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The Generations of Classical Correlations via Quantum Schemes

Zhenyu Chen, Lijinzhi Lin, Xiaodie Lin, Zhaohui Wei, Penghui Yao

TL;DR

It is proved that when the seed is a pure quantum state, solving the problem is equivalent to finding out whether the target classical correlation has some canonical form of positive semi-definite factorizations that matches the seed pure state, revealing an interesting connection between the current problem and optimization theory.

Abstract

Suppose two separated parties, Alice and Bob, share a bipartite quantum state or a classical correlation called a \emph{seed}, and they try to generate a target classical correlation by performing local quantum or classical operations on the seed, i.e., any communications are not allowed. We consider the following fundamental problem about this setting: whether Alice and Bob can use a given seed to generate a target classical correlation. We show that this problem has rich mathematical structures. Firstly, we prove that even if the seed is a pure bipartite state, the above decision problem is already NP-hard and a similar conclusion can also be drawn when the seed is also a classical correlation, implying that this problem is hard to solve generally. Furthermore, we prove that when the seed is a pure quantum state, solving the problem is equivalent to finding out whether the target classical correlation has some diagonal form of positive semi-definite factorizations that matches the seed pure state, revealing an interesting connection between the current problem and optimization theory. Based on this observation and other insights, we give several necessary conditions where the seed pure state has to satisfy to generate the target classical correlation, and it turns out that these conditions can also be generalized to the case that the seed is a mixed quantum state. Lastly, since diagonal forms of positive semi-definite factorizations play a crucial role in solving the problem, we develop an algorithm that can compute them for an arbitrary classical correlation, which has decent performance on the cases we test.

The Generations of Classical Correlations via Quantum Schemes

TL;DR

It is proved that when the seed is a pure quantum state, solving the problem is equivalent to finding out whether the target classical correlation has some canonical form of positive semi-definite factorizations that matches the seed pure state, revealing an interesting connection between the current problem and optimization theory.

Abstract

Suppose two separated parties, Alice and Bob, share a bipartite quantum state or a classical correlation called a \emph{seed}, and they try to generate a target classical correlation by performing local quantum or classical operations on the seed, i.e., any communications are not allowed. We consider the following fundamental problem about this setting: whether Alice and Bob can use a given seed to generate a target classical correlation. We show that this problem has rich mathematical structures. Firstly, we prove that even if the seed is a pure bipartite state, the above decision problem is already NP-hard and a similar conclusion can also be drawn when the seed is also a classical correlation, implying that this problem is hard to solve generally. Furthermore, we prove that when the seed is a pure quantum state, solving the problem is equivalent to finding out whether the target classical correlation has some diagonal form of positive semi-definite factorizations that matches the seed pure state, revealing an interesting connection between the current problem and optimization theory. Based on this observation and other insights, we give several necessary conditions where the seed pure state has to satisfy to generate the target classical correlation, and it turns out that these conditions can also be generalized to the case that the seed is a mixed quantum state. Lastly, since diagonal forms of positive semi-definite factorizations play a crucial role in solving the problem, we develop an algorithm that can compute them for an arbitrary classical correlation, which has decent performance on the cases we test.
Paper Structure (11 sections, 10 theorems, 59 equations, 2 algorithms)

This paper contains 11 sections, 10 theorems, 59 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. The quantum case that $\rho$ is pure
  3. On the computational complexity
  4. The diagonal form of PSD factorizations
  5. Several necessary conditions for that $| \psi \rangle$ generates $P$
  6. The general case that the seed state is mixed
  7. A general necessary condition for the case of mixed seed
  8. When the seed is also a classical correlation
  9. Quantum has no advantage in reachability
  10. The NP-hardness of the case of classical seed
  11. An algorithm that computes diagonal forms of PSD factorizations

Key Result

Lemma 2.1

Let $P = [P(x,y)]_{x,y}$ be a classical correlation, and $\rho=\sum_{x,y}P(x,y)\cdot| x \rangle\langle x | \otimes | y \rangle\langle y |$ be a quantum state in $\mathcal{H}_A \otimes \mathcal{H}_B$. Then it holds that where $\hbox{\tt {S-rank}}\xspace(| \psi \rangle)$ is the Schmidt rank of $| \psi \rangle$ with respect to the partition $AA_1|BB_1$.

Theorems & Definitions (17)

  • Lemma 2.1: jain2013efficient
  • Theorem 2.1
  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.2
  • Proposition 2.1
  • Example 1
  • Proposition 2.2
  • Example 2
  • Proposition 2.3
  • ...and 7 more