## Appendix A Convergence of the hierarchy

In the main text we saw how to construct an infinite sequence of decreasing semidefinite programming outer approximations to the quantum set . In the following, we show that the proposed hierarchy converges to The proof is a straightforward generalisation of the convergence proof from Miguel2008 .

For any integer let be the semidefinite program of size defined by the linear constraints identified in the main text. Let be a feasible solution to . For the convergence proof, it is instructive to think of as an block matrix, where each block has size , i.e.,

(A1) |

where the entries of are indexed by . Furthermore, let be the projection of the set of feasible solutions to to the relevant coordinates, i.e.,

Theorem: We have that

###### Proof.

We have already seen that We now show the other inclusion.

We start with some comments concerning the proof. First, to ease notation, we assume that measurement outcomes uniquely define the measurement they correspond to. We denote by (resp. ) the measurement corresponding to Alice’s outcome (resp. Bob’s outcome ). This convention allows us to write as opposed to , which is used in the main text. A quantum realization of is given by where , if , and . Second, for any the matrix is indexed by the operators in the set . Operators and will be associated with row or column indices, and respectively. Consider a distribution and let be a level- certificate.

Step 1: For any integer , we have that

(A2) |

where is the entry of As is positive semidefinite, it suffices to show that

(A3) |

To get (A2) from (A3), consider the principal submatrix of indexed by and , i.e.,

As this matrix is PSD, its determinant is nonnegative. Combined with (A3), this implies that

Lastly, we prove (A3). Trivially, we have that , and thus, . As is PSD, the principal submatrix of indexed by the words and is also PSD, i.e.,

which in turn implies that for all .

Step 2: Next we embed all matrices in a single normed space (we need to do this as they have different sizes) where the sequence has a convergent subsequence. For this, we append zeros to extend each matrix to an infinite matrix , which is indexed by all words .

Now, by Step 1, all matrices lie in the unit ball of the Banach space . Furthermore, it is well-known that is the dual space of RS . Thus, by the Banach-Alaogly theorem RS , the sequence has a convergent subsequence, with respect to the weak topology, i.e., there exists an infinite matrix such that

(A4) |

Equation (A4) has two important consequences. First, by the definition of the weak* topology, it also implies point-wise convergence, i.e.,

(A5) |

Second, the infinite matrix is a PSD kernel PSDkernel . As is feasible for we have that

(A6) |

which combined with (A5) implies that

(A7) |

Furthermore, as is a PSD kernel, there exists a (possibly infinite dimensional) Hilbert space

and vectors

such that(A8) |

This process is the infinite dimensional analogue of the Cholesky decomposition Horn2013 .

Step 3: Using the vectors from (A8) we construct a quantum realization for . Specifically, for define the quantum states and the measurement operators

(A9) |

It remains to show that such that . From the relation

(A10) |

we get that

(A11) |

Using (A8), Equation (A11) implies that

(A12) |

In particular, (A12) implies that

(A13) |

which implies that

(A14) |

Furthermore, we have that

(A15) |

where for the last equality we use the definition of and for the second to last equality we use that ; This follows from the following chain of implications:

(A16) |

Setting , Equation (A15) implies that

(A17) |

Likewise, we get that

(A18) |

Using induction, it follows from (A17) and (A18) that

(A19) |

which in turn implies that

(A20) |

Lastly, combining (A7) with (A20) we get that:

(A21) |

and furthermore

(A22) |

The last step of the proof is to show that . For this note that which implies that

In turn, using (A20) this is equivalent to

which implies that .

∎

## Appendix B Distributed QRAC

Here we present the semidefinite program for the distributed QRAC protocol. To start with, let us first present the quantum characterisation problem, which is modelled by a set of projective operators and where and a set of transformed quantum code states , where . For clarity, in this section we have changed the notation of the measurement operators from to to distinguish Alice’s measurement operators from Bob’s. We assume nothing about these operators and states, except that the Gram matrix of coincides with the Gram matrix of the vectors . More precisely, we require that , where

Using this quantum model, Alice’s and Bob’s guessing probabilities are given by

where the quantum code states and measurements are assumed to be randomly chosen (this can be easily generalised to an arbitrary distribution). To characterise the quantum behaviour of the protocol, we maximise Alice’s guessing probability given that Bob’s guessing probability is set to some fixed value . More specifically, we consider the following optimisation problem:

(A23) |

However, as mentioned in the main text, this optimisation problem is computationally intractable. To this end, we consider instead the SDP relaxation of (A23) corresponding to the set of operators . Using the label

as a classifier, we can partition any feasible solution

to the SDP into blocks , each having size . We index the rows and columns of by . The reader may refer to the following exposition of for reference:As explained in the main text, the matrix , where , is Hermitian PSD and furthermore, its entries satisfy certain linear relations corresponding to the algebraic constraints of the measurement operators and the Gram matrix of the code states. More specifically, we have that:

(A24) |

Furthermore, Alice’s guessing probability is given by:

(A25) |

and similarly, Bob’s guessing probability is given by:

(A26) |

By optimising Alice’s guessing probability over all PSD matrices satisfying the linear constraints described in (A24), and additionally, satisfying for a fixed scalar , we get the plot of , as illustrated in Figure 2.

## Appendix C Coherent-state QKD

Here we analyse the asymptotic security of the phase-encoding coherent-state QKD protocol assuming collective attacks Scarani2009 ; the extension to coherent attacks is straightforward using either the post-selection technique Christandl2009 or the recently developed entropy accumulation theorem Dupuis2016 . The security analysis of the time-encoding QKD protocol is the same. To this end, we first start with a (hypothetical but equivalent) purified state

which is shared between Alice, Bob and Eve after the transmission. The security of this state, with respect to Alice making the qubit measurement on her share, is given by the Devetak-Winter’s key distillation bound Devetak2005 and the entropic uncertainty relation for quantum memories Berta2010 :

where and

are random variables corresponding to Alice’s measurement outcomes obtained from

and , respectively, and and are random variable associated with Bob’s measurement outcomes given and
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