambda and μ\mu are linear functionals on VV and WW (respectively), then λμ\lambda \otimes \mu is a linear functional on VWV \otimes W. Let the injective cross norm x ϵ{\|x\|_\epsilon} of an element xx of VWV \otimes W be

x ϵsup{|(λμ)x||λ V *,μ W *1}. {\|x\|_\epsilon} \coloneqq \sup \{ {|(\lambda \otimes \mu)x|} \;|\; {\|\lambda\|_{V^*}}, {\|\mu\|_{W^*}} \leq 1 \} .

Then the injective tensor product V^ ϵWV {\displaystyle\hat{\otimes}_\epsilon} W of VV and WW is the completion of VWV \otimes W under the injective cross norm.

Definition (tensor product of Hilbert spaces)

If VV and WW are Hilbert spaces, then their norms determine and are determined by their inner products, so let us discuss inner products. The elements of VWV \otimes W are generated by elements of the form vwv w, so set

v 1w 1,v 2w 2v 1,v 2w 1,w 2 \langle{v_1 w_1, v_2 w_2}\rangle \coloneqq \langle{v_1, v_2}\rangle \langle{w_1, w_2}\rangle

and extend by linearity. We write the norm of an element xx of the inner product space VWV \otimes W as x σ{\|x\|_\sigma}. Then the tensor product V^ σWV {\displaystyle\hat{\otimes}_\sigma} W of the Hilbert spaces VV and WW is the completion of VWV \otimes W under this norm (or inner product).

Cross norms

Besides the specific norms defined above, we can define axioms of a reasonable norm on VWV \otimes W.

Definition (cross norm)

A cross norm on VV and WW is any norm χ\chi on VWV \otimes W such that:

  • vw χ=v Vw W{\|v \otimes w\|_\chi} = {\|v\|_V} {\|w\|_W} for any elements vv and ww of VV and WW (respectively);
  • λμ χ *=λ V *μ W *{\|\lambda \otimes \mu\|_{\chi^*}} = {\|\lambda\|_{V^*}} {\|\mu\|_{W^*}} for any bounded linear functionals λ\lambda and μ\mu on VV and WW (respectively).
Definition (uniform cross norm)

A uniform cross norm is an operation that takes two Banach spaces and returns a norm on their algebraic tensor product, naturally in the two spaces. Equivalently, it's a functor χ:Ban×BanNVect\chi\colon Ban \times Ban \to NVect that makes the following diagram commute (or fills it with a natural isomorphism):

Ban×Ban χ NVect Vect×Vect Vect \array { Ban \times Ban & \overset{\chi}\rightarrow & NVect \\ \downarrow & & \downarrow \\ Vect \times Vect & \underset{\otimes}\rightarrow & Vect }

A uniform cross norm is obviously desirable from the nPOV, but does it meet the analysts' needs for a cross norm? Yes:

Proposition

A uniform cross norm assigns a cross norm to any two Banach spaces.

The specific cross norms from the previous section qualify as much as possible:

Proposition

The projective and injective cross norms are uniform cross norms (and hence are in fact cross norms). The norm on the algebraic tensor product of two Hilbert spaces is also a cross norm.

As far as I can tell, the Hilbert-space cross norm σ\sigma doesn't apply to arbitrary Banach spaces, so it doesn't define a uniform cross norm as defined above; however, it does define a functor on Hilb×HilbHilb \times Hilb, so it's as uniform as could be expected.

Looking only at the general theory of cross norms, the projective and injective cross norms appear naturally:

Proposition

If χ\chi is any uniform cross norm, VV and WW are any Banach spaces, and xx is any element of VWV \otimes W, then

x ϵx χx π. {\|x\|_\epsilon} \leq {\|x\|_\chi} \leq {\|x\|_\pi} .

That is, we have a poset of uniform cross norms, and the projective and injective cross norms are (respectively) the top and bottom of this poset.

Although σ\sigma is not a uniform cross norm, it relates to ϵ\epsilon and π\pi in the same way:

Proposition

If VV and WW are Hilbert spaces and xx is an element of VWV \otimes W, then

x ϵx σx π. {\|x\|_\epsilon} \leq {\|x\|_\sigma} \leq {\|x\|_\pi} .

Actually, this would all be simpler if Propostion 3 applied to arbitrary cross norms and not just uniform ones. Perhaps it does. Or perhaps σ\sigma extends to a uniform cross norm on all of BanBan; that would also make things simpler. I don't know.

Of course, any cross norm χ\chi on VV and WW allows us to form the Banach space V^ χWV {\displaystyle\hat{\otimes}_\chi} W, which may reasonably be called a tensor product of VV and WW; that's why we care.

Schmidt decomposition

The Schmidt decomposition is a way of expressing a pure state in the tensor product of two Hilbert spaces in terms of states of the two components:

Theorem Nielsen and Chuang Theorem 2.7

Let AA and BB be finite-dimensional Hilbert spaces. Let |ψ|\psi\rangle be a pure state of ABA \otimes B. Then there exist orthonormal families? {|i A} i\{ |i_A \rangle \}_i in AA and {|i B} i\{ |i_B \rangle \}_i in BB, and non-negative real numbers λ i\lambda_i, such that

(1)|ψ= iλ i|i A|i B |\psi\rangle = \sum_i \lambda_i |i_A \rangle \otimes |i_B\rangle

and iλ i 2=1\sum_i \lambda_i^2 = 1.

The numbers λ i\lambda_i are called the Schmidt co-efficients of |ψ|\psi\rangle, and the families {|i A}\{ |i_A\rangle \} and {|i B}\{ |i_B\rangle \} the Schmidt bases for AA and BB.

Definition

The Schmidt number of |ψ|\psi\rangle is the number of non-zero Schmidt coefficients of |ψ|\psi\rangle.

Foundational issues

We need the Hahn–Banach theorem for ϵ\epsilon to be a cross norm; but σ\sigma and π\pi work regardless. Possibly some of the other propositions rely on some other form of the axiom of choice; I haven't seen their proofs.

References

  • M. Nielsen and I. Chuang. /Quantum Computation and Quantum Information/. Cambridge University Press. 2000.

Many facts taken from Wikipedia:

Revised on March 22, 2014 05:42:52 by Toby Bartels (98.23.133.242)