Using the leading vector method, we show that any vector $h\in(C^2)^{\otimes l}$ can be decomposed as a sum of at most (and at least in the generic case) $2^l-l$ product vectors using local bitwise unitary transformations. The method is based on representing the vectors by chains of appropriate simplicial complex. This generalizes the Scmidt decomposition of pure states of a 2-bit register to registers of arbitrary length $l$ .Using the leading vector method, we show that any vector $h\in(C^2)^{\otimes l}$ can be decomposed as a sum of at most (and at least in the generic case) $2^l-l$ product vectors using local bitwise unitary transformations. The method is based on representing the vectors by chains of appropriate simplicial complex. This generalizes the Scmidt decomposition of pure states of a 2-bit register to registers of arbitrary length $l$.

arXiv:quant-ph/0010104Oct 30, 20000 citations

Comment on `Decomposition of pure states of a quantum register'

Alexander Yu. Vlasov

I. Raptis and R. Zapatrin in the quant-ph/0010104 show possibility to express general state of $l$ -qubits quantum register as sum at most $2^l-l$ product states. In the comment is suggested more simple construction with possibility of generalization for decomposition of tensor product of Hilbert spaces with arbitrary dimension $n$ (here simplicial complexes used in the article mentioned above would not be applied directly). In this case it is decomposition with $n^l-(n^2-n)l/2$ product states.I. Raptis and R. Zapatrin in the quant-ph/0010104 show possibility to express general state of $l$-qubits quantum register as sum at most $2^l-l$ product states. In the comment is suggested more simple construction with possibility of generalization for decomposition of tensor product of Hilbert spaces with arbitrary dimension $n$ (here simplicial complexes used in the article mentioned above would not be applied directly). In this case it is decomposition with $n^l-(n^2-n)l/2$ product states.

arXiv:quant-ph/0011017Nov 3, 20000 citations

On the Decompositions of a Quantum State

Gianni Cassinelli, Ernesto De Vito, Alberto Levrero

Journal of Mathematical Analysis and ApplicationsJun 1, 199722 citations

Coherence and decoherence in generalized Shor’s algorithm

Linlin 琳琳 Ye 叶, Zhaoqi Wu, Nanrun 南润 Zhou 周

Abstract Quantum coherence constitutes a fundamental physical mechanism essential to the study of quantum algorithms. We study coherence and decoherence in the generalized Shor’s algorithm where the register A is initialized in an arbitrary pure state, or the combined register AB is initialized in a pseudo-pure state, which encompasses the standard Shor’s algorithm as a special case. We derive both lower and upper bounds on the performance of the generalized Shor’s algorithm, and establish the relation between the probability of calculating the order r when register AB is initialized in a pseudo-pure state and that when register A is initialized in an arbitrary pure state. Moreover, we study coherence and decoherence in the noisy Shor’s algorithm and give a lower bound on the probability that we can calculate the order r .Abstract Quantum coherence constitutes a fundamental physical mechanism essential to the study of quantum algorithms. We study coherence and decoherence in the generalized Shor’s algorithm where the register A is initialized in an arbitrary pure state, or the combined register AB is initialized in a pseudo-pure state, which encompasses the standard Shor’s algorithm as a special case. We derive both lower and upper bounds on the performance of the generalized Shor’s algorithm, and establish the relation between the probability of calculating the order r when register AB is initialized in a pseudo-pure state and that when register A is initialized in an arbitrary pure state. Moreover, we study coherence and decoherence in the noisy Shor’s algorithm and give a lower bound on the probability that we can calculate the order r .

arXiv:2508.11962Chinese Physics BFeb 11, 20260 citations

The Structure of Quantum States

Gary E. Bowman

Oxford University Press eBooksNov 1, 20070 citations

Using the leading vector method, we show that any vector

h\in(C^2)^{\otimes l}

can be decomposed as a sum of at most (and at least in the generic case)

2^l-l

product vectors using local bitwise unitary transformations. The method is based on representing the vectors by chains of appropriate simplicial complex. This generalizes the Scmidt decomposition of pure states of a 2-bit register to registers of arbitrary length

l

I. Raptis and R. Zapatrin in the quant-ph/0010104 show possibility to express general state of

l

-qubits quantum register as sum at most

2^l-l

product states. In the comment is suggested more simple construction with possibility of generalization for decomposition of tensor product of Hilbert spaces with arbitrary dimension

n

(here simplicial complexes used in the article mentioned above would not be applied directly). In this case it is decomposition with

n^l-(n^2-n)l/2

product states.