Analytical expression of s‐th power of the Gram matrix is shown when the quantum signals are group covariant. An application of the formula to coded quantum signals by classical linear codes over a finite field is also shown.
In quantum information science, it is very important to solve the eigenvalue problem of the Gram matrix for quantum signals. This allows various quantities to be calculated, such as the error probability, mutual information, channel capacity, and the upper and lower bounds of the reliability function. Solving the eigenvalue problem also provides a matrix representation of quantum signals, which is useful for simulating quantum systems. In the case of symmetric signals, analytic solutions to the eigenvalue problem of the Gram matrix have been obtained, and efficient computations are possible. However, for asymmetric signals, there is no analytic solution and universal numerical algorithms that must be used, rendering the computations inefficient. Recently, we have shown that, for asymmetric signals such as amplitude-shift keying coherent-state signals, the Gram matrix eigenvalue problem can be simplified by exploiting its partial symmetry. In this paper, we clarify a method for simplifying the eigenvalue problem of the Gram matrix for quadrature amplitude modulation (QAM) signals, which are extremely important for applications in quantum communication and quantum ciphers. The results presented in this paper are applicable to ordinary QAM signals as well as modified QAM signals, which enhance the security of quantum cryptography.
The formula of the channel matrix for group co-variant signals for classical-quantum communication has been studied to calculate the channel matrix analytically. However, despite its importance, the formula derived previously cannot be applied to coded q-ary phase shift keying (PSK) signals by linear codes over an extension field F q . To derive the formula for codes over F q , we focus on the 4PSK signals coded by the codes over F 4 . In the present paper, we propose a construction method of a code of length 2n over F 4 from a code of length n. If the original code is a group code, the coded 4PSK signals by the constructed code is narrow sense group covariant.
Efficient computation of eigenvalues and eigenvectors of the Gram matrix for quantum signals is desirable in the field of quantum communication theory. Because various quantities such as the error probability, mutual information, channel capacity, and upper and lower bounds of the reliability function can be obtained by the eigenvalues and eigenvectors. Moreover, solving the eigenvalue problem also provides a matrix representation of quantum signals, which is useful for simulating quantum systems. In the case of symmetric signals, analytic solutions to the eigenvalue problem of the Gram matrix have been obtained and efficient computations are possible. However, for asymmetric signals, there is no analytic solution and universal numerical algorithms must be used. Recently, we have shown that, for certain asymmetric signals, the Gram matrix eigenvalue problem can be simplified and it is sufficient to consider the eigenvalue problem of smaller matrices at half or quarter size. In this paper, we consider nm-ary quantum signals formed by rotating signal points in a circular sector region and show that solving the 1/n-size matrix eigenvalue problem is sufficient to give the eigenvalues and eigenvectors of the Gram matrix.
We develop a quantum message-passing framework for factor graphs over finite abelian groups. Our starting point is the task of discriminating between a collection of quantum states indexed by the elements of a finite abelian group G whose overlaps respect the structure of a group-covariant pure-state channel (PSC). For such channels, we show that the Gram matrix constructed from the output states is diagonalized by the character basis of the dual group G. Hence, the channel is characterized, up to isometric equivalence, by its character-indexed eigen list.
Based on this representation, we analyze the induced classical-quantum channels associated with check, equality, homomorphism, marginalization, and automorphism factors. For each factor, we derive explicit update rules showing that if the incoming messages are heralded mixtures of group-covariant PSCs, then the outgoing message remains in the same class. This provides a closed quantum message-passing framework for tree-structured factor graphs assembled from these primitives.
The framework applies directly to several standard code families over finite abelian groups, including polar codes, LDPC codes, and convolutional and turbo codes. It recovers the previously studied q-ary formulation as the special case (G=Zq), while extending the belief propagation with quantum messages (BPQM) framework introduced by Renes to non-cyclic alphabets and more general factor-graph constraints described by homomorphisms between products of abelian groups.