Symmetric Extendability of Quantum States

Basic Formulation of the Problem

Given some state ρAB, we can always write it in the following form:

symext_20x

where dA is the dimension of Alice’s system and dB the dimension of Bob’s system. The operators {σiA}i=0dA21 and {σjB}j=0dB21 are the generalizations of the familiar Pauli operators in two dimensions:

symext_21x

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 {σx, σy, σz} also form a basis for the Lie algebra su(2). Similarly, in higher dimensions, the operators {σiA}i=0dA21 are such that σ0A = 1A and {σiA}i=1dA21 is a basis for the Lie algebra su(dA). These basis elements of the Lie algebra are also traceless. This immediately implies that if ρAB is a state, that is, has unit trace, then α0,0 = 1.

Now, we want to know whether ρAB has a symmetric extension. This means that we want to know if there exists a state ρABB such that TrB[ρABB] = TrB[ρABB] = ρAB. Just as we can write ρAB in the basis of the generalized Pauli operators, we can write ρABB as

symext_22x

where {βi,j,j} are dA2dB4 real numbers whose values need to be determined so that ρABB is a symmetric extension. This means that ρABB must firstly be a state, that is, a positive semi-definite operator with unit trace, and secondly satisfy TrB[ρABB] = TrB[ρABB] = ρAB. The unit-trace condition immediately allows us to impose β0,0,0 = 1. The symmetric extension conditions allow us to immediately determine many more of the values of the βi,j,j coefficients. In particular,

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where we have used the fact that all the σjB operators are traceless except for the the zeroth one, which is equal to the identity and therefore has trace dB. For this to be equal to ρAB, we need

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Similarly,

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For this to be equal to ρAB, we need

symext_26x

So, among the dA2dB4 open parameters, the symmetric extension condition allows us to fix 2dA2dB2 1 of them. The rest must now be chosen so that ρABB is positive semi-definite, i.e., has non-negative eigenvalues. This is not such an easy thing to do!

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

symext_27xThis was first conjectured by Geir Ove Myhr in his PhD Thesis and later proven by Chen et al. This means that in the case dA = dB = 2, we need not go through the trouble of finding values for the open parameters βi,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 ρAB, 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):

symext_27x

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 ρAB. In particular, if t 0, then there exists a positive semi-definite R that satisfies the symmetric extendability conditions, which means that ρAB 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 ρAB is not symmetrically extendable. To summarize,

symext_28x