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Norms and Dual Norms as Supremums and Infimums

May 26th, 2012

Let \mathcal{H} be a finite-dimensional Hilbert space over \mathbb{R} or \mathbb{C} (the fields of real and complex numbers, respectively). If we let \|\cdot\| be a norm on \mathcal{H} (not necessarily the norm induced by the inner product), then the dual norm of \|\cdot\| is defined by

\displaystyle\|\mathbf{v}\|^\circ := \sup_{\mathbf{w} \in \mathcal{H}}\Big\{ \big| \langle \mathbf{v}, \mathbf{w} \rangle \big| : \|\mathbf{w}\| \leq 1 \Big\}.

The double-dual of a norm is equal to itself (i.e., \|\cdot\|^{\circ\circ} = \|\cdot\|) and the norm induced by the inner product is the unique norm that is its own dual. Similarly, if \|\cdot\|_p is the vector p-norm, then \|\cdot\|_p^\circ = \|\cdot\|_q, where q satisfies 1/p + 1/q = 1.

In this post, we will demonstrate that \|\cdot\|^\circ has an equivalent characterization as an infimum, and we use this characterization to provide a simple derivation of several known (but perhaps not well-known) formulas for norms such as the operator norm of matrices.

For certain norms (such as the “separability norms” presented at the end of this post), this ability to write a norm as both an infimum and a supremum is useful because computation of the norm may be difficult. However, having these two different characterizations of a norm allows us to bound it both from above and from below.

The Dual Norm as an Infimum

Theorem 1. Let S \subseteq \mathcal{H} be a bounded set satisfying {\rm span}(S) = \mathcal{H} and define a norm \|\cdot\| by

\displaystyle\|\mathbf{v}\| := \sup_{\mathbf{w} \in S}\Big\{ \big| \langle \mathbf{v}, \mathbf{w} \rangle \big| \Big\}.

Then \|\cdot\|^\circ is given by

\displaystyle\|\mathbf{v}\|^\circ = \inf\Big\{ \sum_i |c_i| : \mathbf{v} = \sum_i c_i \mathbf{v}_i, \mathbf{v}_i \in S \ \forall \, i \Big\},

where the infimum is taken over all such decompositions of \mathbf{v}.

Before proving the result, we make two observations. Firstly, the quantity \|\cdot\| described by Theorem 1 really is a norm: boundedness of S ensures that the supremum is finite, and {\rm span}(S) = \mathcal{H} ensures that \|\mathbf{v}\| = 0 \implies \mathbf{v} = 0. Secondly, every norm on \mathcal{H} can be written in this way: we can always choose S to be the unit ball of the dual norm \|\cdot\|^\circ. However, there are times when other choices of S are more useful or enlightening (as we will see in the examples).

Proof of Theorem 1. Begin by noting that if \mathbf{w} \in S and \|\mathbf{v}\| \leq 1 then \big| \langle \mathbf{v}, \mathbf{w} \rangle \big| \leq 1. It follows that \|\mathbf{w}\|^{\circ} \leq 1 whenever \mathbf{w} \in S. In fact, we now show that \|\cdot\|^\circ is the largest norm on \mathcal{H} with this property. To this end, let \|\cdot\|_\prime be another norm satisfying \|\mathbf{w}\|_{\prime}^{\circ} \leq 1 whenever \mathbf{w} \in S. Then

\displaystyle \| \mathbf{v} \| = \sup_{\mathbf{w} \in S} \Big\{ \big| \langle \mathbf{w}, \mathbf{v} \rangle \big| \Big\} \leq \sup_{\mathbf{w}} \Big\{ \big| \langle \mathbf{w}, \mathbf{v} \rangle \big| : \|\mathbf{w}\|_{\prime}^{\circ} \leq 1 \Big\} = \|\mathbf{v}\|_\prime.

Thus  \| \cdot \| \leq \| \cdot \|_\prime, so by taking duals we see that \| \cdot \|^\circ \geq \| \cdot \|_\prime^\circ, as desired.

For the remainder of the proof, we denote the infimum in the statement of the theorem by \|\cdot\|_{{\rm inf}}. Our goal now is to show that: (1) \|\cdot\|_{{\rm inf}} is a norm, (2) \|\cdot\|_{{\rm inf}} satisfies \|\mathbf{w}\|_{{\rm inf}} \leq 1 whenever \mathbf{w} \in S, and (3) \|\cdot\|_{{\rm inf}} is the largest norm satisfying property (2). The fact that \|\cdot\|_{{\rm inf}} = \|\cdot\|^\circ will then follow from the first paragraph of this proof.

To see (1) (i.e., to prove that \|\cdot\|_{{\rm inf}} is a norm), we only prove the triangle inequality, since positive homogeneity and the fact that \|\mathbf{v}\|_{{\rm inf}} = 0 if and only if \mathbf{v} = 0 are both straightforward (try them yourself!). Fix \varepsilon > 0 and let \mathbf{v} = \sum_i c_i \mathbf{v}_i, \mathbf{w} = \sum_i d_i \mathbf{w}_i be decompositions of \mathbf{v}, \mathbf{w} with \mathbf{v}_i, \mathbf{w}_i \in S for all i, satisfying \sum_i |c_i| \leq \|\mathbf{v}\|_{{\rm inf}} + \varepsilon and \sum_i |d_i| \leq \|\mathbf{w}\|_{{\rm inf}} + \varepsilon. Then

\displaystyle \|\mathbf{v} + \mathbf{w}\|_{{\rm inf}} \leq \sum_i |c_i| + \sum_i |d_i| \leq \|\mathbf{v}\|_{{\rm inf}} + \|\mathbf{w}\|_{{\rm inf}} + 2\varepsilon.

Since \varepsilon > 0 was arbitrary, the triangle inequality follows, so \|\cdot\|_{{\rm inf}} is a norm.

To see (2) (i.e., to prove that \|\mathbf{v}\|_{{\rm inf}} \leq 1 whenever \mathbf{v} \in S), we simply write \mathbf{v} in its trivial decomposition \mathbf{v} = \mathbf{v}, which gives the single coefficient c_1 = 1, so \|\mathbf{v}\|_{{\rm inf}} \leq \sum_i c_i = c_1 = 1.

To see (3) (i.e., to prove that \|\cdot\|_{{\rm inf}} is the largest norm on \mathcal{H} satisfying condition (2)), begin by letting \|\cdot\|_\prime be any norm on \mathcal{H} with the property that \|\mathbf{v}\|_{\prime} \leq 1 for all \mathbf{v} \in S. Then using the triangle inequality for \|\cdot\|_\prime shows that if \mathbf{v} = \sum_i c_i \mathbf{v}_i is any decomposition of \mathbf{v} with \mathbf{v}_i \in S for all i, then

\displaystyle\|\mathbf{v}\|_\prime = \Big\|\sum_i c_i \mathbf{v}_i\Big\|_\prime \leq \sum_i |c_i| \|\mathbf{v}_i\|_\prime = \sum_i |c_i|.

Taking the infimum over all such decompositions of \mathbf{v} shows that \|\mathbf{v}\|_\prime \leq \|\mathbf{v}\|_{{\rm inf}}, which completes the proof.

The remainder of this post is devoted to investigating what Theorem 1 says about certain specific norms.

Injective and Projective Cross Norms

If we let \mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2, where \mathcal{H}_1 and \mathcal{H}_2 are themselves finite-dimensional Hilbert spaces, then one often considers the injective and projective cross norms on \mathcal{H}, defined respectively as follows:

\displaystyle \|\mathbf{v}\|_{I} := \sup\Big\{ \big| \langle \mathbf{v}, \mathbf{a} \otimes \mathbf{b} \rangle \big| : \|\mathbf{a}\| = \|\mathbf{b}\| = 1 \Big\} \text{ and}

\displaystyle \|\mathbf{v}\|_{P} := \inf\Big\{ \sum_i \| \mathbf{a}_i \| \| \mathbf{b}_i \| : \mathbf{v} = \sum_i \mathbf{a}_i \otimes \mathbf{b}_i \Big\},

where \|\cdot\| here refers to the norm induced by the inner product on \mathcal{H}_1 or \mathcal{H}_2. The fact that \|\cdot\|_{I} and \|\cdot\|_{P} are duals of each other is simply Theorem 1 in the case when S is the set of product vectors:

\displaystyle S = \big\{ \mathbf{a} \otimes \mathbf{b} : \|\mathbf{a}\| = \|\mathbf{b}\| = 1 \big\}.

In fact, the typical proof that the injective and projective cross norms are duals of each other is very similar to the proof of Theorem 1 provided above (see [1, Chapter 1]).

Maximum and Taxicab Norms

Use n to denote the dimension of \mathcal{H} and let \{\mathbf{e}_i\}_{i=1}^n be an orthonormal basis of \mathcal{H}. If we let S = \{\mathbf{e}_i\}_{i=1}^n then the norm \|\cdot\| in the statement of Theorem 1 is the maximum norm (i.e., the p = ∞ norm):

\displaystyle\|\mathbf{v}\|_\infty = \sup_i\Big\{\big|\langle \mathbf{v}, \mathbf{e}_i \rangle \big| \Big\} = \max \big\{ |v_1|,\ldots,|v_n|\big\},

where v_i = \langle \mathbf{v}, \mathbf{e}_i \rangle is the i-th coordinate of \mathbf{v} in the basis \{\mathbf{e}_i\}_{i=1}^n. The theorem then says that the dual of the maximum norm is

\displaystyle \|\mathbf{v}\|_\infty^\circ = \inf \Big\{ \sum_i |c_i| : \mathbf{v} = \sum_i c_i \mathbf{e}_i \Big\} = \sum_{i=1}^n |v_i|,

which is the taxicab norm (i.e., the p = 1 norm), as we expect.

Operator and Trace Norm of Matrices

If we let \mathcal{H} = M_n, the space of n \times n complex matrices with the Hilbert–Schmidt inner product

\displaystyle \big\langle A, B \big\rangle := {\rm Tr}(AB^*),

then it is well-known that the operator norm and the trace norm are dual to each other:

\displaystyle \big\| A \big\|_{op} := \sup_{\mathbf{v}}\Big\{ \big\|A\mathbf{v}\big\| : \|\mathbf{v}\| = 1 \Big\} \text{ and}

\displaystyle \big\| A \big\|_{op}^\circ = \big\|A\big\|_{tr} := \sup_{U}\Big\{ \big| {\rm Tr}(AU) \big| : U \in M_n \text{ is unitary} \Big\},

where \|\cdot\| is the Euclidean norm on \mathbb{C}^n. If we let S be the set of unitary matrices in M_n, then Theorem 1 provides the following alternate characterization of the operator norm:

Corollary 1. Let A \in M_n. Then

\displaystyle \big\|A\big\|_{op} = \inf\Big\{ \sum_i |c_i| : A = \sum_i c_i U_i \text{ and each } U_i \text{ is unitary} \Big\}.

As an application of Corollary 1, we are able to provide the following characterization of unitarily-invariant norms (i.e., norms \|\cdot\|_{\prime} with the property that \big\|UAV\big\|_{\prime} = \big\|A\big\|_{\prime} for all unitary matrices U, V \in M_n):

Corollary 2. Let \|\cdot\|_\prime be a norm on M_n. Then \|\cdot\|_\prime is unitarily-invariant if and only if

\displaystyle \big\|ABC\big\|_\prime \leq \big\|A\big\|_{op}\big\|B\big\|_{\prime}\big\|C\big\|_{op}

for all A, B, C \in M_n.

Proof of Corollary 2. The “if” direction is straightforward: if we let A and C be unitary, then

\displaystyle \big\|B\big\|_\prime = \big\|A^*ABCC^*\big\|_\prime \leq \big\|ABC\big\|_\prime \leq \big\|B\big\|_{\prime},

where we used the fact that \big\|A\big\|_{op} = \big\|C\big\|_{op} = 1. It follows that \big\|ABC\big\|_\prime = \big\|B\big\|_\prime, so \|\cdot\|_\prime is unitarily-invariant.

To see the “only if” direction, write A = \sum_i c_i U_i and C = \sum_i d_i V_i with each U_i and V_i unitary. Then

\displaystyle \big\|ABC\big\|_\prime = \Big\|\sum_{i,j}c_i d_j U_i B V_j\Big\|_\prime \leq \sum_{i,j} |c_i| |d_j| \big\|U_i B V_j\big\|_\prime = \sum_{i,j} |c_i| |d_j| \big\|B\big\|_\prime.

By taking the infimum over all decompositions of A and C of the given form and using Corollary 1, the result follows.

An alternate proof of Corollary 2, making use of some results on singular values, can be found in [2, Proposition IV.2.4].

Separability Norms

As our final (and least well-known) example, let \mathcal{H} = M_m \otimes M_n, again with the usual Hilbert–Schmidt inner product. If we let

\displaystyle S = \{ \mathbf{a}\mathbf{b}^* \otimes \mathbf{c}\mathbf{d}^* : \|\mathbf{a}\| = \|\mathbf{b}\| = \|\mathbf{c}\| = \|\mathbf{d}\| = 1 \},

where \|\cdot\| is the Euclidean norm on \mathbb{C}^m or \mathbb{C}^n, then Theorem 1 tells us that the following two norms are dual to each other:

\displaystyle \big\|A\big\|_s := \sup\Big\{ \big| (\mathbf{a}^* \otimes \mathbf{c}^*)A(\mathbf{b} \otimes \mathbf{d}) \big| : \|\mathbf{a}\| = \|\mathbf{b}\| = \|\mathbf{c}\| = \|\mathbf{d}\| = 1 \Big\} \text{ and}

\displaystyle \big\|A\big\|_s^\circ = \inf\Big\{ \sum_i \big\|A_i\big\|_{tr}\big\|B_i\big\|_{tr} : A = \sum_i A_i \otimes B_i \Big\}.

There’s actually a little bit of work to be done to show that \|\cdot\|_s^\circ has the given form, but it’s only a couple lines – consider it an exercise for the interested reader.

Both of these norms come up frequently when dealing with quantum entanglement. The norm \|\cdot\|_s^\circ was the subject of [3], where it was shown that a quantum state \rho is entangled if and only if \|\rho\|_s^\circ > 1 (I use the above duality relationship to provide an alternate proof of this fact in [4, Theorem 6.1.5]). On the other hand, the norm \|\cdot\|_s characterizes positive linear maps of matrices and was the subject of [5, 6].

References

  1. J. Diestel, J. H. Fourie, and J. Swart. The Metric Theory of Tensor Products: Grothendieck’s Résumé Revisited. American Mathematical Society, 2008. Chapter 1: pdf
  2. R. Bhatia. Matrix Analysis. Springer, 1997.
  3. O. Rudolph. A separability criterion for density operators. J. Phys. A: Math. Gen., 33:3951–3955, 2000. E-print: arXiv:quant-ph/0002026
  4. N. Johnston. Norms and Cones in the Theory of Quantum Entanglement. PhD thesis, University of Guelph, 2012.
  5. N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information TheoryJournal of Mathematical Physics, 51:082202, 2010.
  6. N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory IIQuantum Information & Computation, 11(1 & 2):104–123, 2011.

MATLAB Scripts for Computing Completely Bounded Norms via Semidefinite Programming

July 23rd, 2011

In operator theory, the completely bounded norm of a linear map on complex matrices \Phi : M_m \rightarrow M_n is defined by \|\Phi\|_{cb} := \sup_{k \geq 1} \| id_k \otimes \Phi \|, where \|\Phi\| is the usual norm on linear maps defined by \|\Phi\| := \sup_{X \in M_m} \{ \|\Phi(X)\| : \|X\| \leq 1\} and \|X\| is the operator norm of X [1]. The completely bounded norm is particularly useful when thinking of M_m and M_n as operator spaces.

The dual of the completely bounded norm is called the diamond norm, which plays an important role in quantum information theory, as it can be used to measure the distance between quantum channels. The diamond norm of \Phi is typically denoted \|\Phi\|_{\diamond}. For properties of the completely bounded and diamond norms, see [1,2,3].

A method for efficiently computing the completely bounded and diamond norms via semidefinite programming was recently presented in [4]. The purpose of this post is to provide MATLAB scripts that implement this algorithm and demonstrate its usage.

Download and Install

In order to make use of these scripts to compute the completely bounded or diamond norm, you must download and install two things: the SeDuMi semidefinite program solver and the MATLAB scripts themselves.

  1. SeDuMi – Please follow the instructions on the SeDuMi website to download and install it. If possible, you should install SeDuMi 1.1R3, not SeDuMi 1.21 or SeDuMi 1.3, since there is a bug with the newer versions when dealing with complex matrices.
  2. CB Norm MATLAB Package – Once SeDuMi is installed, download the CB norm MATLAB scripts, unzip them, and place them in your MATLAB scripts directory. The zip file contains 10 MATLAB scripts.

Once the scripts are installed, type “help CBNorm” or “help DiamondNorm” at the MATLAB prompt to learn how to use the CBNorm and DiamondNorm functions. Several usage examples are provided below.

Usage Examples

The representation of the linear map \Phi that the CBNorm and DiamondNorm functions take as input is a pair of arrays of its left- and right- generalized Choi-Kraus operators. That is, an array of operators \{A_i\} and \{B_i\} such that \Phi(X) = \sum_i A_i X B_i for all X.

Basic Examples

If we wanted to compute the completely bounded and diamond norms of the map

the MATLAB input and output would be as follows:

>> PhiA(:,:,1) = [1,1;1,0];
>> PhiA(:,:,2) = [1,0;1,2];
>> PhiB(:,:,1) = [1,0;0,1];
>> PhiB(:,:,2) = [1,2;1,1];
>> CBNorm(PhiA,PhiB)

ans =

    7.2684

>> DiamondNorm(PhiA,PhiB)

ans =

    7.4124

So we see that its completely bounded norm is 7.2684 and its diamond norm is 7.4124.

If we instead want to compute the completely bounded or diamond norm of a completely positive map, we only need to provide its Kraus operators – i.e., operators \{A_i\} such that \Phi(X) = \sum_i A_i X A_i^\dagger for all X. Furthermore, in this case semidefinite programming isn’t used at all, since [1, Proposition 3.6] tells us that \|\Phi\|_{cb} = \|\Phi(I)\| and \|\Phi\|_{\diamond} = \|\Phi^\dagger(I)\|, and computing \|\Phi(I)\| is trivial. The following example demonstrates the usage of these scripts in this case, via a completely positive map \Phi : M_3 \rightarrow M_2 with four (essentially random) Kraus operators:

>> PhiA(:,:,1) = [1 0 0;0 1 1];
>> PhiA(:,:,2) = [-3 0 1;5 1 1];
>> PhiA(:,:,3) = [0 2 0;0 0 0];
>> PhiA(:,:,4) = [1 1 3;0 2 0];
>> CBNorm(PhiA)

ans =

   42.0000

>> DiamondNorm(PhiA)

ans =

   38.7303

Transpose Map

Suppose we want to compute the completely bounded or diamond norm of the transpose map on M_n. A generalized Choi-Kraus representation is given by defining A_{ij} = B_{ij} = e_i e_j^\dagger, where \{e_i\} is the standard basis of \mathbb{C}^n (i.e., A_{ij} and B_{ij} are the operators with matrix representation in the standard basis with a one in the (i,j)-entry and zeroes elsewhere). It is known that the completely bounded and diamond norms of the n-dimensional transpose map are both equal to n, which can be verified in small dimensions as follows:

>> % 2-dimensional transpose
>> PhiA(:,:,1) = [1 0;0 0];
>> PhiA(:,:,2) = [0 1;0 0];
>> PhiA(:,:,3) = [0 0;1 0];
>> PhiA(:,:,4) = [0 0;0 1];
>> PhiB = PhiA;
>> CBNorm(PhiA,PhiB)

ans =

    2.0000

>> DiamondNorm(PhiA,PhiB)

ans =

    2.0000
>> % 3-dimensional transpose
>> I = eye(3);
>> for i=1:3
for j=1:3
PhiA(:,:,3*(i-1)+j) = I(:,i)*I(j,:);
end
end
>> PhiB = PhiA;
>> CBNorm(PhiA,PhiB)

ans =

    3.0000

>> DiamondNorm(PhiA,PhiB)

ans =

    3.0000

Difference of Unitary Channels

Now consider the map \Phi : M_2 \rightarrow M_2 defined by \Phi(X) = X - UXU^\dagger, where U is the following unitary matrix:

We know from [2, Theorem 12] that the CB norm and diamond norm of \Phi are both equal to the diameter of the smallest closed disc containing all of the eigenvalues of U. Because the eigenvalues of U are (1 \pm i)/\sqrt{2}, the smallest closed disc containing its eigenvalues has diameter \sqrt{2}, so \|\Phi\|_{cb} = \|\Phi\|_{\diamond} = \sqrt{2}. This result can be verified as follows:

>> PhiA(:,:,1) = [1 0;0 1];
>> PhiA(:,:,2) = [1 1;-1 1]/sqrt(2);
>> PhiB(:,:,1) = [1 0;0 1];
>> PhiB(:,:,2) = -[1 -1;1 1]/sqrt(2);
>> CBNorm(PhiA,PhiB)

ans =

    1.4142

>> DiamondNorm(PhiA,PhiB)

ans =

    1.4142

References

  1. V. I. Paulsen. Completely bounded maps and operator algebras. Cambridge University Press, 2003.
  2. N. Johnston, D. W. Kribs, and V. I. Paulsen. Computing stabilized norms for quantum operations via the theory of completely bounded maps. Quantum Inf. Comput., 9:16-35, 2009.
  3. J. Watrous. Theory of quantum information lecture notes.
  4. J. Watrous. Semidefinite programs for completely bounded norms. Theory Comput., 5:217–238, 2009.

Separability-Preserving Operators in Entanglement Theory

June 14th, 2011

One of the key concepts in quantum information theory is the difference between separable states and entangled states. A pure quantum state (that is, a unit vector) v ∈ CnCn is said to be separable if it can be written as v = a ⊗ b for some a,b ∈ Cn; otherwise v is called entangled. In this post we will investigate what operators preserve the set of separable pure states, as well as what operators entangle all separable pure states.

Separable Pure State Preservers and Entangling Gates

In the design of quantum algorithms, entangling gates play a very important role. Entangling gates are unitary operators that are able to generate entanglement. A bit more specifically, a unitary operator U ∈ Mn ⊗ Mn (where Mn is the space of n × n complex matrices) is called an entangling gate if there exists a separable pure state v = a ⊗ b ∈ CnCn such that Uv is entangled. Conversely, we will say that a unitary operator U preserves separability if Uv is separable whenever v is separable.

In order to answer the question of what unitaries preserve separability, it is instructive to consider some simple examples (this is often a useful way to formulate conjectures regarding preserver problems). For example, it is clear that if U = A ⊗ B for some unitary operators A, B ∈ Mn, then U preserves separability (because U(a ⊗ b) = Aa ⊗ Bb is separable). Another example of a unitary operator that preserves separability is the swap (or flip) operator S defined on separable states by S(a ⊗ b) = b ⊗ a (the action of S on the rest of CnCn is determined by extending linearly). It turns out that these are essentially the only operators that preserve separability [1,2,3]:

Theorem 1. Let U ∈ Mn ⊗ Mn be a unitary operator. Then U preserves separability (i.e., U is not an entangling gate) if and only if there exist unitary operators A, B ∈ Mn such that either U = A ⊗ B or U = S(A ⊗ B).

As we already saw, the “if” direction of the above result is trivial – the meat and potatoes of the theorem comes from the “only if” direction (as is typically the case with results about linear preservers). Theorem 1 was first proved in [1] essentially by case analysis and checking the action of a separability-preserving unitary on a basis of CnCn, and was subsequently re-proved using similar techniques (but with different motivations and connections) in [2]. The result was proved in [3] by using the vector-operator isomorphism and the fact that a linear map Φ : Mn → Mn preserves the set of rank-1 operators if and only if there exist A, B ∈ Mn such that either Φ(X) ≡ AXB or Φ(X) ≡ AXtB [4].

Theorem 1 also follows as a simple corollary of several related results that have recently been proved in [5,6]. A version of Theorem 1 for multipartite systems (i.e., systems that are the tensor product of more than two copies of Cn) can be found in [3] and [7].

Universal Entangling Gates

A universal entangling gate is, as its name suggests, a stronger form of an entangling gate – it is a unitary operator U such that U(a ⊗ b) is entangled for all a, b ∈ Cn (contrast this with entangling gates, which require only that U(a ⊗ b) is entangled for some a, b ∈ Cn). The structure of universal entangling gates is much less well-understood than that of entangling gates, though we can still at least say when they exist.

It is not difficult to convince yourself that universal entangling gates can’t exist in small dimensions. Let’s begin by supposing n = 2. The set of pure states in C2C2 can be regarded as a 7-dimensional real manifold (7 = 2 × (n × n) – 1, where we subtract one because pure states all have unit length), while the set of separable pure states in C2C2 can be regarded as a 5-dimensional real manifold (5 = (2 × n – 1) + (2 × n – 1) – 1, where the final one is subtracted because the overall phase of the first system relative to the second system is irrelevant). Thus, if U ∈ M2 ⊗ M2 were a universal entangler, it would have to send a 5-dimensional manifold into the 7 – 5 = 2 remaining dimensions of the space, which seems unlikely. Similarly, if n = 3 and U ∈ M3 ⊗ M3 were a universal entangler, it would have to send a 9-dimensional manifold into the 17 – 9 = 8 remaining dimensions of the space, which also seems unlikely.

Indeed, this type of argument was made rigorous via methods of algebraic geometry in [8], where the following result was proved:

Theorem 2. There exists a universal entangling gate in Mn ⊗ Mn if and only if n ≥ 4.

Despite knowing when universal entangling gates exist, we still don’t have a characterization of such operators, nor do we even have many explicit examples (does anyone have an explicit example for 3 ⊗ 4 or 4 ⊗ 4 systems?). Similar techniques to those used in the proof of Theorem 2 should also shed light on when universal entangling gates exist in multipartite systems Mn1 ⊗ Mn2 ⊗ … ⊗ Mnk, but to my knowledge this calculation has not been explicitly carried out.

References:

  1. M. Marcus and B. N. Moyls, Transformations on tensor product spaces. Pacific Journal of Mathematics 9, 1215–1221 (1959).
  2. F. Hulpke, U. V. Poulsen, A. Sanpera, A. Sen De, U. Sen, and M. Lewenstein, Unitarity as preservation of entropy and entanglement in quantum systems. Foundations of Physics 36, 477–499 (2006). E-print: arXiv:quant-ph/0407118
  3. N. Johnston, Characterizing Operations Preserving Separability Measures via Linear Preserver Problems. To appear in Linear and Multilinear Algebra (2011). E-print: arXiv:1008.3633 [quant-ph]
  4. L. Beasley, Linear operators on matrices: the invariance of rank k matrices. Linear Algebra and its Applications 107, 161–167 (1988).
  5. E. Alfsen and F. Shultz, Unique decompositions, faces, and automorphisms of separable states. Journal of Mathematical Physics 51, 052201 (2010). E-print: arXiv:0906.1761 [math.OA]
  6. S. Friedland, C.-K. Li, Y.-T. Poon, and N.-S. Sze, The automorphism group of separable states in quantum information theory. Journal of Mathematical Physics 52, 042203 (2011). E-print: arXiv:1012.4221 [quant-ph]
  7. R. Westwick, Transformations on tensor spaces. Pacific Journal of Mathematics 23, 613–620 (1967).
  8. J. Chen, R. Duan, Z. Ji, M. Ying, J. Yu, Existence of Universal Entangler. Journal of Mathematical Physics 49, 012103 (2008). E-print: arXiv:0704.1473 [quant-ph]

The Other Superoperator Isomorphism

November 20th, 2009

A few months ago, I spent two posts describing the Choi-Jamiolkowski isomorphism between linear operators from Mn to Mm (often referred to as “superoperators“) and linear operators living in the space Mn ⊗ Mm. However, there is another isomorphism between superoperators and regular operators — one that I’m not sure of any name for but which has just as many interesting properties.

Recall from Section 1 of this post that any superoperator Φ can be written as

\Phi(X)=\sum_iA_iXB_i.for some operators {Ai} and {Bi}. The isomorphism that I am going to focus on in this post is the one given by associating Φ with the operator

M_\Phi:=\sum_iA_i\otimes B_i^{T}.

The main reason that MΦ can be so useful is that it retains the operator structure of Φ. In particular, if you define vec(X) to be the vectorization of the operator X, then

{\rm vec}(\Phi(X))=M_\Phi{\rm vec}(X).

In other words, if you treat X as a vector, then MΦ is the operator describing the action of Φ on X. From this it becomes simple to compute some basic quantities describing Φ. For example, the induced Frobenius norm,

\big\|\Phi\big\|_F:=\sup_{\|X\|_F=1}\Big\{\big\|\Phi(X)\big\|_F\Big\},

is equal to the standard operator norm of MΦ. If n = m then we can define the eigenvalues {λ} and the eigenmatrices {V} of Φ in the obvious way via

\Phi(V)=\lambda V.

Then the eigenvalues of Φ are exactly the eigenvalues of MΦ, and the corresponding eigenvectors of MΦ are the vectorizations of the eigenmatrices of Φ. It is similarly easy to check whether Φ is invertible (by checking whether or not det(MΦ) = 0), find the inverse if it exists, or find the nullspace (and a pseudoinverse) if it doesn’t.

Finally, here’s a question for the interested reader to think about: why is the transpose required on the Bi operators for this isomorphism to make sense? That is, why can we not define an isomorphism between Φ and the operator

\sum_iA_i\otimes B_i?

Approximating the Distribution of Schmidt Vector Norms

November 6th, 2009

Recently, a family of vector norms [1,2] have been introduced in quantum information theory that are useful for helping classify entanglement of quantum states. In particular, the Schmidt vector k-norm of a vector v ∈ CnCn, for an integer 1 ≤ k ≤ n, is defined by

\|v\|_{s(k)}:=\sup_w\Big\{\big|\langle v,w\rangle\big|:\|w\|=1,SR(w)\leq k\Big\}.

In the above definition, SR(w) refers to the Schmidt rank of the vector w and so these norms are in some ways like a measure of entanglement for pure state vectors. One of the results of [2] shows how to compute these norms efficiently, so with that in mind we can perform all sorts of fun numerical analysis on them. Analytic results are provided in the paper, so I’ll provide more hand-wavey stuff and pictures here. In particular, let’s look at what the distributions of the Schmidt vector norms look like.

Figure 1: The Schmidt 1 and 2 vector norms in 3 ⊗ 3 dimensional space

Figure 1: The distribution of the Schmidt 1 and 2 vector norms in (3 ⊗ 3)-dimensional space

Figure 1 shows the distributions of the Schmidt 1 and 2 norms of unit vectors distributed according to the Haar measure in C3C3, based on 5×105 vectors generated randomly via MATLAB. Note that the Schmidt 3-norm just equals the standard Euclidean norm so it always equals 1 and is thus not shown. Figures 2 and 3 show similar distributions in C4C4 and C5C5.

Figure 2: The distribution of the Schmidt 1, 2, and 3 vector norms in (4 ⊗ 4)-dimensional space

Figure 2: The distribution of the Schmidt 1, 2, and 3 vector norms in (4 ⊗ 4)-dimensional space

Schmidt vector 1, 2, 3, and 4 norms for n = 5

Figure 3: The distribution of the Schmidt 1, 2, 3, and 4 vector norms in (5 ⊗ 5)-dimensional space

The following table shows various basic statistics about the above distributions. I suppose the natural next step is to ask whether or not we can analytically determine the distribution of the Schmidt vector norms. Since these norms are essentially just the singular values of an operator that is associated with the vector, it seems like this might even already be a (partially) solved problem, since many results are known about the distribution of the singular values of random matrices. The difficulty comes in trying to interpret the Haar measure (or any other natural measure on pure states, such as the Hilbert-Schmidt measure) on the associated operators.

Space k Mean Median Std. Dev.
C3C3 1 0.8494 0.8516 0.0554
2 0.9811 0.9860 0.0171
C4C4 1 0.7799 0.7792 0.0501
2 0.9411 0.9435 0.0247
3 0.9921 0.9943 0.0074
C5C5 1 0.7240 0.7225 0.0444
2 0.8976 0.8987 0.0268
3 0.9707 0.9722 0.0129
4 0.9960 0.9971 0.0039

References:

  1. D. Chruscinski, A. Kossakowski, G. Sarbicki, Spectral conditions for entanglement witnesses vs. bound entanglement, Phys. Rev A 80, 042314 (2009). arXiv:0908.1846v2 [quant-ph]
  2. N. Johnston and D. W. Kribs, A Family of Norms With Applications in Quantum Information Theory. Journal of Mathematical Physics 51, 082202 (2010). arXiv:0909.3907 [quant-ph]

The Equivalences of the Choi-Jamiolkowski Isomorphism (Part II)

October 23rd, 2009

This is a continuation of this post.
Please read that post to learn what the Choi-Jamiolkowski isomorphism is.

In part 1, we learned about hermicity-preserving linear maps, positive maps, k-positive maps, and completely positive maps. Now let’s see what other types of linear maps have interesting equivalences through the Choi-Jamiolkowski isomorphism. Recall that the notation CΦ is used to represent the Choi matrix of the linear map Φ.

6. Entanglement Breaking Maps / Separable Quantum States

An entanglement breaking map is defined as a completely positive map Φ with the property that (idn ⊗ Φ)(ρ) is a separable quantum state whenever ρ is a quantum state (i.e., a density operator). A separable quantum state σ is one that can be written in the form

\sigma=\sum_ip_i\sigma_i\otimes\tau_i,

where {pi} forms a probability distribution (i.e., pi ≥ 0 for all i and the pi‘s sum to 1) and each σi and τi is a density operator. It turns out that the Choi-Jamiolkowski equivalence for entanglement-breaking maps is very natural — Φ is entanglement breaking if and only if CΦ is separable. Because it is known that determining whether or not a given state is separable is NP-HARD [1], it follows that determining whether or not a given linear map is entanglement breaking is also NP-HARD. Nonetheless, there are several nice characterizations of entanglement breaking maps. For example, Φ is entanglement breaking if and only if it can be written in the form

\Phi(X)=\sum_iA_iXA_i^*,

where each operator Ai has rank 1 (recall from Section 4 of the previous post that every completely positive map can be written in this form for some operators Ai — the rank 1 condition is what makes the map entanglement breaking). For more properties of entanglement breaking maps, the interested reader is encouraged to read [2].

7. k-Partially Entanglement Breaking Maps / Quantum States with Schmidt Number at Most k

The natural generalization of entanglement breaking maps are k-partially entanglement breaking maps, which are completely positive maps Φ with the property that (idn ⊗ Φ)(ρ) always has Schmidt number [3] at most k for any density operator ρ. Recall that an operator has Schmidt number 1 if and only if it is separable, so the k = 1 case recovers exactly the entanglement breaking maps of Section 6. The set of operators associated with the k-partially entanglement breaking maps via the Choi-Jamiolkowski isomorphism are exactly what we would expect: the operators with Schmidt number no larger than k. In fact, pretty much all of the properties of entanglement breaking maps generalize in a completely natural way to this situation. For example, a map is k-partially entanglement breaking if and only if it can be written in the form

\Phi(X)=\sum_iA_iXA_i^*,

where each operator Ai has rank no greater than k. For more information about k-partially entanglement breaking maps, the interested reader is pointed to [4]. Additionally, there is an interesting geometric relationship between k-positive maps (see Section 5 of the previous post) and k-partially entanglement breaking maps that is explored in this note and in [5].

8. Unital Maps / Operators with Left Partial Trace Equal to Identity

A linear map Φ is said to be unital if it sends the identity operator to the identity operator — that is, if Φ(In) = Im. It is a simple exercise in linear algebra to show that Φ is unital if and only if

{\rm Tr}_1(C_\Phi)=I_m,

where Tr1 denotes the partial trace over the first subsystem. In fact, it is not difficult to show that Tr1(CΦ) always equals exactly Φ(In).

9. Trace-Preserving Maps / Operators with Right Partial Trace Equal to Identity

In quantum information theory, maps that are trace-preserving (i.e., maps Φ such that Tr(Φ(X)) = Tr(X) for every operator X ∈ Mn) are of particular interest because quantum channels are modeled by completely positive trace-preserving maps (see Section 4 of the previous post to learn about completely positive maps). Well, some simple linear algebra shows that the map Φ is trace-preserving if and only if

{\rm Tr}_2(C_\Phi)=I_n,

where Tr2 denotes the partial trace over the second subsystem. The reason for the close relationship between this property and the property of Section 8 is that unital maps and trace-preserving maps are dual to each other in the Hilbert-Schmidt inner product.

10. Completely Co-Positive Maps / Positive Partial Transpose Operators

A map Φ such that T○Φ is completely positive, where T represents the transpose map, is called a completely co-positive map. Thanks to Section 4 of the previous post, we know that Φ is completely co-positive if and only if the Choi matrix of T○Φ is positive semi-definite. Another way of saying this is that

(id_n\otimes T)(C_\Phi)\geq 0.

This condition says that the operator CΦ has positive partial transpose (or PPT), a property that is of great interest in quantum information theory because of its connection with the problem of determining whether or not a given quantum state is separable. In particular, any quantum state that is separable must have positive partial transpose (a condition that has become known as the Peres-Horodecki criterion). If n = 2 and m ≤ 3, then the converse is also true: any PPT state is necessarily separable [6]. It follows via our equivalences of Sections 4 and 6 that any entanglement breaking map is necessarily completely co-positive. Conversely, if n = 2 and m ≤ 3 then any map that is both completely positive and completely co-positive must be entanglement breaking.

11. Entanglement Binding Maps / Bound Entangled States

A bound entangled state is a state that is entangled (i.e., not separable) yet can not be transformed via local operations and classical communication to a pure maximally entangled state. In other words, they are entangled but have zero distillable entanglement. Currently, the only states that are known to be bound entangled are states with positive partial transpose — it is an open question whether or not other such states exist.

An entanglement binding map [7] is a completely positive map Φ such that (idn ⊗ Φ)(ρ) is bound entangled for any quantum state ρ. It turns out that a map is entanglement binding if and only if its Choi matrix CΦ is bound entangled. Thus, via the result of Section 10 we see that a map is entanglement binding if it is both completely positive and completely co-positive. It is currently unknown if there exist other entanglement binding maps.

References:

  1. L. Gurvits, Classical deterministic complexity of Edmonds’ Problem and quantum entanglement, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, 10-19 (2003). arXiv:quant-ph/0303055v1
  2. M. Horodecki, P. W. Shor, M. B. Ruskai, General Entanglement Breaking Channels, Rev. Math. Phys 15, 629–641 (2003). arXiv:quant-ph/0302031v2
  3. B. Terhal, P. Horodecki, A Schmidt number for density matrices, Phys. Rev. A Rapid Communications Vol. 61, 040301 (2000). arXiv:quant-ph/9911117v4
  4. D. Chruscinski, A. Kossakowski, On partially entanglement breaking channels, Open Sys. Information Dyn. 13, 17–26 (2006). arXiv:quant-ph/0511244v1
  5. L. Skowronek, E. Stormer, K. Zyczkowski, Cones of positive maps and their duality relations, J. Math. Phys. 50, 062106 (2009). arXiv:0902.4877v1 [quant-ph]
  6. M. Horodecki, P. Horodecki, R. Horodecki, Separability of Mixed States: Necessary and Sufficient Conditions, Physics Letters A 223, 1–8 (1996). arXiv:quant-ph/9605038v2
  7. P. Horodecki, M. Horodecki, R. Horodecki, Binding entanglement channels, J.Mod.Opt. 47, 347–354 (2000). arXiv:quant-ph/9904092v1