N-qudit SLOCC equivalent W states are determined by their bipartite reduced density matrices with tree form

It has been proved that N-qudit (i.e., d-level subsystems) generalized W states are determined by their bipartite reduced density matrices. In this paper, we prove that only $$(N-1)$$ of the bipartite reduced density matrices are sufficient. Furthermore, we find that N-qudit W states preserve their determinability under stochastic local operation and classical communication (SLOCC). That is, all multipartite pure states that are SLOCC equivalent to N-qudit W states can be uniquely determined (among pure, mixed states) by their $$(N-1)$$ of the bipartite reduced density matrices, if the $$(N-1)$$ pairs of qudits constitute a tree graph on N vertices, where each pair of qudits represents an edge.

[1]  M. Guta,et al.  A comparative study of estimation methods in quantum tomography , 2019, Journal of Physics A: Mathematical and Theoretical.

[2]  W. Wootters,et al.  The parts determine the whole in a generic pure quantum state. , 2002, Physical review letters.

[3]  W. Wootters,et al.  Almost every pure state of three qubits is completely determined by its two-particle reduced density matrices. , 2002, Physical review letters.

[4]  Preeti Parashar,et al.  Reducible correlations in Dicke states , 2009, 0910.1422.

[5]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.

[6]  Preeti Parashar,et al.  N-Qubit W States are Determined by their Bipartite Marginals , 2008, 0809.4394.

[7]  Georgios M. Nikolopoulos,et al.  Error tolerance of two-basis quantum key-distribution protocols using qudits and two-way classical communication , 2006 .

[8]  Bei Zeng,et al.  Tomography is Necessary for Universal Entanglement Detection with Single-Copy Observables. , 2015, Physical review letters.

[9]  Bei Zeng,et al.  Quantum State Tomography via Reduced Density Matrices. , 2016, Physical review letters.

[10]  Determination of W states equivalent under stochastic local operations and classical communication by their bipartite reduced density matrices with tree form , 2014 .

[11]  Optimal reducibility of all W states equivalent under stochastic local operations and classical communication , 2011 .

[12]  C. Schwemmer,et al.  Systematic errors in current quantum state tomography tools. , 2013, Physical review letters.

[13]  S. Massar,et al.  Bell inequalities for arbitrarily high-dimensional systems. , 2001, Physical review letters.

[14]  Qiao-Yan Wen,et al.  Quantum private query: A new kind of practical quantum cryptographic protocol , 2019, Science China Physics, Mechanics & Astronomy.

[15]  N. Vitanov,et al.  Quantum phase estimation and quantum counting with qudits , 2016 .

[16]  J. Eisert,et al.  Efficient and feasible state tomography of quantum many-body systems , 2012, 1204.5735.

[17]  D. Gross,et al.  Efficient quantum state tomography. , 2010, Nature communications.

[18]  G. Cañas,et al.  Experimental implementation of an eight-dimensional Kochen-Specker set and observation of its connection with the Greenberger-Horne-Zeilinger theorem , 2013, 1308.6336.

[19]  B. Zeng,et al.  Quantum state tomography for generic pure states , 2017, Science China Physics, Mechanics & Astronomy.

[20]  Lajos Diosi.  Three-party pure quantum states are determined by two two-party reduced states , 2004 .

[21]  Matthias Christandl,et al.  Reliable quantum state tomography. , 2011, Physical review letters.

[22]  Zhengfeng Ji,et al.  Uniqueness of quantum states compatible with given measurement results , 2012, 1212.3503.

[23]  J. Cirac,et al.  Three qubits can be entangled in two inequivalent ways , 2000, quant-ph/0005115.

[24]  David W. Lyons,et al.  Only n-Qubit Greenberger-Horne-Zeilinger states are undetermined by their reduced density matrices. , 2007, Physical review letters.

[25]  Bin Liu,et al.  QKD-based quantum private query without a failure probability , 2015, Science China Physics, Mechanics & Astronomy.

[26]  G. Cañas,et al.  Experimental quantum tomography assisted by multiply symmetric states in higher dimensions , 2018, Physical Review A.

[27]  N. Linden,et al.  Parts of quantum states , 2004, quant-ph/0407117.

[28]  D. L. Zhou,et al.  Irreducible multiparty correlations in quantum states without maximal rank. , 2008, Physical review letters.