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International Computer Conference on Wavelet Active Media Technology and Information Processing (ICCWAMTIP)

Matlab Programming Implementation and Application of PPT Criteria

Fan Zhiyi

Published Dec 19, 2025• 0 citations
Publisher

Abstract

Quantum entanglement is an important form of quantum non-locality, and it is a very important physical resource in quantum computing, quantum cryptography and other applications. It can be said that quantum entanglement is the core of the whole quantum information technology. Therefore, determining whether a quantum state is entangled or not is the core of the whole problem. This thesis introduces some basic quantum entanglement criteria, including the Positive Partial Transpose Criterion (PPT Criterion), Computable Cross Norm Realignment Criterion (CCNR Criterion). This thesis focuses on the PPT criterion given by Peres and Horodecki, which is recognized as the strongest operable criterion at present and has quite strong judgment ability. This thesis elaborates the implementation process of PPT criteria and gives the specific process of PPT criteria MATLAB programming. In addition, the programmed PPT criterion is applied to the entanglement judgment of some quantum states, and the difference of the entanglement judgment ability between the PPT criterion and the CCNR criterion and their respective limitations are reflected in the entanglement judgment problem.

DOI
10.1109/iccwamtip68645.2025.11352640

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Quantum entanglement is an important form of quantum non-locality, and it is a very important physical resource in quantum computing, quantum cryptography and other applications. It can be said that quantum entanglement is the core of the whole quantum information technology. Therefore, determining whether a quantum state is entangled or not is the core of the whole problem. This thesis introduces some basic quantum entanglement criteria, including the Positive Partial Transpose Criterion (PPT Criterion), Computable Cross Norm Realignment Criterion (CCNR Criterion). This thesis focuses on the PPT criterion given by Peres and Horodecki, which is recognized as the strongest operable criterion at present and has quite strong judgment ability. This thesis elaborates the implementation process of PPT criteria and gives the specific process of PPT criteria MATLAB programming. In addition, the programmed PPT criterion is applied to the entanglement judgment of some quantum states, and the difference of the entanglement judgment ability between the PPT criterion and the CCNR criterion and their respective limitations are reflected in the entanglement judgment problem.
Given two bipartite quantum states and the convex linear combination of them, we discuss the relation between the entanglement of the convex linear combination state and the entanglement of states being combined. This is achieved by characterizing quantum states quantitatively via the positive partial transpose (PPT) criterion and the computable cross-norm or realignment (CCNR) criterion. Inspired by the Horodecki's 3 ⊗ 3 quantum states, we also give explicit examples to illustrate all possible cases of convex linear combination. Finally, as an application of this method, we show how to construct new bipartite PPT entangled states from known PPT entangled states by convex linear combination.
The efficient detection of quantum entanglement is a central problem in quantum information processing. This paper systematically proposes a quantum circuit implementation scheme based on the Positive Partial Transpose (PPT) and the Computable Cross-Norm Realignment (CCNR) criteria, providing a complete quantum algorithmic pathway for efficient and computable entanglement detection. By encoding quantum states into specific forms and utilizing SWAP operations, complex matrix operations such as partial transpose and realignment are transformed into executable quantum circuits. Furthermore, by integrating an improved variational quantum singular value decomposition subroutine, the scheme enables the efficient estimation of the trace norm, thereby determining the existence of entanglement. Designed to operate within a hybrid quantum-classical framework, this scheme exhibits excellent scalability and practicality, offering a theoretical tool and methodological support for analyzing the entanglement structure of complex quantum systems on future intermediate-scale quantum devices.
The correlation matrix (CM) criterion is a recently derived powerful sufficient condition for the presence of entanglement in bipartite quantum states of arbitrary dimensions. It has been shown that it can be stronger than the positive partial transpose (PPT) criterion, as well as the computable cross norm or realignment (CCNR) criterion in different situations. However, it remained as an open question whether there existed sets of states for which the CM criterion could be stronger than both criteria simultaneously. Here, we give an affirmative answer to this question by providing examples of entangled states that scape detection by both the PPT and CCNR criteria whose entanglement is revealed by the CM condition. We also show that the CM can be used to measure the entanglement of pure states and obtain lower bounds for the entanglement measure known as tangle for general (mixed) states.
We show how to design families of operational criteria that distinguish entangled from separable quantum states. The simplest of these tests corresponds to the well-known Peres-Horodecki positive partial transpose (PPT) criterion, and the more complicated tests are strictly stronger. The new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs. We successfully applied the results to many low-dimensional states from the literature where the PPT test fails. As a by-product of the criteria, we provide an explicit construction of the corresponding entanglement witnesses.
Summary: \nEntanglement is a strange feature contained in the quantum mechanical framework, first observed theoretically by Erwin Schrödinger and Albert Einstein. An inherent attribute of entanglement is non-locality, and in the continuation of this, several non-reductionist features, which are conceptually challenging and philosophically interesting. \n \nA considerable growth of the interest in quantum entanglement has occurred since the early 1990s, and much of the reason for this is that the tools of quantum optics and modern instrumentation have made it possible to measure and manipulate entangled quantum systems. Today, several applications of entanglement exist, including quantum computers and several areas in quantum communications. \n \nA quantum state is either separable or entangled, it cannot be none or both. A separable state do not exhibit the non-local features found in entangled states, and are in this sense comparable to classic states. An important issue here is the separability problem, which is the problem of determining whether a given state is entangled or not. \n \nWe study these quantum states as Hermitian matrices with positive eigenvalues in convex spaces. The PPT criterion is based on the partial transposition operation, which essentially transposes one of the two subsystems in the quantum system. For some states the result is another quantum (positive) state, and these are PPT (Positive Partial Transpose) states, while others that gain negative eigenvalues are called NPT states. \n \nFor systems of dimensions 2 × 2 and 2 × 3, the PPT states are identical to the separable states, and the NPT states are identical to the entangled states, so the PPT criterion is always conclusive. But for larger dimensions, such as for instance the 3 × 3 system, there always exist PPT states that are entangled, so for this case the PPT criterion can only be used in a contrapositive way. Since entangled PPT states in this sense seem to have great significance, their characteristics are worth studying. \nAs mentioned, the PPT criterion uses the transposition map on one of the two subsystems of a composite quantum system. Transposition is an example of a positive map, in that it maps all quantum states into quantum states. It is possible to use positive maps to formulate a much more general criterion for the separability problem.There is a link (an isomorphism) between positive maps and a subset of Hermitian operators called entanglement witnesses. \n \nThe so-called Horodecki criterion states that a quantum state is separable if and only if the expectation value of all entanglement witnesses in the state are non-negative. If the expectation value of one (or possibly several) entanglement witnesses in a given quantum state is negative, then the state is always entangled. This criterion is both necessary and sufficient, but demands that we can characterize the entire set of entanglement witnesses. \n \nThe approach in this thesis is through geometry. Of fundamental importance is the convex nature of the sets in question. The knowledge and understanding of the extremal points in these sets is very relevant, since they describe the sets completely. Undertaking completely analytic studies of the geometry at hand is in general very hard, especially as the dimension of the systems grow. The development and use of numerical tools is therefore essential.
The computable cross-norm (CCN) criterion is a powerful analytical and computable separability criterion for bipartite quantum states, which is also known to systematically detect bound entanglement. In certain aspects this criterion complements the well-known Peres positive partial transpose (PPT) criterion. In the present paper we study important analytical properties of the CCN criterion. We show that in contrast to the PPT criterion it is not sufficient in dimension $2\ifmmode\times\else\texttimes\fi{}2.$ In higher dimensions, theorems connecting the fidelity of a quantum state with the CCN criterion are proved. We also analyze the behavior of the CCN criterion under local operations and identify the operations that leave it invariant. It turns out that the CCN criterion is in general not invariant under local operations.