Symmetric Extendability of Quantum States

Basic Formulation of the Problem

Given some state ρ^AB, we can always write it in the following form:

where d_A is the dimension of Alice’s system and d_B the dimension of Bob’s system. The operators {σ_i^A}_i=0^d_A2−1 and {σ_j^B}_j=0^d_B2−1 are the generalizations of the familiar Pauli operators in two dimensions:

These operators form a basis for the space of all 2 × 2 Hermitian matrices. This means that any 2 × 2 matrix can be written as a linear combination of the Pauli operators with real number coefficients. The operators {σ_x, σ_y, σ_z} also form a basis for the Lie algebra su(2). Similarly, in higher dimensions, the operators {σ_i^A}_i=0^d_A2−1 are such that σ₀^A = 1_A and {σ_i^A}_i=1^d_A2−1 is a basis for the Lie algebra su(d_A). These basis elements of the Lie algebra are also traceless. This immediately implies that if ρ^AB is a state, that is, has unit trace, then α_0,0 = 1.

Now, we want to know whether ρ^AB has a symmetric extension. This means that we want to know if there exists a state ρ^ABB′ such that Tr_B[ρ^ABB′] = Tr_B′[ρ^ABB′] = ρ^AB. Just as we can write ρ^AB in the basis of the generalized Pauli operators, we can write ρ^ABB′ as

where {β_i,j,j′} are d_A²d_B⁴ real numbers whose values need to be determined so that ρ^ABB′ is a symmetric extension. This means that ρ^ABB′ must firstly be a state, that is, a positive semi-definite operator with unit trace, and secondly satisfy Tr_B[ρ^ABB′] = Tr_B′[ρ^ABB′] = ρ^AB. The unit-trace condition immediately allows us to impose β_0,0,0 = 1. The symmetric extension conditions allow us to immediately determine many more of the values of the β_i,j,j′ coefficients. In particular,

where we have used the fact that all the σ_j^B operators are traceless except for the the zeroth one, which is equal to the identity and therefore has trace d_B. For this to be equal to ρ^AB, we need

Similarly,

For this to be equal to ρ^AB, we need

So, among the d_A²d_B⁴ open parameters, the symmetric extension condition allows us to fix 2d_A²d_B² − 1 of them. The rest must now be chosen so that ρ^ABB′ is positive semi-definite, i.e., has non-negative eigenvalues. This is not such an easy thing to do!

Remarkably, when d_A = d_B = 2, that is, both Alice and Bob’s systems are two-dimensional, there exists the following condition on ρ^AB that is necessary and sufficient for it to be symmetrically extendable:

This was first conjectured by Geir Ove Myhr in his PhD Thesis and later proven by Chen et al. This means that in the case d_A = d_B = 2, we need not go through the trouble of finding values for the open parameters β_i,j,j′ – we can simply plug in our given state into the above inequality, which is pretty easy to evaluate, and see if it holds! If it does, then we know our state is symmetrically extendable.

It would be great to have a corresponding condition for any dimension of Alice and Bob’s systems. Unfortunately, such conditions don’t exist except in very special cases (see, e.g., Terhal et al., Ranade, and Johnson and Viola). In fact, such a simple condition as the one above is not expected to exist in general since that condition depends only on the spectrum (that is, the eigenvalues) of ρ^AB, and for higher dimensions it has been shown that simply considering the spectrum is not sufficient to determine symmetric extendability (see Myhr and Lütkenhaus).

Formulation as a Semi-Definite Program

Fortunately, it is always possible numerically to determine if a given state is symmetrically extendable using the following semi-definite program (SDP):

SDPs are a particular kind of optimization problem. You can find a general introduction to them in Watrous.

In this particular SDP, the “optimization variable” is the matrix R and the function that we are minimizing is t. The bottom two constraints on R are precisely the symmetric extendability conditions, and the first constraint on R is the positive semi-definite constraint. This SDP finds the smallest possible value of t such that R is a symmetric extension of ρ^AB. In particular, if t ≤ 0, then there exists a positive semi-definite R that satisfies the symmetric extendability conditions, which means that ρ^AB is symmetrically extendable with R a symmetric extension. On the other hand, if t > 0 then there does not exist a positive semi-definite R satisfying the symmetric extendability conditions, which means that ρ^AB is not symmetrically extendable. To summarize,