Two- and three-mode squeezing in a three-qubit entangled system

Quantum Information Processing (2025) 24:339 https://doi.org/10.1007/s11128-025-04940-1 Twoand three-mode squeezing in a three-qubit entangled system Joanna K. Kalaga1·Jan Peˇrina Jr.2 Received: 21 July 2025 / Accepted: 16 September 2025 © The Author(s) 2025 Abstract The states of a three-mode bosonic system with the restricted Hilbert space are discussed in the context of quantum entanglement and squeezing of quantum fluctuations. The states exhibiting nonzero tripartite entanglement are considered. Mutual relations between the twoand three-mode entanglement quantified by the corresponding negativities and the squeezing described by the corresponding principal squeeze variances are revealed. Entangled three-qubit states exhibiting squeezing are identified. Keywords Quantum entanglement ·Tripartite entanglement ·Quantum squeezing · Three-qubit system 1 Introduction Entanglement and squeezing are two fundamental concepts in quantum mechanics that are both essential to quantum information processing. Although squeezing is usually linked to continuous-variable (CV) systems with infinitely large Hilbert spaces, this concept can also be naturally applied to the cases in which the Hilbert spaces are restricted, e.g., using the method of quantum scissors. In analogy, this concept can be defined and studied in qubit systems, where, as we demonstrate, squeezing similarly as the entanglement offers valuable insights into the quantum properties of these systems and their correlations. Three-qubit quantum states play an important role in various areas of quantum information processing. These states, which exhibit entanglement, are a resource BJoanna K. Kalaga [email protected] Jan Peˇrina Jr. [email protected] 1Quantum Optics and Engineering Division, Institute of Physics, University of Zielona Góra, Prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland 2Joint Laboratory of Optics of Palacký University and Institute of Physics of AS CR, Faculty of Science, Palacký University, 17. listopadu 12, 779 00 Olomouc, Czech Republic 0123456789().: V,-vol 123 339 Page 2 of 22 J.K. Kalaga, J. Peˇ rina Jr. for quantum computation and quantum communication [1–9]. The Greenberger–Horne–Zeilinger (GHZ) states and W states play a special role. These states are characterized by the full tripartite entanglement [10–15]. However, due to differences in the occurrence of bipartite entanglement, they have found different applications. Entanglement of three qubits can be used in quantum cryptography to detect eavesdropping. Quantum secret sharing protocols make use of multi-qubit states. These protocols, based on the idea that a secret is shared among several parties via entanglement, ensure that no party can access the secret alone. This allows to use the entangled states in secure quantum communication protocols [1,3,16–20]. Three-qubit entangled states are also exploited in quantum error correction schemes [21–24]. Both quantum computers and reliable quantum communication protocols require protocols that protect quantum information. For instance, the authors of [25] demonstrated a three-qubit quantum error correction code based on super-conducting circuits. Other compelling quantum states, besides the entangled states, are the squeezed states [26–29]. These states are, in general, characterized by the reduced uncertainty in one field quadrature compared to the vacuum state, but having increased the uncertainty in the conjugate quadrature. Such states, with respect to light, were first generated in 1985 by Slusher et al. [30], who achieved this through four-wave mixing in sodium atoms placed in an optical cavity. Shortly afterward, in 1986, Shelby et al. also generated squeezed light using four-wave mixing in an optical fiber [31]. It has been also produced using degenerate parametric down-conversion in a second-order nonlinear crystal placed in an optical cavity [32]. Similar to entangled states, squeezed states have a wide range of applications. As they are related to entanglement in multi-mode systems, they are a valuable resource [33–37]. Squeezed states, for instance, are used in quantum cryptography protocols [38–40] and in quantum error correction [41–43]. Three-qubit systems exhibit a rich structure that supports both bipartite and genuine tripartite entanglement. Examples of such states include the GHZ state and the W state, which represent distinct classes of multipartite entanglement [14]. It is crucial to understand how squeezing manifests in such three-qubit states and what is its relation to the entanglement considering different entanglement classes of the three-qubit system. Moreover, considering multi-qubit systems described by qubit-chain models or collective qubit interactions, the three-qubit states frequently act as elementary building blocks for larger entangled ensembles. The structure of the paper is as follows. In Sect. 2, classification of three-qubit states is given. Sect. 3provides definitions of the parameters of twoand three-mode entanglement and squeezing. Section 4is devoted to the analysis of the relations among the parameters quantifying twoand three-mode entanglement and squeezing considering different types of three-qubit states exhibiting full tripartite entanglement. Our findings are summarized in the concluding Sec. 5. 2 Three-qubit states The analysis of squeezing in the three-qubit system is based on the classification of three-qubit entangled states proposed by Sabín and García-Alcaine in [14]. This 123 Twoand three-mode squeezing... Page 3 of 22 339 j j k k i i j j k k i i j j k k i i a a ) ) b b ) ) d d ) ) c c ) ) j j k k i i Fig. 1 Graphical representation of the states that are classified as type III: (a) subtype III-0, (b) subtype III-1, (c) subtype III-2, (d) subtype III-3 classification includes both pure and mixed states and divides the three-qubit states into three types: I —fully separable states, II —states with only bipartite entanglement, III —states with nonzero full tripartite entanglement. We further concentrate on the entangled states exhibiting the full tripartite entanglement, i.e., belonging to type III. Graphical representation of the states classified as type III is presented in Fig. 1, where the qubits are represented by spheres labeled i,j, and k. The lines connecting these qubits represent two-mode entanglement, whereas the outer circle denotes the full three-mode entanglement. According to the classification of [14], the states with nonzero full tripartite entanglement can be further divided into the following four subtypes: III-0 —two-mode entanglement is missing, i.e., all reduced negativities are zero (Nij =Nik =Njk =0)—see Fig. 1a, 123 339 Page 4 of 22 J.K. Kalaga, J. Peˇ rina Jr. III-1 —two-mode entanglement between two qubits is observed, i.e., one reduced negativity is nonzero (Nij =Nik =0, Njk = 0)—Fig. 1b, III-2 —two-mode entanglement inside two pairs of qubits is observed, i.e., two reduced negativities are nonzero (Njk =0, Nij and Nik = 0)—Fig. 1c, III-3 —two-mode entanglement inside all three pairs of qubits is observed, i.e., all three reduced negativities are nonzero (Nij,Nik, and Njk = 0)—Fig. 1d. 3 Measures of squeezing and entanglement As a measure of entanglement of two modes, we use negativity. This measure is based on the negative-partial-transposition (NPT) criterion [44,45]. Negativity Nij is defined as the sum of the absolute values of all negative eigenvalues λlcalculated for the two-qubit matrix ρij after partial transposition with respect to one of the modes. A two-mode density matrix ρij is derived from the three-mode density matrix ρijk by tracing out subsystem k Nij =−2 l λlρTi ij .(1) We have Nij =0 for separable states. To quantify full tripartite entanglement, we use the geometric average of three negativities [14,46] Nijk =Ni−jkNj−ikNk−ij1 3,(2) where Ni−jk,Nj−ik, and Nk−ij are the bipartite negativities for the three-mode density matrix ρijk, and the partial transposition is performed in turn for the modes i,j, and k. The negativity Nijk is always equal to zero for the states belonging to classes I and II. On the other hand, Nijk stands nonzero for the states of class III. For instance, assuming the state (|000+|111)/√2, the negativity Nijk =1. On the other hand, for the state (|001+|010+|100)/√3, the negativity Nijk =0.94. To reveal squeezing in two-mode states, we introduce the following two-mode quadrature operators: ˆ Xij =ˆai+ˆa† i+ˆaj+ˆa† j, ˆ Yij =−iˆai−ˆa† i+ˆaj−ˆa† j.(3) For these operators, the uncertainty relation takes the form: ˆ Xij2ˆ Yij2≥4.(4) Suitably rotating the field operators in the definitions (3), we reveal the principal squeeze variance λij that quantifies the maximal attainable squeezing of the field quadrature operators identified by the following inequality [26,47,48]: λij =21+ˆa† iˆai+ˆa† jˆaj+2Reˆa† iˆaj −| ˆai2+ˆaj2+2ˆaiˆaj|<2(5) 123 Twoand three-mode squeezing... Page 5 of 22 339 and ˆa† iˆai=ˆa† iˆai−ˆa† iˆai. Similarly, we define the three-mode quadrature operators ˆ Xijk and ˆ Yijk to judge three-mode squeezing: ˆ Xijk =ˆai+ˆa† i+ˆaj+ˆa† j+ˆak+ˆa† k, ˆ Yijk =−iˆai−ˆa† i+ˆaj−ˆa† j+ˆak−ˆa† k.(6) The corresponding three-mode principal squeeze variance satisfies the following inequality: λijk =3+2ˆa† iˆai+ˆa† jˆaj+ˆa† kˆak +4Re ˆa† iˆaj+ˆa† iˆak+ˆa† jˆak −2|ˆai2+ˆaj2+ˆak2+2ˆaiˆaj +ˆaiˆak+ˆajˆak|<3,(7) We note that, in this case, the quadrature uncertainty relation takes the form: ˆ Xijk2ˆ Yijk2≥9.(8) In continuous-variable (CV) systems, squeezing refers to the situation in which the noise in one quadrature of an optical field is reduced below the standard quantum limit. This is accompanied by the increases in the noise in the conjugated quadrature. Though the description of CV systems reached by using the quantum scissors [49] is restricted to few states and its dynamics resembles that of discrete systems with couple of quantum levels, physical meaning of squeezing remains the same. We note that, for qubits, the general concept of squeezing results in the spin squeezing. In these systems, that can be considered as spin 1/2 particles, squeezing refers to reducing the uncertainty (variance) in one component of the collective spin operator below the standard quantum limit. This is achieved at the expense of increased uncertainty in another complementary component and enables enhanced precision in estimating small rotations around an orthogonal axis. This is analogous to standard CV quadrature squeezing. Spin squeezing arises naturally in ensembles of qubits (e.g., atoms or ions) and, similarly as in the analyzed system, it can indicate the presence of non-classical correlations [50,51]. Simplicity of the projection measurements in qubit systems motivate us to study squeezing in truncated bosonic systems where it can be a useful resource for ‘qubitbased’ quantum information processing. The analyzed qubit-squeezed states and the spin-squeezed states in particular have a number of practical applications in atomic clocks and quantum sensors. They help to overcome the limitations imposed by the projection noise. For small cold-atom ensembles and related metrological platforms, these states can significantly increase precision [52]. 123 339 Page 6 of 22 J.K. Kalaga, J. Peˇ rina Jr. Moreover, the truncated bosonic systems serve as a source of strong quantum correlations, including entanglement, quantum steering and Bell nonlocality [53,54]. Nair and Flebus demonstrated in [55] that spin squeezing can form multiparty entanglement, which is a crucial resource in quantum metrology and information processing. We note that the standard quantum limit is derived from the Heisenberg uncertainty principle and equals to 2 (3) for two- (three-) mode variances. 4 Results Considering the states exhibiting tripartite entanglement (Nijk >0) and using the above division of these states into sub-classes III-0, III-1, III-2, and III-3, we systematically examine the twoand three-mode squeezing and their relation to twoand three-mode negativity. 4.1 Type III-0 For the states belonging to type III-0, all two-mode negativities are zero (Nij =Nik = Njk =0) (see Fig. 1a) and the states are parameterized as follows: |ψ=C000|000+C111|111(9) where C000 and C111 are the probability amplitudes of the states |000and |111. In our numerical investigations, we have randomly generated the three-qubit states described by the density matrix ρ≡|ψψ|approximately ∼104times. In detail, we generated ∼104pairs of numbers in which the first number is C000 and the second one C111. These numbers satisfy the condition C000C∗ 000 +C111C∗ 111 =1. Then, we created a state ψaccording to the formula (9) and calculated the corresponding density matrix ρfor each pair. Finally, the negativities and the principal squeeze variances were determined for each realization. We express the twoand three-mode principal squeeze variances for the system described by the wave function given in equation (9) using formulas (5) and (7) and the probability P111 =|C111|2as follows: λij =λjk =λik =2+4P111, λijk =3+6P111.(10) For simplicity, we assume that the probability amplitudes are real. This assumption is used in this and the subsequent analysis. It reveals that the two-mode principal squeeze variance for each pair of qubits never reaches the values smaller than 2, and the three-mode principal squeeze variance is always equal to or greater than 3. Therefore, twoand three-mode squeezing does not occur for states belonging to subtype III-0. We note that the symmetry of the states ensures that λij =λik =λjk 123 Twoand three-mode squeezing... Page 7 of 22 339 0 0.2 0.4 0.6 0.8 1 Ni j k 3 4 5 6 7 8 9 ijk Fig. 2 Three-mode principal squeeze variance λijk as it depends on three-mode negativity Nijk for the states of the type III-0 and Nij =Nik =Njk =0. Interestingly, the three-mode negativity Nijk and threemode principal squeeze variance λijk form a specific relation described in the graph of Fig. 2. 4.2 Type III-1 The states belonging to type III-1 (Nij =Nik =0, Njk = 0, see Fig. 1b), can be divided into the following two subtypes: . the subtype III-1A |ψ=C000|000+C|1i|00jk|1i⊗|00jk +C111|111(11) . the subtype III-1B |ψ=C000|000+C|0i|11jk|0i⊗|11jk +C111|111.(12) To introduce the notation used in Eqs. (11) and (12), the states belonging to subtype III-1A are explicitly expressed as C000|000+C100|100+C111|111,C000|000+ C010|010+C111|111, and C000|000+C001|001+C111|111. We note that the states of subtype III-1A can be mapped to the states of subtype III-1B by mutual replacing the states |0and |1in each mode. Nevertheless, as these states belong to the states of quantum harmonic oscillator, the corresponding states of subtype III-1A and III-1b exhibit different squeezing properties. To reveal the properties of the states given in Eqs. (11) and (12), we have randomly generated ∼105states. In the following figures, in which we analyze the behavior of twoand three-mode squeezing, yellow regions denote the occurrence of 123 339 Page 8 of 22 J.K. Kalaga, J. Peˇ rina Jr. states exhibiting squeezing, i.e., λjk <2orλijk <3. The regions encompassing the remaining states are drawn in green. To achieve better understanding, in these figures we also draw the curves belonging to the states in which one of the three probabilities equals zero. The black, red, and blue curves belong in turn to the states in which P000 =|C000|2=0, P111 =0, or P|1i|00jk =0. Considering the states described by the wave function |ψ=C000|000+C100|100+C111|111,(13) the twoand three-mode principal squeeze variances are defined as: λij =λik =2(1+P100 +2P111 −2P000 P100), λjk =21+2P111 −2P100 P111, λijk =3+2(P100 −P000 P100 +3P111)−2|2P100 P111 −P000 P100 |.(14) Analyzing these formulas, we can see that the principal squeeze variances for two and three modes can be lower than 2 and 3, respectively. Therefore, twoand three-mode squeezing occurs for some states characterized by Eq (13). A more detailed analysis of the states classified as subtype III-1A is given in Fig. 3. In Fig. 3a, we analyze the relation between the two-mode squeezing λjk and the only two-mode nonzero negativity Njk. The black solid curve gives the minimal and maximal border values of the squeezing parameter λjk. Whereas the minimal values of λjk are obtained for the probabilities P000 =0 and P111 ∈0;0.5, the maximal values occur when P000 =0 and P111 ∈0.5;1. In this specific case, the wave function (11) simplifies into the form |ψ=C|1i|00jk|1i⊗|00jk +C111|111. According to the curves in Fig. 3a, the smallest value of the principal squeeze variance λjk ≈1.17 occurs for the negativity Njk ≈0.71 and the probability P111 =1 42−√2≈0.15. We note that, at this point, λij =λik =21 42−√2+2≈4.29 (see the black curve in Fig. 3b, and λijk =−22−√21 4√2−2+1−√2+7≈4.17 indicates no other form of squeezing in the analyzed state. Nevertheless, two-mode squeezing in modes i−jand i−kis also reached, as documented in Fig. 3b. In this case, the principal squeeze variances λij =λik = 1.75, λjk =2 and the corresponding states attain the form |ψ=C000|000+ C|1i|00jk|1i⊗|00jk. The relation between three-mode squeezing and tripartite entanglement is illustrated in Fig. 3c. Only a small number of states exhibit tripartite squeezing, specifically those with rather weak tripartite entanglement. The three-mode negativity Nijk cannot exceed ≈0.19 to observe the three-mode squeezing. The smallest value of three-mode principal squeeze variance λijk =2.75 occurs for Nijk =0. The two-mode principal squeeze variances equal λjk =2 and λij =λik =1.75 at this point. Figure 3d and e illustrate the relation between twoand three-mode squeezing. In the yellow area the states exhibiting three-mode squeezing (λijk <3) are found. 123 Twoand three-mode squeezing... Page 9 of 22 339 Fig. 3 Graphs revealing the relations among twoand three-mode principal squeeze variances λjk,λij = λik,andλijk and twoand three-mode negativities Njk and Nijk for the states of the type III-1A The strongest three-mode squeezing occurs when the parameters λij and λik reach their minimal value. The strongest two-mode squeezing also occurs for two pairs of qubits. However, the remaining pair of qubits, j−k, is not squeezed (λjk ≈2). Importantly, three-mode squeezing occurs when at least two of the three two-mode squeezing parameters are smaller than 2. Now let us analyze the states classified as subtype III-1B and expressed as C000|000+C011|011+C111|111,C000|000+C101|101+C111|111, and C000|000+C110|110+C111|111. 123 339 Page 16 of 22 J.K. Kalaga, J. Peˇ rina Jr. Fig. 9 Graphs revealing the relations among twoand three-mode principal squeeze variances λij,λjk, λik,andλijk and twoand three-mode negativities Nij,Nik,andNijk for the states of the type III-3 The relation between two-mode squeezing in the pair of modes i−kand the pair of modes j−kis revealed in the graph of Fig. 7c that closely resembles that of Fig. 4b devoted to type III-1B states. Nevertheless, in the graph in Fig. 7c we depict also the states exhibiting two-mode squeezing in the pair of modes j−k(λjk <2). The type III-2 states exhibit three-mode squeezing quantified by the three-mode principal squeeze variance λijk <3. The comparison of graphs in Figs. 3c, 4c, and 7d reveals that the range of three-mode negativities Nijk allowing three-mode squeezing is the broadest for the type III-2 states (Nijk <0.8). 123 Twoand three-mode squeezing... Page 17 of 22 339 Figures 7e7g provide the relation between twoand three-mode squeezing. The yellow area belongs to the states exhibiting three-mode squeezing. Similarly to the states classified as III-1B, the strongest three-mode squeezing for the states of subtype III-2 occurs when the two-mode squeezing is also the strongest for one pair of qubits (namely, qubits i−k). On the other hand, the remaining two pairs of qubits do not show squeezing. Importantly, the three-mode principal squeeze variance λijk reaches values lower than 3 in various cases in which the two-mode squeezing is observed in one, two, or three pairs of qubits. Surprisingly, three-mode squeezing can also arise in the absence of two-mode squeezing in all three pairs of modes. This resembles the situation found in the states having three-mode entanglement without two-mode entanglement and belonging to subtype III-0. Furthermore, if three-mode squeezing occurs while one of the qubit pairs does not exhibit squeezing, the two-mode principal squeeze variance of this pair remains below 3. 4.4 Type III-3 Finally, we analyze the states belonging to type III-3, which exhibit both twoand three-mode entanglement among all three modes, i.e., Nij >0, Nik >0, Njk >0, and Nijk >0(seeFig.1d). They are parameterized as |ψ=C000|000+C|0k|11ij|0k⊗|11ij +C|0j|11ik|0j⊗|11ik (20) and include, e.g., the states C000|000+C101|101+C110|110and C000|000+ C011|011+C110|110. The relation between twoand three-mode negativities for III-3 states is elucidated in Fig. 8where the maximal three-mode entanglement is obtained provided that all three two-mode negativities equal to 1 3√5−1≈0.41. However, the three-mode entanglement disappears when one of the two-mode negativities equals 1 and the remaining two-mode negativities are zero. Moreover, if two qubits are maximally entangled, the other pairs of qubits cannot be entangled. In Fig. 8a, the black curve giving the maximal value of Nijk for the fixed two-mode negativities corresponds to the situation in which two appropriate probabilities equal (P000 =P|0k|11ij). Otherwise, the three-mode negativity Nijk attains its maxima for fixed Njk (Nik) provided that P|0j|11ik =P|0k|11ij (P000 =P|0j|11ik). The states belonging to subtype III-3 include, among others, those described by the following wave function: |ψ=C000|000+C101|101+C110|110.(21) 123 339 Page 18 of 22 J.K. Kalaga, J. Peˇ rina Jr. For these states, the principal squeeze variances attain their forms: λij =21+P101 +2P110 −2P000 P110, λik =21+P110 +2P101 −2P000 P101, λjk =21+P101 +P110 +2P101 P110, λijk =3+2(2P101 +2P110)−4P000 P101 +P000 P110+4P101 P110. (22) These formulas express that the three-mode principal squeeze variance can be lower than 3 on one side, not all two-mode principal squeeze variances can be lower than 2 on the other side. Further analysis of the states classified as subtype III-3 is provided in Fig. 9. The relation between the two-mode principal squeeze variances λij and λik and the corresponding two-mode negativities Nij and Nik plotted in the graph in Fig. 9a reflects the corresponding relations found for the III-1A (Fig. 3a) and III-1B (Fig. 4a) states, and also for the III-2 states (Fig. 7a). In Fig. 9a similarly as above, the black solid curve determines the minima (maxima) of the principal squeeze variances λij and λik reached under the condition P|0j|11ik = 0(P|0k|11ij =0). The lowest two-mode principal squeeze variances λij =λik ≈1.17 occur for the two-mode negativities Nij =Nik ≈0.71 and P000 =1 42+√2≈0.85. For the lowest value of λij (λik), the remaining principal squeeze variances obey λik >2 and λjk >2(λij >2 and λjk >2). Interestingly, no squeezing is observed for the pair of modes j−k, as illustrated in Fig. 9b. Similarly as type III-1 and type III-2 states, also the type III-3 states exhibit threemode squeezing, as illustrated in Fig. 9d. The range of negativity Nijk allowing threemode squeezing is even broader and the three-mode principal squeeze variance λijk can take on smaller values. The minimal principal squeeze variance λijk ≈1.8is found for Nijk ≈0.6. Three-mode squeezed states are observed up to Nijk ≈0.89. The relations between the twoand three-mode principal squeeze variances are revealed in Figs. 9e and 9f where the yellow area represents the states exhibiting three-mode squeezing (λijk <3). For the III-3 states, the largest three-mode squeezing occurs for other states than those endowed with the largest two-mode squeezing. The largest three-mode squeezing is achieved if λij =λik attains their smallest possible values. Importantly, the three-mode principal squeeze variance λijk is smaller than 3 only when both pairs of qubits exhibit two-mode squeezing, or when only one of them does. Also, when three-mode squeezing is generated, the two-mode principal squeeze variance of the unsqueezed qubit pair is lower than 3, as noted earlier. We summarize the above results in Table 1that qualitatively shows the presence or absence of twoand three-mode entanglement and squeezing in all subgroups of the class III states. 123 Twoand three-mode squeezing... Page 19 of 22 339 Table 1 Twoand three-mode negativities and principal squeeze variances for the states classified as type III III-0 III-1A III-1B III-2 III-3 Nij = 0no no no yesyes Njk = 0noyes yes noyes Nik = 0no no no yesyes Nijk = 0 yes yes yes yes yes λij <2 no yes no yes yes λjk <2 no yes yes yes no λik <2 no yes no yes yes λijk <3 no yes yes yes yes 5 Conclusions In this paper, we have analyzed various three-qubit states that exhibit full three-mode entanglement. Following the classification of three-qubit states proposed by Sabín and GarcíaAlcaine in [14], we have investigated the relations between twoand three-mode entanglement and twoand three-mode squeezing for all four groups of the states exhibiting three-mode entanglement. We have demonstrated that, except of the GHZ states forming type III-0, the states exhibit both twoand three-mode squeezing. The GHZ states are specific as, though endowed with three-mode entanglement, they do not show any form of two-mode entanglement. For the remaining types of states, we have identified the minima of the twoand three-mode principal squeeze variances in the corresponding space of the states. The same lowest two-mode principal squeeze variances are reached in type III-1, III-2, and III-3 groups of states. On the other hand, the strongest three-mode squeezing is observed for the type III-3 states that belong to the W-class states. This group of states also allows to observe the three-mode squeezed states that exhibit high level of three-mode entanglement. Our results show that some states are simultaneously strongly entangled and highly squeezed, both in twoand three modes. This may seem surprising. However, Wang and Sanders demonstrated in [56] that spin squeezing is associated with pairwise entanglement and the increase in squeezing results in the increase of entanglement. Nevertheless, it should be noted that the strongest two-mode entangled states of the analyzed types are characterized by either no squeezing or weak squeezing. In the case of three-mode entanglement, the situation is complex owing to various multipartite entanglement structures and this requires further analysis. We have revealed that three-mode squeezing occurs alongside with two-mode squeezing found in one, two, or three pairs of qubits. However, for specific states two-mode squeezing is absent. The smallest three-mode principal squeeze variance for the states of types III-1 and III-2 is reached together with the smallest two-mode principal squeeze variances of one or two pairs of qubits. Acknowledgements J.K.K. acknowledges the support provided by the program of the National Science Center (NCN) entitled MINIATURA8 program, project no. 2024/08/X/ST2/00753. J.P. acknowledges support by the project No. 25-15775S of the Czech Science Foundation and ITI 123 339 Page 20 of 22 J.K. Kalaga, J. Peˇ rina Jr. CZ.02.01.01/00/23_021/0008790 of the Ministry of Education, Youth, and Sports of the Czech Republic and EU. Author Contributions All authors wrote the main manuscript text and reviewed the manuscript. J.K.K. prepared figures. Data Availability Data sets generated during the current study are available from the corresponding author on reasonable request. Declarations Conflict of interest The authors declare no Conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829–1834 (1999). https://doi.org/10.1103/PhysRevA.59.1829 2. Lance, A.M., Symul, T., Bowen, W.P., Sanders, B.C., Lam, P.K.: Tripartite quantum state sharing. Phys. Rev. Lett. 92, 177903 (2004). https://doi.org/10.1103/PhysRevLett.92.177903 3. Chen, Y.A., Zhang, A.N., Zhao, Z., Zhou, X.Q., Lu, C.Y., Peng, C.Z., Yang, T., Pan, J.W.: Experimental quantum secret sharing and third-man quantum cryptography. Phys. Rev. Lett. 95, 200502 (2005). https://doi.org/10.1103/PhysRevLett.95.200502 4. Kimble, H.J.: The quantum internet. Nature 453, 1023–1030 (2008). https://doi.org/10.1038/ nature07127 5. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010). https://doi.org/10.1017/CBO9780511976667 6. Chitambar, E., Hsieh, M.H.: Relating the resource theories of entanglement and quantum coherence. Phys. Rev. Lett. 117, 020402 (2016). https://doi.org/10.1103/PhysRevLett.117.020402 7. Figgatt, C., Maslov, D., Landsman, K.A., Linke, N.M., Debnath, S., Monroe, C.: Complete 3-qubit grover search on a programmable quantum computer. Nat. Commun. 8, 1918 (2017). https://doi.org/ 10.1038/s41467-017-01904-7 8. Chitambar, E., Gour, G.: Quantum resource theories. Rev. Mod. Phys. 91, 025001 (2019). https://doi. org/10.1103/RevModPhys.91.025001 9. Wang, Y., Chen, Y., Bui, H.T., Wolf, C., Haze, M., Mier, C., Kim, J., Choi, D.J., Lutz, C.P., Bae, Y., Phark, S.H., Heinrich, A.J.: An atomic-scale multi-qubit platform. Science 382, 87–92 (2023). https:// doi.org/10.1126/science.ade5050 10. Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going Beyond Bell’s Theorem, pp. 69–72. Springer, Netherlands, Dordrecht (1989) 11. Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000). https://doi.org/10.1103/PhysRevA.62.062314 12. Acín, A., Bruß, D., Lewenstein, M., Sanpera, A.: Classification of mixed three-qubit states. Phys. Rev. Lett. 87, 040401 (2001). https://doi.org/10.1103/PhysRevLett.87.040401 13. Sharma, S.S., Sharma, N.K.: Two-way and three-way negativities of three-qubit entangled states. Phys. Rev. A 76, 012326 (2007). https://doi.org/10.1103/PhysRevA.76.012326 14. Sabín, C., García-Alcaine, G.: A classification of entanglement in three-qubit systems. Eur. Phys. J. D 48, 435–442 (2008) 123 Twoand three-mode squeezing... Page 21 of 22 339 15. Enríquez, M., Delgado, F., Zyczkowski, K.: Entanglement of three-qubit random pure states. Entropy 20(10) (2018). https://doi.org/10.3390/e20100745 16. Joo, J., Park, Y.J., Lee, J., Jang, J., Kim, I.: Quantum secure communication via a w state. J. Korean Phys. Soc. 46(4), 763 (2005) 17. Fu-Guo, D., Ping, Z., Xi-Han, L., Chun-Yan, L., Hong-Yu, Z.: Efficient multiparty quantum secret sharing with greenberger–horne–zeilinger states. Chin. Phys. Lett. 23(5), 1084 (2006). https://doi.org/ 10.1088/0256-307X/23/5/006 18. Jian, W., Quan, Z., Chao-Jing, T.: Quantum secure communication scheme with w state. Commun. Theor. Phys. 48(4), 637 (2007). https://doi.org/10.1088/0253-6102/48/4/013 19. Tsai, C.W., Hwang, T.: New deterministic quantum communication via symmetric w state. Opt. Commun. 283(21), 4397–4400 (2010). https://doi.org/10.1016/j.optcom.2010.06.039 20. Li, G.D., Cheng, W.C., Wang, Q.L., Cheng, L., Mao, Y., Jia, H.Y.: Quantum Secret Sharing Enhanced: Utilizing W States for Anonymous and Secure Communication (2024). arXiv:abs/2402.02413 21. Yang, C.P., Gea-Banacloche, J.: Three-qubit quantum error-correction scheme for collective decoherence. Phys. Rev. A 63, 022311 (2001). https://doi.org/10.1103/PhysRevA.63.022311 22. Tornberg, L., Wallquist, M., Johansson, G., Shumeiko, V.S., Wendin, G.: Implementation of the threequbit phase-flip error correction code with superconducting qubits. Phys. Rev. B 77, 214528 (2008). https://doi.org/10.1103/PhysRevB.77.214528 23. Ottaviani, C., Vitali, D.: Implementation of a three-qubit quantum error-correction code in a cavity-qed setup. Phys. Rev. A 82, 012319 (2010). https://doi.org/10.1103/PhysRevA.82.012319 24. Sohn, I., Tarucha, S., Choi, B.S.: Analysis of physical requirements for simple three-qubit and ninequbit quantum error correction on quantum-dot and superconductor qubits. Phys. Rev. A 95, 012306 (2017). https://doi.org/10.1103/PhysRevA.95.012306 25. Reed, M.D., DiCarlo, L., Nigg, S.E., Sun, L., Frunzio, L., Girvin, S.M., Schoelkopf, R.J.: Realization of three-qubit quantum error correction with superconducting circuits. Nature 482, 382–385 (2012). https://doi.org/10.1038/nature10786 26. Lukš, V.P.A., Peˇrina, J.: Principal squeezing of vacuum fluctuations. Opt. Commun. 67, 149 (1988). https://doi.org/10.1016/0030-4018(88)90322-7 27. Dodonov, V.V.: Nonclassical states in quantum optics: a squeezed review of the first 75 years. J. Opt. B: Quantum Semiclass. Opt. 4, R1–R33 (2002) 28. Dodonov, V.V., Man´ ko, V.I.: Nonclassical states in quantum physics: brief historical review, in Theory of Nonclassical States of Light, ed. by V.V. Dodonov, M.V. Man´ ko (Taylor & Francis, 2003), pp. 1—80 29. Lvovsky, A.I., Raymer, M.G.: Continuous-variable optical quantum state tomography. Rev. Mod. Phys. 81, 299–332 (2009) 30. Slusher, R.E., Hollberg, L.W., Yurke, B., Mertz, J.C., Valley, J.F.: Observation of squeezed states generated by four-wave mixing in an optical cavity. Phys. Rev. Lett. 55, 2409–2412 (1985). https:// doi.org/10.1103/PhysRevLett.55.2409 31. Shelby, R.M., Levenson, M.D., Walls, D.F., Aspect, A., Milburn, G.J.: Generation of squeezed states of light with a fiber-optic ring interferometer. Phys. Rev. A 33, 4008–4025 (1986). https://doi.org/10. 1103/PhysRevA.33.4008 32. Wu, L.A., Kimble, H.J., Hall, J.L., Wu, H.: Generation of squeezed states by parametric down conversion. Phys. Rev. Lett. 57, 2520–2523 (1986). https://doi.org/10.1103/PhysRevLett.57.2520 33. Dell’Anno, F., De Siena, S., Adesso, G., Illuminati, F.: Teleportation of squeezing: optimization using non-gaussian resources. Phys. Rev. A 82, 062329 (2010). https://doi.org/10.1103/PhysRevA. 82.062329 34. Goldberg, A.Z., Heshami, K.: How squeezed states both maximize and minimize the same notion of quantumness. Phys. Rev. A 104, 032425 (2021). https://doi.org/10.1103/PhysRevA.104.032425 35. Renger, M., Pogorzalek, S., Fesquet, F., Honasoge, K., Kronowetter, F., Chen, Q., Nojiri, Y., Inomata, K., Nakamura, Y., Marx, A., Deppe, F., Gross, R., Fedorov, K.G.: Flow of quantum correlations in noisy two-mode squeezed microwave states. Phys. Rev. A 106, 052415 (2022). https://doi.org/10. 1103/PhysRevA.106.052415 36. García-Beni, J., Giorgi, G.L., Soriano, M.C., Zambrini, R.: Squeezing as a resource for time series processing in quantum reservoir computing. Opt. Express 32(4), 6733–6747 (2024). https://doi.org/ 10.1364/OE.507684 37. Nguyen, H.Q., Derkach, I., Hajomer, A.A.E., Chin, H.M., Oruganti, A.n., Andersen, U.L., Usenko, V., Gehring, T.: Digital reconstruction of squeezed light for quantum information processing. npj Quantum Information 11, 71 (2025). https://doi.org/10.1038/s41534-025-01018-9 123 339 Page 22 of 22 J.K. Kalaga, J. Peˇ rina Jr. 38. Madsen, L.S., Usenko, V.C., Lassen, M., Filip, R., Andersen, U.L.: Continuous variable quantum key distribution with modulated entangled states. Nat. Commun. 3, 1083 (2012). https://doi.org/10.1038/ ncomms2097 39. Gehring, T., Händchen, V., Duhme, J., Furrer, F., Franz, T., Pacher, C., Werner, R.F., Schnabel, R.: Implementation of continuous-variable quantum key distribution with composable and one-sideddevice-independent security against coherent attacks. Nat. Commun. 6, 8795 (2015). https://doi.org/ 10.1038/ncomms9795 40. Alexander, B.J., Bollinger, J.J., Tame, M.S.: Robustness of the projected squeezed state protocol. Phys. Rev. A 109, 052614 (2024). https://doi.org/10.1103/PhysRevA.109.052614 41. Hillmann, T., Quijandría, F.: Quantum error correction with dissipatively stabilized squeezed-cat qubits. Phys.Rev.A107, 032423 (2023). https://doi.org/10.1103/PhysRevA.107.032423 42. Xu, Q., Zheng, G., Wang, Y.X., Zoller, P., Clerk, A.A., Jiang, L.: Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits. npj Quantum Information 9,78 (2023). https://doi.org/10.1038/s41534-023-00746-0 43. Korolev, S.B., Bashmakova, E.N., Golubeva, T.Y.: Error correction using squeezed fock states. Quantum Inf. Process. 23, 354 (2024). https://doi.org/10.1007/s11128-024-04549-w 44. Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996). https:// doi.org/10.1103/PhysRevLett.77.1413 45. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996). https://doi.org/10.1016/S0375-9601(96)00706-2 46. Kumar, S., Cárdenas-López, F., Hegade, N., Albarrán-Arriagada, F., Solano, E., Barrios, G.A.: Tripartite entanglement in quantum memristors. Phys. Rev. Appl. 18, 034004 (2022). https://doi.org/10.1103/ PhysRevApplied.18.034004 47. Korolkova, N., Peˇrina, J.: Quantum statistics and dynamics of kerr nonlinear couplers. Opt. Commun. 136, 135–149 (1996). https://doi.org/10.1016/S0030-4018(96)00676-1 48. Ariunbold, G., Peˇrina, J.: Quantum statistics of contradirectional kerr nonlinear couplers. Opt. Commun. 176, 149–154 (2000) 49. Leo´nski, W., Kowalewska-Kudłaszyk, A.: in Progress in Optics, vol. 56, ed. by E. Wolf (Elsevier, 2011), pp. 131–185. https://doi.org/10.1016/B978-0-444-53886-4.00003-4 50. Korbicz, J.K., Cirac, J.I., Lewenstein, M.: Spin squeezing inequalities and entanglement of nqubit states. Phys. Rev. Lett. 95, 120502 (2005). https://doi.org/10.1103/PhysRevLett.95.120502 51. Ma, J., Wang, X., Sun, C., Nori, F.: Quantum spin squeezing. Phys. Rep. 509(2), 89–165 (2011). https:// doi.org/10.1016/j.physrep.2011.08.003 52. Gross, C.: Spin squeezing, entanglement and quantum metrology with bose–einstein condensates. J. Phys. B: At. Mol. Opt. Phys. 45(10), 103001 (2012). https://doi.org/10.1088/0953-4075/45/10/103001 53. Kalaga, J.K., Kowalewska-Kudłaszyk, A., Jarosik, M.W., Szcz¸e´sniak, R., Leo´nski, W.: Enhancement of the entanglement generation via randomly perturbed series of external pulses in a nonlinear bose– hubbard dimer. Nonlinear Dyn. 97(2), 1619–1633 (2019). https://doi.org/10.1007/s11071-019-050845 54. Kalaga, J.K., Kowalewska-Kudłaszyk, A.: Frequency variations in impulse excitations as a way of entanglement increase in the two-mode bose-hubbard model. J. Opt. Soc. Am. B 36(8), 2140–2146 (2019). https://doi.org/10.1364/JOSAB.36.002140 55. Nair, J.M.P., Flebus, B.: Engineering long-lived entanglement through dissipation in quantum hybrid solid-state platforms (2025). arXiv:abs/2410.15588 56. Wang, X., Sanders, B.C.: Spin squeezing and pairwise entanglement for symmetric multiqubit states. Phys.Rev.A68, 012101 (2003). https://doi.org/10.1103/PhysRevA.68.012101 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123