# Few-shot Dialogue Strategy Learning
for Motivational Interviewing via Inductive Reasoning

Zhouhang Xie1 Bodhisattwa Prasad Majumder2 Mengjie Zhao3
Yoshinori Maeda3 Keiichi Yamada1 Hiromi Wakaki3 Julian McAuley1
1 UC San Diego 2 Allen Institute for AI 3 Sony Group Corporation
{zhx022<EMAIL_ADDRESS>
###### Abstract

We consider the task of building a dialogue system that can motivate users to
adopt positive lifestyle changes: Motivational Interviewing. Addressing such a
task requires a system that can infer how to motivate a user effectively. We
propose DIIR, a framework that is capable of learning and applying
conversation strategies in the form of natural language inductive rules from
expert demonstrations. Automatic and human evaluation on instruction-following
large language models show natural language strategy descriptions discovered
by DIIR can improve active listening skills, reduce unsolicited advice, and
promote more collaborative and less authoritative responses, outperforming
various demonstration utilization methods.

Few-shot Dialogue Strategy Learning
for Motivational Interviewing via Inductive Reasoning

Zhouhang Xie1 Bodhisattwa Prasad Majumder2 Mengjie Zhao3 Yoshinori Maeda3
Keiichi Yamada1 Hiromi Wakaki3 Julian McAuley1 1 UC San Diego 2 Allen
Institute for AI 3 Sony Group Corporation {zhx022<EMAIL_ADDRESS>

##  1 Introduction

We study Motivational Interviewing (MI), where a conversational agent assists
users in finding motivation and reducing ambivalence in adopting positive
lifestyle changes such as quitting drug abuse and seeking medical assistance
Moyers et al. (2016). A successful MI dialogue system must be able to infer
how to best motivate a user to adopt positive changes, which is a nontrivial
task. For example, as shown in Figure 1, one way to convince a user to workout
is to educate the user about its benefits, yet immediately offering
suggestions without understanding the context can risk creating resistance and
confronting users Brehm (1981). While prior work studied rule-based dialogue
systems for MI Welch et al. (2020); Park et al. (2019) and MI dialogue
response rephrasing Welivita and Pu (2023), another important problem is to
build dialogue systems that automatically discover and learn MI skills.

Figure 1: DIIR retrieves learned dialogue strategy descriptions to generate a
response that leads to positive outcomes. See Table 9 in the Appendix for
example strategies.

In this work, we introduce DIIR (Dialogue Strategy Learning via Inductive
Reasoning), a framework for aligning LLMs to dialogue behavior of professional
MI interviewers given a few demonstration dialogues. DIIR does not require
training and instead employs LLMs to generate a set of natural language
dialogue strategy descriptions, such as ‘‘when the user is hesitant about
change, the interviewer should ask open questions to learn more about the
user’s concern" by analyzing demonstration dialogues. These strategies are
hypothesis about how professional interviewers respond in various situations,
and are reused at inference time as instructions for LLMs to mimic MI
interviewers. This process of identifying underlying principles from
observations and generalizing them for inference is a form of inductive
reasoning Feeney and Heit (2007).

Different from common practice that use available demonstrations as in-
context-learning (ICL) examples Brown et al. (2020); Xu et al. (2023a),
strategies descriptions discovered by our framework explicitly state the
desired behavior for the model, such as ‘‘ask open questions" and their
applicable situations, such as ‘‘when the client is hesitant about change".
These automatically generated strategy descriptions are a more precise nudge
to the LLM being instructed than ICL examples. Existing work that employs
language models for generating and reusing natural language strategies from
observations Majumder et al. (2023); Nottingham et al. (2023); Wang et al.
(2024a) (i.e. Inductive Reasoning with LLMs) require interactive environments
to verify and improve the generated statements. In comparison, our method
learns from a static demonstration dialogue dataset which is more commonly
available than interactive environments for dialogue models.

To evaluate DIIR, we automate and adapt a set of metrics derived from clinical
psychology literature for evaluating human MI practitioners. We experiment
with DIIR on a publicly available MI dataset. Automatic and human evaluation
show strategies discovered by DIIR can help instruction-following downstream
language models produce more collaborative, respectful, and least resistance-
inducing responses with as few as five annotated dialogues of expert
demonstrations. Our contributions are as follows: (1) To our best knowledge,
we are the first to work to build and evaluate a dialogue system that
automatically learns from dialogue data for MI. (2) We develop DIIR, a
framework that learns and applies dialogue strategies as natural language
inductive rules from MI experts for only a few dialogues. (3) We automate
evaluations from the clinical psychology literature on MI to facilitate future
research on MI.

##  2 Dialogue Strategy Learning via Inductive Reasoning

The goal of an MI interviewer is to convince a client to make a positive
change of interest, such as to start working out. Given a dialogue history $h$
consisting of previous user-system turns, an ideal MI dialogue system produces
the next system response $a$ that best convinces the user to make a positive
change. We solve this optimal response generation problem by using LLMs to
analyze demonstration dialogues and generate dialogue strategy descriptions
stating how to respond in various situations, such as “When the user is
hesitant about change, ask open questions.” These descriptions can then be
reused to improve an instruction-following LLM to promote demonstration-
aligned dialogue behavior, such as asking open questions when the client is
hesitant. We now describe DIIR’s learning and inference process (see Appendix
D for step-by-step concrete examples).

Algorithm 1 EnhanceStrategy

1:Dialogue history $h$, gold response $a^{*}$, generator LLM $G$,
discriminator LLM $D$, executor LLM $E$, maximum number of trials $N$

2:Strategy $e$ in natural language, description of its applicable situation
$k$

3:$e\leftarrow$ ""

4:for i in $N$ do

5: $a^{\prime}\leftarrow$ E.generate_response (h, e)

6: if D.confirms_is_similar ($a^{\prime}$, $a^{*}$) then

7: break

8: $e\leftarrow$ G.improve_strategy ($h$, $e$, $a^{*}$, $a^{\prime}$)

9:$k\leftarrow$ G.describe_situation ($h$)

10:return $e,k$

Learning Dialogue Strategy by Analyzing Demonstration Dialogues Given a
dialogue history $h$ and a gold interviewer response $a^{*}$, a corresponding
dialogue strategy description $e$ should contain sufficient information such
that an LLM could closely reproduce $a^{*}$ given $h$ and $e$. Intuitively,
one could instruct an LLM to analyze the gold response and its corresponding
dialogue history to generate descriptions containing information about “When
<description about $h$>, one should <description about $a^{*}$>,” which can
then be used as strategy descriptions. However, there is no guarantee that
LLM-generated strategy descriptions are valid.

To address this issue, we propose to employ a discriminator LLM $D$ and an
executor LLM $E$ in place of the interactive environment in previous works to
give feedback during the strategy statement generation process. The intuition
is that to verify a dialogue strategy such as ‘‘When the user is hesitant
about change, ask open questions", we could instruct an generator LLM $G$ to
generate a response given the dialogue history where a user is hesitant and
this instruction, and check whether the generated response matches the gold
interviewer response.

This verification process eliminates two sources of error: the generated
strategy is incorrect or is correct but is not detailed enough to instruct an
LLM to reproduce the groundtruth response. Existing work typically addresses
this statement-verification problem by verifying the statements in interactive
environments such as MineCraft Nottingham et al. (2023). However, there are no
environments to verify statements about dialogue strategies.

The process for generating and verifying a strategy description given a
context-and-gold-response pair is as shown in Algorithm 1. Concretely, given a
hypothesized dialogue strategy $e$ (initialized to be an empty string), the
executor generates a response $a^{\prime}$ given the dialogue history $h$ and
the strategy $e$. If the discriminator checks and marks $a^{\prime}$ as
similar to gold response $a^{*}$, we accept the strategy $e$ as valid.
Otherwise, we instruct the generator LLM to regenerate a better strategy given
the dialogue history, current strategy, generated response, and the gold
response. This generate-and-verify process repeats until a valid strategy
description has been found or a fixed number of search steps $N$ has been
reached.

Finally, to be able to reuse this strategy in the future, we prompt the
generator to generate a description of the situation $k$ such as “the user is
hesitant about change.” This situation description can be derived from $e$
since the dialogue strategy already contains information about when the
strategy is applicable, but we use an LLM to generate it in practice for ease
of implementation. We repeat the process introduced above for all context-
response pairs available in the demonstration dialogues to create suitable
strategies for various scenarios. While drawing inspirations from adversarial
training Goodfellow et al. (2014) and reference games Fried et al. (2018), our
algorithm is purely language-based.

#### Inference via Strategy Reuse

After learning, DIIR accumulates a set of strategies $\\{e_{i}\\}$ and their
corresponding applicable situations $\\{k_{i}\\}$. Thus, given an inference-
time dialogue history $h_{\text{inference}}$, we can simply generate a
description of the situation $k_{\text{inference}}$ using the generator LLM,
and then use $k_{\text{inference}}$ to retrieve the most suitable strategy
$e_{i}$ by identifying $k_{i}$ that is most similar to $k_{\text{inference}}$.
We then instruct the executor to generate a response given
$h_{\text{inference}}$ and the retrieved $e_{i}$. In practice, we use a text
embedding model $\Phi$ to encode all situational descriptions and find the top
ten most suitable demonstration situation by maximizing the dot product of
$\Phi(k_{i})$ and $\Phi(k_{\text{inference}})$ following Lewis et al. (2020);
Guu et al. (2020). After this, we instruct the Generator LLM to pick a
strategy that is most appropriate to the dialogue history from the ten
retrieved strategies and follow its description. This process is illustrated
in Figure 1.

##  3 Experiments

#### Models and Baselines

We demonstrate DIIR’s few-shot learning ability by learning from 5 dialogues
using GPT3.5 and GPT4. We chose these two models due to their strong reasoning
ability, similar to various recent works on LLM-based reasoning Wang et al.
(2024b); Majumder et al. (2023); Nottingham et al. (2023). For baselines, we
consider three variants of in-context learning (ICL) in addition to the
vanillan LLM: ICL-RAND with 5 random demonstrations, ICL-KNN Xu et al. (2023a)
with 5 demonstrations with most similar dialogue context to the inference-time
context computed using a text embedding model, and ICL-ALL with all
demonstration dialogues.

#### Dataset

We experiment on the AnnoMI dataset Wu et al. (2022), an MI dataset
transcribed from YouTube videos with 133 MI transcripts for various topics
labeled as high and low-quality interviewer demonstrations. We discard all
dialogues labeled as low-quality demonstrations, randomly sample 80 dialogues
with 1192 turns for our main evaluation on GPT3.5, and use the rest for
learning. For all high-cost experiments involving GPT4 or human evaluation, we
further randomly sub-sample 10 dialogues with 148 turns.

##  4 Evaluation

For automatic evaluation, we evaluate the alignment of our model by obtaining
dialogue action counts with a dialogue act classifier trained on crowd-sourced
annotations Welivita and Pu (2022). We report the following behavior-count-
based evaluation adopted from the Motivational Interviewing Integrity
Treatment (MITI; Moyers et al. (2016)) and Motivational Interviewing Skill
Code (MISC; Miller et al. (2003)) used to evaluate professional interviewers
from clinical psychology literature.

Ratio of MI-guideline-inconsist Behavior (%MI-i) The ratio of MI-inconsistent
behaviors (confront, direct, warn, and advice without permission) in model
responses.

Complex over Simple Reflections (C/S) The ratio of complex over simple
reflections in model responses, indicating how often the model produces in-
depth feedback to the client.

Reflection over Questions (R/Q) The count of reflections over questions in the
models’ responses.

Percent of Active Listening Behaviors (%AL) The ratio of the sum of Questions
and Reflections among all behavior counts, showing how often does the modal
employ active listening skills.

Percent of Non-authoritatitive Behaviors (%NA) The frequency of responses that
are not confront, warn, direct, advice (with or without permission), or give
information.

We verify the effectiveness of our automatic evaluation by running the
evaluation on high and low-quality interviewer demonstrations in the AnnoMI
dataset Wu et al. (2022), results show our method can effectively tell apart
high versus low quality demonstrations (see Appendix C for additional
discussions and details of the dialogue act classifier).

For human evaluation, we evaluate our models’ performance against in-context
baselines by asking human evaluators on Amazon Mechanical Turk to read four
context-response pairs produced by both models and choose the one that is best
aligned with a description of MI (details in Appendix C).

##  5 Results and Analysis

Method | %MI-i↓ | C/S↑ | R/Q↑ | %AL↑ | %NA↑
---|---|---|---|---|---
W/o Strategies
GPT3.5-BESTBASE | 2.1 | 44.8 | 1.53 | 54.3 | 82.8
GPT4-BESTBASE | 1.4 | 11.8 | 1.06 | 45.8 | 71.8
With Strategies (Ours)
GPT3.5-DIIR | 1.3 | 115 | 0.99 | 61.1 | 95.1
GPT4-DIIR | 0.6 | 135 | 1.04 | 59.6 | 94.4
Gold | 1.4 | 2.3 | 1.51 | 59.0 | 81.7
Table 1: Alignment performance against best possible result selected per
dimension from all baselines. We found open-source LLMs cannot consistently
follow self-generated statements, coherent with recent findings Qiu et al.
(2024), and thus are less performant (See Appendix E).

Our main results are as shown in Table 1. Due to space constraints, we report
the best performance among the four baselines (base model and three ICL
variants) for each dimension under the name BESTBASE, and show detailed
performance in Table 4 in the Appendix. Although BESTBASE has an unfair
advantage due to combining best baseline results for each dimension, our
method outperforms the baselines in all categories except the Reflection-to-
Question ratio, sometimes outperforming the ground truth responses. Meanwhile,
MI guideline defines an R/Q ratio of 1.0 to be good and 0.5 to be satisfactory
Moyers et al. (2016), showing that our model maintains competitive performance
in this dimension.

Overall, our method produces fewer MI-inconsistent responses, provides more
in-depth reflections, and better leverages active listening skills and the MI
spirit of not assuming expert roles. These trends are coherent with human
evaluation results, as shown in Table 2.

DIIR vs. ICL | RAND | ALL | KNN
---|---|---|---
GPT3.5 |  21* |  7* |  23*
GPT4 |  15* | 5 | -5
Table 2: Human evaluation results. We report number of additional times DIIR
win against baseline; with statistical difference with p$<$0.05 denoted in *.

##  6 Related Work

#### Motivational Interviewing in NLP

Prior work in NLP about MI primarily studied the task of annotating Welivita
and Pu (2022); Tanana et al. (2015) and predicting Pérez-Rosas et al. (2017);
Huang et al. (2018) dialogue actions. Recently, there are attempts on building
systems for evaluating Min et al. (2022) and rewriting Welivita and Pu (2023)
responses for MI but does not focus on response generation. A few pioneering
works build MI dialogue systems with hard-coded rules or dialogue templates
Welch et al. (2020); Park et al. (2019) but focuses specific domains such as
COVID-19. To the best of our knowledge, we are the first to build an MI
dialogue system that automatically learns from expert demonstrations.

#### LLMs for Generating and Reusing Natural Language Strategies

Generating hypothesis from observations and reusing the generated hypothesis
with LLMs have shown a promising direction for various tasks, such as planning
Majumder et al. (2023); Nottingham et al. (2023); Wang et al. (2023), prompt-
generation Wang et al. (2024b), and code generation Wang et al. (2024a). This
is a form of inductive reasoning Wang et al. (2023) where LLMs draw conclusion
from obvervations. However, prior works require interactive environments for
feedback which are unsuitable for learning from static datasets, while our
work use an LLM for feedback and lifts the requirement for interactive
environments.

##  7 Conclusion

We propose DIIR, a method for few-shot aligning large language models to MI
techniques. Our method infers natural language strategies from expert
demonstrations and reuses these strategies at inference time. Automatic and
human evaluation show our method is better aligned with the behavior of
motivational interviewing experts, produces more in-depth reflections, and
better leverages active listening skills, outperforming a variety of
demonstration-based methods.

##  8 Limitations

We note that our framework is developed for Motivational Interviewing and
benefits from domain-specific knowledge such as MI-specific dialogue actions.
Meanwhile, our framework assumes access to a strong instruction following
language model, which is usually restricted by API access or computational
resources. Finally, we note that while we adopt clinical psychology literature
verified metrics and human evaluation to measure the performance of DIIR, the
ideal evaluation is deploying MI dialogue systems for continued intervention.

##  9 Ethical Concerns

Following previous work on conversational persuasion Wang et al. (2019), we
note Motivational Interviewing shares similarities with persuasion dialogues.
While the technique for Motivational Interviewing is developed for inducing
positive changes in users, system deployers should carefully analyze the
specific topic for Motivational Interviewing to eliminate risks of harm.
Meanwhile, when deploying a system equipped with MI abilities, the user should
be informed about the system’s identity so that they can make an informed
choice when interacting with such a system.

##  10 Acknowledgments

We would like to thank Tim Althoff, Jessica Echterhoff, and Yilan Chen for
their helpful comments. This work is funded by the SONY Research Award
Program.

## References

  * Brehm (1981) Jack W. Brehm. 1981.  A theory of psychological reactance.
  * Brown et al. (2020) Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. 2020.  Language models are few-shot learners.  In _Neurips_ , volume 33, pages 1877–1901. Curran Associates, Inc.
  * Feeney and Heit (2007) Aidan Feeney and Evan Heit. 2007.  Inductive reasoning: Experimental, developmental and computational approaches.
  * Fried et al. (2018) Daniel Fried, Jacob Andreas, and Dan Klein. 2018.  Unified pragmatic models for generating and following instructions.  In _NAACL-HLT_ , pages 1951–1963, New Orleans, Louisiana. Association for Computational Linguistics.
  * Goodfellow et al. (2014) Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron C. Courville, and Yoshua Bengio. 2014.  Generative adversarial nets.  In _NeurIPS_ , pages 2672–2680.
  * Guu et al. (2020) Kelvin Guu, Kenton Lee, Zora Tung, Panupong Pasupat, and Ming-Wei Chang. 2020.  Realm: Retrieval-augmented language model pre-training.  In _ICML_ , ICML’20. JMLR.org.
  * Huang et al. (2018) Xiaolei Huang, Lixing Liu, Kate Carey, Joshua Woolley, Stefan Scherer, and Brian Borsari. 2018.  Modeling temporality of human intentions by domain adaptation.  In _EMNLP_ , pages 696–701, Brussels, Belgium. Association for Computational Linguistics.
  * Ji et al. (2022) Shaoxiong Ji, Tianlin Zhang, Luna Ansari, Jie Fu, Prayag Tiwari, and Erik Cambria. 2022.  MentalBERT: Publicly available pretrained language models for mental healthcare.  In _LREC_ , pages 7184–7190, Marseille, France. European Language Resources Association.
  * Lewis et al. (2020) Patrick S. H. Lewis, Ethan Perez, Aleksandra Piktus, Fabio Petroni, Vladimir Karpukhin, Naman Goyal, Heinrich Küttler, Mike Lewis, Wen-tau Yih, Tim Rocktäschel, Sebastian Riedel, and Douwe Kiela. 2020.  Retrieval-augmented generation for knowledge-intensive NLP tasks.  In _Neurips_.
  * Majumder et al. (2023) Bodhisattwa Prasad Majumder, Bhavana Dalvi Mishra, Peter Jansen, Oyvind Tafjord, Niket Tandon, Li Zhang, Chris Callison-Burch, and Peter Clark. 2023.  Clin: A continually learning language agent for rapid task adaptation and generalization.
  * Miller et al. (2003) William R Miller, Theresa B Moyers, Denise Ernst, and Paul Amrhein. 2003.  Manual for the motivational interviewing skill code (misc).  _Unpublished manuscript. Albuquerque: Center on Alcoholism, Substance Abuse and Addictions, University of New Mexico_.
  * Min et al. (2022) Do June Min, Verónica Pérez-Rosas, Kenneth Resnicow, and Rada Mihalcea. 2022.  PAIR: Prompt-aware margIn ranking for counselor reflection scoring in motivational interviewing.  In _EMNLP_ , pages 148–158, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
  * Moyers et al. (2016) Theresa B. Moyers, Lauren N. Rowell, Jennifer K Manuel, Denise B. Ernst, and Jon M. Houck. 2016.  The motivational interviewing treatment integrity code (miti 4): Rationale, preliminary reliability and validity. _Journal of substance abuse treatment_ , 65:36–42.
  * Nottingham et al. (2023) Kolby Nottingham, Prithviraj Ammanabrolu, Alane Suhr, Yejin Choi, Hannaneh Hajishirzi, Sameer Singh, and Roy Fox. 2023.  Do embodied agents dream of pixelated sheep: Embodied decision making using language guided world modelling.  In _ICML_ , volume 202 of _Proceedings of Machine Learning Research_ , pages 26311–26325. PMLR.
  * Park et al. (2019) Sohyun Park, Jeewon Choi, Sungwoo Lee, Changhoon Oh, Changdai Kim, Soohyun La, Joonhwan Lee, and Bongwon Suh. 2019.  Designing a chatbot for a brief motivational interview on stress management: Qualitative case study.  _Journal of Medical Internet Research_ , 21.
  * Pérez-Rosas et al. (2016) Verónica Pérez-Rosas, Rada Mihalcea, Kenneth Resnicow, Satinder Singh, and Lawrence An. 2016.  Building a motivational interviewing dataset.  In _Proceedings of the Third Workshop on Computational Linguistics and Clinical Psychology_ , pages 42–51, San Diego, CA, USA. Association for Computational Linguistics.
  * Pérez-Rosas et al. (2017) Verónica Pérez-Rosas, Rada Mihalcea, Kenneth Resnicow, Satinder Singh, Lawrence An, Kathy J. Goggin, and Delwyn Catley. 2017.  Predicting counselor behaviors in motivational interviewing encounters.  In _EACL_ , pages 1128–1137, Valencia, Spain. Association for Computational Linguistics.
  * Qiu et al. (2024) Linlu Qiu, Liwei Jiang, Ximing Lu, Melanie Sclar, Valentina Pyatkin, Chandra Bhagavatula, Bailin Wang, Yoon Kim, Yejin Choi, Nouha Dziri, and Xiang Ren. 2024.  Phenomenal yet puzzling: Testing inductive reasoning capabilities of language models with hypothesis refinement.  In _ICLR_.
  * Shah et al. (2022) Raj Sanjay Shah, Faye Holt, Shirley Anugrah Hayati, A. Agarwal, Yi-Chia Wang, Robert E. Kraut, and Diyi Yang. 2022.  Modeling motivational interviewing strategies on an online peer-to-peer counseling platform.  _Proceedings of the ACM on Human-Computer Interaction_ , 6:1 – 24.
  * Tanana et al. (2015) Michael Tanana, Kevin Hallgren, Zac Imel, David Atkins, Padhraic Smyth, and Vivek Srikumar. 2015.  Recursive neural networks for coding therapist and patient behavior in motivational interviewing.  In _Proceedings of the 2nd Workshop on Computational Linguistics and Clinical Psychology: From Linguistic Signal to Clinical Reality_ , pages 71–79, Denver, Colorado. Association for Computational Linguistics.
  * Tavabi et al. (2021) Leili Tavabi, Trang Tran, Kalin Stefanov, Brian Borsari, Joshua Woolley, Stefan Scherer, and Mohammad Soleymani. 2021.  Analysis of behavior classification in motivational interviewing.  In _Proceedings of the Seventh Workshop on Computational Linguistics and Clinical Psychology: Improving Access_ , pages 110–115, Online. Association for Computational Linguistics.
  * Vaswani et al. (2017) Ashish Vaswani, Noam M. Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  In _Neural Information Processing Systems_.
  * Wang et al. (2023) Ruocheng Wang, E. Zelikman, Gabriel Poesia, Yewen Pu, Nick Haber, and Noah D. Goodman. 2023.  Language models meet world models: Embodied experiences enhance language models.  In _NeurIPS_.
  * Wang et al. (2024a) Ruocheng Wang, E. Zelikman, Gabriel Poesia, Yewen Pu, Nick Haber, and Noah D. Goodman. 2024a.  Hypothesis search: Inductive reasoning with language models.  In _ICLR_.
  * Wang et al. (2024b) Xinyuan Wang, Chenxi Li, Zhen Wang, Fan Bai, Haotian Luo, Jiayou Zhang, Nebojsa Jojic, Eric P. Xing, and Zhiting Hu. 2024b.  Promptagent: Strategic planning with language models enables expert-level prompt optimization.
  * Wang et al. (2019) Xuewei Wang, Weiyan Shi, Richard Kim, Yoojung Oh, Sijia Yang, Jingwen Zhang, and Zhou Yu. 2019.  Persuasion for good: Towards a personalized persuasive dialogue system for social good.  In _ACL_ , pages 5635–5649, Florence, Italy. Association for Computational Linguistics.
  * Welch et al. (2020) Charles Welch, Allison Lahnala, Veronica Perez-Rosas, Siqi Shen, Sarah Seraj, Larry An, Kenneth Resnicow, James Pennebaker, and Rada Mihalcea. 2020.  Expressive interviewing: A conversational system for coping with COVID-19.  In _Proceedings of the 1st Workshop on NLP for COVID-19 (Part 2) at EMNLP 2020_ , Online. Association for Computational Linguistics.
  * Welivita and Pu (2022) Anuradha Welivita and Pearl Pu. 2022.  Curating a large-scale motivational interviewing dataset using peer support forums.  In _Proceedings of the 29th International Conference on Computational Linguistics_ , pages 3315–3330, Gyeongju, Republic of Korea. International Committee on Computational Linguistics.
  * Welivita and Pu (2023) Anuradha Welivita and Pearl Pu. 2023.  Boosting distress support dialogue responses with motivational interviewing strategy.  In _Findings of ACL_ , pages 5411–5432, Toronto, Canada. Association for Computational Linguistics.
  * West et al. (2024) Peter West, Ximing Lu, Nouha Dziri, Faeze Brahman, Linjie Li, Jena D. Hwang, Liwei Jiang, Jillian Fisher, Abhilasha Ravichander, Khyathi Chandu, Benjamin Newman, Pang Wei Koh, Allyson Ettinger, and Yejin Choi. 2024.  The generative ai paradox: "what it can create, it may not understand.  In _ICLR_.
  * Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush. 2020.  Transformers: State-of-the-art natural language processing.  In _EMNLP Demos_ , pages 38–45, Online. Association for Computational Linguistics.
  * Wu et al. (2022) Zixiu Wu, Simone Balloccu, Vivek Kumar, Rim Helaoui, Ehud Reiter, Diego Reforgiato Recupero, and Daniele Riboni. 2022.  Anno-mi: A dataset of expert-annotated counselling dialogues.  In _ICASSP_ , pages 6177–6181.
  * Xu et al. (2023a) Benfeng Xu, Quan Wang, Zhendong Mao, Yajuan Lyu, Qiaoqiao She, and Yongdong Zhang. 2023a.  $k$NN prompting: Beyond-context learning with calibration-free nearest neighbor inference.  In _ICLR_.
  * Xu et al. (2023b) Canwen Xu, Yichong Xu, Shuohang Wang, Yang Liu, Chenguang Zhu, and Julian McAuley. 2023b.  Small models are valuable plug-ins for large language models.

##  Appendix A Building the Dialogue Act Classifier

Following prior work in the Computer Science and NLP community on annotating
motivational interviewing dialogues Shah et al. (2022); Welivita and Pu
(2022); Pérez-Rosas et al. (2016); Tavabi et al. (2021), we train a dialogue
classifier to map sentences to motivational interviewing behavior codes. We
train the dialogue act classifier on a publicly available dataset Welivita and
Pu (2022) with annotated dialogue actions specific to MI, since the AnnoMI
dataset does not contain fine-graned dialogue action labels. This dataset
contains single turn interactions only, and thus are suitable for training
dialogue act classifiers only but not the dialogue model itself. We fine-tune
two variants of pre-trained mask language models, BERT Vaswani et al. (2017)
and Mental-BERT Ji et al. (2022), a fine-tuned version of the base BERT model
on psychotherapy dialogues. We use a $0.6-0.2-0.2$ train-validation-test
split. We fine-tune both models using the Huggingface Transformers Library
Wolf et al. (2020) for a maximum of 10 epoch with early stopping at a learning
rate of $5e-5$. While the absolute accuracy of both BERT
111https://huggingface.co/bert-base-uncased and Mental-BERT
222https://huggingface.co/mental/mental-bert-base-uncased are not optimal
(Recall at 1 at 0.60 and 0.61, respectively), we fine the Recall at 5 out of
12 classes are high (at 0.95). Thus, we use the classifier to retrieve N=5
relevant labels given a input sentence, and use a GPT-3.5-turbo to decide the
final label, following recent findings in NLP that combining an LLM with small
classifier for classification has superior performance to both base models Xu
et al. (2023b).

##  Appendix B Other Experiment and Implementation Details

For DIIR’s learning process, we set the maximum number of optimization steps
to 3, and break the reference game between the generator and the discriminator
afterwards regardless of success. To enhance the feedback quality of the
discriminator, we also use the same dialogue act classifier as described in
Appendix A to provide distant labels of what potential actions are in the
ground truth response (this means predicting the string labels of dialogue
actions, and append this to the instruction for the LLM).

For all experiments, the specific model names we used in OpenAI is
gpt-3.5-turbo-1106 and gpt-4-1106-preview. These models are available as of
December 2023. For the off-the-shelf retriever for retrieving strategies, we
use a publicly available light-weight model
333https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2 and retrieve
usiang dot product scores. All experiments are conducting using 1 RTX A6000
GPU. Learning and Inference of DIIR on the specified data size finishes in 3
hours and 1.5 hours, respectively. The trainable component of the dialogue act
classifier (based on BERT) contains 12-layer, 768-hidden, 12-heads, and 110M
parameters. The retriever model contains 12-layer, 384-hidden, 12-heads, 33M
parameters.

##  Appendix C Details on Evaluation

###  C.1 Discussion on Evaluation Metrics

We provide discussions on the derived metrics in this section. In particular,
the evaluation process in MISC evaluates a interviewer on three dimensions:
(1) Acceptance: being non-judgemental and do not direct or confront the user,
(2) Empathy (different from the definition as used in Empathetic Dialogues in
NLP): showing genuine interest in understanding the client’s situation and
thoughts with active listening skills, and (3) Motivational Interviewing
Spirit: the interviewer should be collaborative rather than assuming expert
roles. In practice, we found all models rarely direct or confront the user
using our dialogue action classifier, trivially satisfying the Acceptance
criteria. We use %AL as a proxy for the Empathy criteria, and %NA as a proxy
to measure the Motivational Interviewing Spirit criteria. These metrics are
proven to reduce client resistance and improve intervention outcome in MI
literature Moyers et al. (2016).

###  C.2 Verifying the Effectiveness of Automatic Evaluation

| %MI-i↓ | C/S↑ | R/Q↑ | %AL↑ | %NA↑
---|---|---|---|---|---
Low | 20 | 2.09 | 1.0 | 34.4 | 51.5
High | 0.9 | 2.75 | 1.875 | 62.4 | 82.6
Table 3: Our automatic evaluation can tell apart between high quality (“High"
in the table) and low quality (“Low" in the table) MI demonstrations

to validate our evaluation, we run our classifier-based evaluation pipeline on
231 datapoints (dialogue context + gold response) of good vs. bad
demonstration of MI interviewers (the binary high v.s. low quality labels are
available in the original AnnoMI dataset). Bad demonstrations here means the
response are produced by interviewer that does not follow MI guideline. We use
231 pairs from each class (i.e. 231 high and 231 low-quality demonstrations)
since that is the number of all bad demonstrations available, and we
downsampled good demonstrations to balance classes. Specifically, the desired
behavior here is our evaluation pipeline should be able to tell apart good
versus bad MI interviewers.

As shown in Table 3, our evaluation pipeline does indicate that high quality
MI interviews are much better than low quality ones - this shows that our
evaluation method can tell apart good vs bad MI interviewers.

###  C.3 Human Evaluation Details

We use Amazon Mechanical Turk for our human
evaluation444https://www.mturk.com/. The specific prompt we use is ‘‘Which
response does not claim an expert role and make overt and frequent attempt to
persuade personB, doesn’t push for persuation when giving factual information,
and act more like a companion that show genuin interest in understanding
personB’s thoughts and situation? Pick the one that you thinks is best
described by the style above". To unsure quality response, we request works to
have an lifetime approval rate of 98% or above, and filter out all response
that failed an attention check (for example, picking an animal that is not
often kept as pet from ragdoll cat, golden retriever, elephant, and gold
poodle). We do not impose any constraints (other than English-speaking), and
pay at least 0.1 dollar per task (8 responses).

##  Appendix D Concrete Example of DIIR’s Learning and Inference Process

###  D.1 Learning

We provide a concrete example of the learning process in the “Inferring
Strategy” section in Table 7. The comments denoted in ‘‘##" in the table
should make it self-contained, but we provide more details here. The input to
our model is the dialogue state (dialogue history $h$). First, the generator
(GPT-3.5-Turbo) generates an Initial Response ‘‘It is normal to…” prompted
with the dialogue context; this is just a vanilla response generated by the
LLM. Then, the discriminator model (another LLM) checks and gives feedback to
the generator LLM on its flaws compared to the ground truth, in this case
‘‘when the client acknowledges the difficulty…”. The generator will then be
prompted to regenerate the response based on Dialogue State as well as the
feedback from the discriminator model. In this case, the regenerated response
is ‘‘It sounds like it’s been…” At this point, the discriminator model will be
prompted to check whether the response is matching ground truth - here, both
the generated response ‘‘It sounds like it’s been…” and the gold response
‘‘It’s something that’s always on your mind” are reflecting on what the client
said, and thus the discriminator marked the process as complete. We then save
this instruction ‘‘when the client acknowledges the difficulty…” as a stategy
decription $e$ for reuse at inference time. The discriminator might not always
generate ideal feedback on the first trial, so we prompt the discriminator
model to refine the feedback and repeatedly try the new feedback prompt on the
generator until success (or some predefined maximum number of refinement steps
reached).

###  D.2 Inference

We provide a concrete example of inference in Table 8. The comments denoted as
‘‘##" should make the table self contained, but we provide more details here.
The input to the model is the Dialogue State (dialogue history $h$). After
this, we prompt an LLM to generate a dialogue situation description of the
dialogue state, for example, ‘‘The client is hesitant and unsure about
changing yet". Recall that at training time we assign a dialogue situation
description like the one above to each training dialogue state, so at
inference time we can use a dense retriever to pick out the training data that
has the most similar client mental states to the current one. We then prompt a
separate instance of LLM with the current dialogue state (context) and the
N=10 natural language strategies associated with the retrieved dialogue
contexts, and let it determine the most relevant one, i.e. the reranking step.
In the example shown in Table 5, the selected strategy is ‘‘when the client
seems hesitant and uncertain about making a positive change…" We give this
strategy to the generator model and let it improve the response, producing the
final output ‘‘It’s important to consider the potential risks…".

Method | %MI-i↓ | C/S↑ | R/Q↑ | %AL↑ | %NA↑
---|---|---|---|---|---
W/o Strategies
GPT3.5 | 3.1 | 44.8 | 1.29 | 43.0 | 70.6
GPT3.5-ICL-ALL | 3.4 | 6.25 | 1.53 | 28.4 | 70.6
GPT3.5-ICL-RAND | 2.8 | 7.05 | 0.98 | 51.7 | 82.3
GPT3.5-ICL-KNN | 2.1 | 4.28 | 0.93 | 54.3 | 82.8
GPT4 | 1.4 | 11.8 | 0.86 | 41.1 | 71.3
GPT4-ICL-ALL | 2.4 | 10.8 | 0.91 | 38.0 | 64.8
GPT4-ICL-RAND | 3.0 | 10.55 | 1.01 | 45.1 | 71.8
GPT4-ICL-KNN | 3.1 | 4.18 | 1.06 | 45.8 | 71.8
With Strategies (Ours)
GPT3.5-DIIR | 1.3 | 115 | 0.99 | 61.1 | 95.1
GPT4-DIIR | 0.6 | 135 | 1.04 | 59.6 | 94.4
Gold | 1.4 | 2.3 | 1.51 | 59.0 | 81.7
Table 4: Alignment performance (full table). ↑ and ↓ marks denote positive and
negative metrics.

##  Appendix E Discussion on Open Source Model

Method | %MI-i↓ | C/S↑ | R/Q↑ | %AL↑ | %NA↑
---|---|---|---|---|---
Mistral | 5.4 | 22.5 | 1.15 | 44.4 | 67.7
Mistral-RAND | 5.4 | 2.1 | 1.57 | 45.5 | 74.5
Mistral-KNN | 5.5 | 1.97 | 1.55 | 45.0 | 73.8
Mistral-DIIR | 4.8 | 25.2 | 1.14 | 45.0 | 68.4
Table 5: We additionally report alignment performance of Mistral, and open
source LLM. ↑ and ↓ marks denote positive and negative metrics.

In our exploration phase, we test DIIR on open source models, but find they
cannot consistently follow the generated strategy statements. This is coherent
with recent findings that LLMs are better at generation than understanding Qiu
et al. (2024); West et al. (2024). We provide the performance of
Mistral555https://huggingface.co/mistralai/Mistral-7B-v0.1 in Table 5, as
shown in results, our method is comparable to various baselines on a weaker-
instruction following model (than GPT3.5 and GPT4).

##  Appendix F Definition of Dialogue Actions

We show definition of each dialogue action as in Table 6.

Dialogue Action | Definition
---|---
Give Information |  Gives information, educates, provides feedback, or expresses a professional opinion without persuading, advising, or warning. Self-discose of objective information also goes here.
Question |  All questions from clinicians (open, closed, evocative, fact-finding, etc.)
Simple Reflection |  Reflect (repeat or reword) on what the client have said, without adding further meaning to it.
Complex Reflection |  Reflect (repeat or reword) on what the client have said, but adding further meaning (or make explicit some hidden impliciation) of it.
Affirm |  States something positive or complimentary about the client’s strengths, efforts, intentions, or worth.
Emphasize Autonomy |  Highlights a client’s sense of control, freedom of choice, personal autonomy, ability, and obligation about change.
Confront |  Directly and unambiguously disagreeing, arguing, correcting, shaming, blaming, criticizing, labeling, warning, moralizing, ridiculing, or questioning a client’s honesty.
Seek Collaboration |  Attempts to share power or acknowledge the expertise of a client.
Support |  These are generally sympathetic, compassionate, or understanding comments, with the quality of siding with the client.
Advise with Permission |  Attempts to change a client’s opinions, attitudes, or behaviors, but have obtained the client’s permission to do so, or clearly indicates the decision is the clients’.
Advise without Permission |  Attempts to change a client’s opinions, attitudes, or behaviors using tools such as logic, compelling arguments, self-disclosure, facts, biased information, advice, suggestions, tips, opinions, or solutions to problems.
Other |  Filler words, such as ’mm-hmm’, ’mm’, ’yeah’, ’okay’, ’hmm’, ’uh-huh’, ’huh’, ’right’, ’yep’, etc.
Table 6: Dialogue Actions and Their Definition  Example of Learning w/
DIIR+GPT3.5
---
Dialogue State $h$
Topic: reducing drug use
[therapist]: You don’t want to be here.

[client]: I really want to get off the program.
[therapist]: You seem motivated to follow the rules.
[client]: Mm-hmm. I still crave smoking, even though I’ve stopped.

Gold Response $a^{*}$:It’s something that’s always on your mind.

Inferring Strategy…
## Executor $E$ generates a response $a^{\prime}$ based on dialogue history
only (the strategy $e$ is an empty string at this point):

Executor Response $a^{\prime}$: It’s normal to still have cravings even after
stopping. Let’s talk about some strategies to help you manage those cravings.
—
## Discriminator $D$ checks the generated response vs. the gold response. The
generated response immediately offers solution, which is way different from
gold response.

Discriminator Checks Success Result: No success.
—
## Now generator tries to generate a better strategy than the current one (the
current one is empty string).

Generator: when the client is acknowledging the difficulty of resisting the
craving for the bad habit despite having stopped, the therapist should use a
reflective statement to acknowledge the client’s struggle and show empathy.
The therapist should not jump straight into providing strategies or solutions.
The therapist should use 1 sentence for the reflective statement, which should
focus on acknowledging the client’s struggle and showing empathy.
—
## Now the executor generates a response based on the new strategy description
above and the dialogue history.

Executor Response $a^{\prime}$: It sounds like it’s been really tough for you
to resist the cravings, even though you’ve stopped.
—
## Both generated response and the gold response are reflecting on what the
client have said now.

Discriminator Check Sucess Result: Yes.
—
## Now we can save the strategy above (“when the client is acknowledging…").
## We further generate a description of the situation for reusing this
strategy in the future.

Situation Description $k$ from Generator: The client is acknowledging the
difficulty of change despite making progress.
Table 7: Example of DIIR’s learning process.  Example of Inference w/
DIIR+GPT3.5
---
Dialogue State $h$
Topic: reducing alcohol consumption
[therapist]: Yeah, you’re surprised to hear that?
[client]: Yes. What-what kind of health problems?
[therapist]: Well things like heart disease, cancer, liver problems, uh,
stomach pains, insomnia. Unfortunately, uh, people who drink at a risky level
are more likely to be diagnosed with depression and alcohol can make
depression worse or harder to treat.
[client]: Hmm. Well, that’s not good news.

Inferred Client Mental State (Situation Description) $k_{inference}$:
## Note we will use the dialogue situation description below to retrieve
strategies.
The client is hesitant and unsure about changing yet.

Retrieved Strategy $e_{i}$:
## The associated situation description $k_{i}$ for the strategy below would
be “the client seems hesitant and uncertain about making a positive change"
## Note how the current dialogue state is also about a situation where client
is unsure about making a change.
## In this way, if we encode both $k_{inference}$ and $k_{i}$, they would have
large dot product and thus can be retrieved.
Strategy: when the client seems hesitant and uncertain about making a positive
change in their behavior or bad habit, the therapist should advise the client
on the potential risks and benefits of their behavior in one sentence. Then,
the therapist should ask the client if they would be open to exploring further
information or options in one sentence. The therapist should not immediately
seek collaboration or suggest a plan in the first response.

## Now the executor LM can generate the final response given the retrieved
strategy.
Updated Response:It’s important to consider the potential risks and benefits
of your alcohol consumption. Would you be open to exploring further
information or options?
Table 8: Example of DIIR’s inference process.  Example of Learned strategy
descriptions w/ DIIR+GPT3.5
---
when the client is open and willing to engage in the conversation, the
therapist should first give information about the confidentiality of the
conversation in 1 sentence. Then, the therapist should give information about
the purpose of the discussion in 1 sentence. Finally, the therapist should ask
a question to ensure the client understands and is comfortable with the
conversation in 1 sentence. The therapist should not immediately jump into
discussing the client’s bad habit without setting the context and ensuring the
client’s understanding and comfort.

when the client downplays the significance of their bad habit, the therapist
should ask a specific question to gather more information about the habit and
its impact. The therapist should not make general statements about the
importance of understanding the habit’s role in the client’s life without
directly addressing the client’s behavior. The therapist should ask a specific
question to gather more information about the habit and its impact, which is 1
sentence.

when the client expresses enjoyment or positive aspects of their bad habit,
the therapist should reflect the client’s feelings in one sentence. The
therapist should not immediately suggest alternative activities or changes.
Instead, the therapist should follow up with a question to explore any
negative aspects or ambivalence the client may have about their bad habit in a
second sentence. This approach allows the therapist to fully understand the
client’s perspective before exploring potential changes.
Table 9: Three examples of the strategy descriptions learned.  Prompt for
Client Mental Stage Inference
---
You are a dialogue analyst and your job is to help us understanding
motivational interviewing dialogues. You will be given a dialogue context, and
you will help us determine which of the 5 stages of change the client is at:
Precontemplation, Contemplation, Preparation, Action, or Maintenance.

1\. Precontemplation: At this stage, the individual is not yet considering
making a change and may be unaware of the need for change.
2\. Contemplation: In this stage, the individual is aware of the need for
change and is considering the possibility of making a change in the near
future.
3\. Preparation: During this stage, the individual is actively preparing to
make a change and may be taking small steps toward behavior change.
4\. Action: At this stage, the individual has made a specific, observable
change in their behavior and is actively working to maintain this change.
5\. Maintenance: In the maintenance stage, the individual has successfully
made the desired change and is working to prevent relapse and sustain the new
behavior over time.
Look at the following dialogue snippet, which of the 5 stage is the client in?
@snippet@
Format you answer in this format: ’prediction’: "your answer", you do not have
to explain anything.
Table 10: The prompt we used for client mental stage inference.

##  Appendix G License Statement

We plan to release the code base upon acceptance of this work under CC BY
license.


# Entanglement swapping and swapped entanglement

S. M. Zangi 111Corresponding author Chitra Shukla  Shenzhen Institute for
Quantum Science and Engineering and Department of Physics, Southern University
of Science and Technology, Shenzhen 518055, China  Atta Ur Rahman  School of
Physics, University of Chinese Academy of Science, Yuquan Road 19A, Beijing
100049, China  Bo Zheng 222Corresponding author

###### Abstract

Entanglement swapping is gaining widespread attention due to its application
in entanglement distribution among different parts of quantum appliances. We
investigate the entanglement swapping for pure and noisy systems, and argue
different entanglement quantifiers for quantum states. We explore the
relationship between the entanglement of initial states and the average
entanglement of final states in terms of concurrence and negativity. We find
that if initial quantum states are maximally entangled and we make
measurements in the Bell basis, then average concurrence and average
negativity of final states give similar results. In this case, we simply
obtain the average concurrence (average negativity) of the final states by
taking the product of concurrences (negativities) of the initial states.
However, the measurement in non-maximally entangled basis during entanglement
swapping degrades the average swapped entanglement. Further, the product of
the entanglement of the initial mixed states provides an upper bound to the
average swapped entanglement of final states obtained after entanglement
swapping. The average concurrence of finally obtained states provides an upper
bound to the average negativity of these states. We also discuss how
successfully the output state can be used as a channel for the teleportation
of an unknown qubit.

##  1 Introduction

Quantum networks play a very important role in quantum information science and
have applications in quantum communication [1], computation [2], metrology [3]
and fundamental tests [4]. Quantum networks are comprised of a large number of
nodes interconnected by quantum channels. The stationary qubits at the
separated nodes develop, store, and manipulate quantum states while the flying
qubits constitute quantum channels and can be realized by photons. These
channels teleport quantum states between the nodes with high fidelity,
allowing the distribution of quantum entanglement across the whole network.
Thus, the task of building quantum networks requires the ability to establish
quantum entanglement between distant quantum nodes. H. J. Briegel et al.
proposed a quantum repeater protocol that enables the transmission of
entanglement over long distances [5]. The ability to distribute and manipulate
entanglement between distant parties serves as the basis for quantum
applications. Entanglement swapping is one such tool that helps us to connect
many separable nodes for long-distance communication in a quantum network.
Specifically, entanglement swapping is a protocol by which quantum systems
that have never interacted in the past can become entangled [6, 7]. The
nomenclature “entanglement swapping” describes the transfer of entanglement
from a priori entangled systems to a priori separable systems [8]. It is a
very useful tool for entanglement purification [9], teleportation [10], and
plays an important role in quantum computing and quantum cryptography [11,
12]. We can also use entanglement swapping for the creation of multipartite
entangled states from bipartite entanglement [13].

Let us describe the phenomenon of entanglement swapping. Suppose two entangled
particles $(A,B)$ are shared between Alice and Bob. Similarly Cara and Danny
also share another entangled pair of particles $(C,D)$. Initially there is no
entanglement between Alice’s and Danny’s particles $(A,D)$, shown in Fig.
(1(a)). If Bob and Cara who are situated in the same laboratory make
measurement in a suitable basis on the pair $(B,C)$ and classically
communicate the outcome with distant partners then Alice’s and Danny’s
particles who are at very large distance become entangled as shone in Fig.
(1(b)). This entanglement swapping protocol can be generalized in different
ways: by modifying the initial states, or by modifying the measurement
performed by Bob and Cara, or by extending the number of parties [13, 14].

(a)

(b)

Figure 1: Entanglement swapping. Initially in Fig. (a) entangled pairs are
shared between Alice and Bob, and between Cara and Danny. There is no
entanglement between Alice and Danny. However, in Fig (b), the measurement on
Bob and Cara’s qubits project the entanglement between Alice and Danny.

An ever-increasing body of literature shows that the entanglement swapping and
purification of quantum systems need specific protocols. Smaller changes may
bring huge changes to the output state because of the relative sensitivity of
the operation and quantum systems. For this reason, several previous studies
suggest entanglement swapping of initial states into maximally entangled
states. For example, in Ref. [15], the authors provided a scheme of
entanglement swapping of initial states into biqubit maximally entangled
states when influenced by an amplitude damping channel. The concurrence of the
measuring basis for entanglement swapping caused a two-fold entanglement
matching effect has been witnessed in Ref. [16]. Recently, Ref. [17]
investigated that hyper-entangled states produce deterministic entanglement
swapping while considering projections of biqubit systems on symmetric, and
iso-entangled states. It is found that biqubit entanglement generated through
entanglement swapping, will depart from a Bell-type inequality even for
visibilities smaller than 50% [18]. From the above literature, we found that
entanglement swapping has been previously considered using different
procedures, and various important results have been achieved. This research
work constitutes a relatively more generalized entanglement swapping protocol
that covers the maximum possible cases of swapping. We consider pure, mixed as
well as noisy systems for entanglement swapping. Moreover, we explore the
application of the final state as a channel for the teleportation of an
unknown qubit. We also investigate the fidelity of teleported qubit with the
initial unknown qubit.

The article is organized as follows. In Section 2, we present the mathematical
tools that are useful for the present research. The section 3 describes the
details of the entanglement swapping scenario for non-noisy systems and the
application of swapped entanglement. Next, in Section 4, we demonstrate the
entanglement swapping among noisy qubits and teleportation using a noisy
quantum channel. Finally, we conclude with a short discussion in Section 5.

##  2 Preliminaries

In this section, we explain in brief the Schmidt decomposition, entangled
quantum states and famous entanglement quantifiers. We shall quantify the
swapped entanglement with these quantifiers and compare them in next sections.

###  2.1 Schmidt decomposition

Suppose $\ket{\psi}_{XY}$ is a pure state of a composite system $XY$.
According to Schmidt decomposition [19] there exist orthonormal basis
$\ket{i}_{X}$ for system X, and orthonormal basis $\ket{i}_{Y}$ of system Y
such that

$\ket{\psi}=\Sigma_{i}\sqrt{p_{i}}\ket{i}_{X}\ket{i}_{Y}\;,$ (1)

where $\sqrt{p_{i}}$ are non-negative real numbers satisfying
$\Sigma_{i}p_{i}=1$ and known as Schmidt coefficients.

###  2.2 Entangled states

A quantum state of a compound system is entangled if it cannot be written as a
convex combination of product states. A bipartite quantum state represented by
product states $\rho_{i}^{X}\otimes\rho_{i}^{Y}$ with convex weights $p_{i}$
as

$\rho_{XY}=\sum_{i}^{L}p_{i}\rho_{i}^{X}\otimes\rho_{i}^{Y}\;,$ (2)

is called a separable state [20].

###  2.3 Entanglement quantifiers

Many entanglement quantifiers have been proposed [21, 22, 23, 24], however
concurrence and negativity are among the well-defined measures of entanglement
[25, 26, 27, 28].

####  2.3.1 Concurrence

For a finite-dimensional bipartite state $\ket{\psi}\in H_{X}\otimes H_{Y}$,
where $H_{X}(H_{Y})$ denotes the $M(N)$-dimensional vector space associated
with the subsystem $X(Y)$ such that $M\leq N$, the concurrence is

$\textbf{C}(\psi)=\sqrt{2\left(1-\operatorname{Tr}\left(\rho_{X}^{2}\right)\right)}\;,$
(3)

where the reduced density operator $\rho_{X}=Tr_{Y}(\ket{\psi}\bra{\psi})$.
Sometimes for $n-$dimensional system, the factor 2 is replaced by
$\frac{n}{n-1}$, nonetheless this difference in normalization is not
essential. In order to compute the concurrence from the Schmidt decomposition
of biqubit state $\ket{\psi}$, we can use the following simple relation:

$\textbf{C}(\psi)=2\sqrt{\sum_{j<k}p_{j}p_{k}}\;.$ (4)

The above equation implies that concurrence is the root mean square of pair-
wise products of the Schmidt coefficients.

####  2.3.2 Negativity

Entanglement of a bipartite state $\rho_{XY}$ can also be quantified by using
the concept of negativity defined as

$\mathcal{N}(\rho)=\frac{\left\|\rho^{T_{Y}}\right\|_{1}-1}{M-1}$ (5)

where $\rho^{T_{Y}}$ is partial transpose of the total state $\rho_{XY}$ with
respect to the subsystem $Y$, defined as $\left\langle
i_{X},j_{Y}\left|\rho^{T_{B}}\right|k_{X},l_{Y}\right\rangle=\left\langle
k_{X},j_{Y}\left|\rho\right|i_{X},l_{Y}\right\rangle$. The huge advantage of
the negativity is that it can easily be computed even for a mixed state.

If a biqubit state $\ket{\psi}$ given in the Schmidt decomposition form then
the negativity can be computed by taking the sum of pair-wise products of the
Schmidt coefficients as

$\mathcal{N}(\psi)=2\sum_{j<k}\sqrt{p_{j}}\sqrt{p_{k}}\;.$ (6)

Here, we use the convention given in Ref. [29] for the measurement of
negativity, which is twice the value of the original definition. It is
important to note that $C\geq\mathcal{N}$ [30]. The maximum value of $C$ and
$\mathcal{N}$ is one for maximally entangled states.

##  3 Entanglement swapping among qubit systems

We consider two pairs of qubits for entanglement swapping. The entangled
qubits $A$ and $B$ make the first pair and the second pair consist of
entangled qubits $C$ and $D$ (the qubits’ names are chosen such that they make
the initial of Alice, Bob, Cara and Danny, shown in Fig.(1)). In terms of
Schmidt decomposition, these subsystems can be written as

$\begin{split}\ket{\phi}_{AB}&=\sqrt{p_{0}}\ket{00}_{AB}+\sqrt{p_{1}}\ket{11}_{AB},\\\
\ket{\phi}_{CD}&=\sqrt{p_{0}^{\prime}}\ket{00}_{CD}+\sqrt{p_{1}^{\prime}}\ket{11}_{CD}\;.\end{split}$
(7)

The concurrence and negativity of these systems are
$\textbf{C}(\ket{\phi}_{AB})=2\sqrt{p_{0}p_{1}}$,
$\textbf{C}(\ket{\phi}_{CD})=2\sqrt{p_{0}^{\prime}p_{1}^{\prime}}$ and
$\mathcal{N}(\ket{\phi}_{AB})=2\sqrt{p_{0}}\sqrt{p_{1}}$,
$\mathcal{N}(\ket{\phi}_{CD})=2\sqrt{p_{0}^{\prime}}\sqrt{p_{1}^{\prime}}$. It
means for these bi-dimensional systems concurrence and negativity produce
similar results. The initial state of our four-qubit system can be written as

$\begin{split}\ket{\Phi}&=\ket{\phi}_{AB}\ket{\phi}_{CD}\\\
&=\sqrt{p_{0}p_{0}^{\prime}}\ket{0000}_{ABCD}+\sqrt{p_{0}p_{1}^{\prime}}\ket{0011}_{ABCD}+\sqrt{p_{1}p_{0}^{\prime}}\ket{1100}_{ABCD}+\sqrt{p_{1}p_{1}^{\prime}}\ket{1111}_{ABCD}\;.\end{split}$
(8)

Let us do some simple algebra. We rearrange the qubits in each term so that we
can write qubits $A$ and $D$ together and qubits $B$ and $C$ together:

$\ket{\Phi}=\sqrt{p_{0}p_{0}^{\prime}}\ket{00}_{AD}\ket{00}_{BC}\sqrt{p_{0}p_{1}^{\prime}}\ket{01}_{AD}\ket{01}_{BC}+\sqrt{p_{1}p_{0}^{\prime}}\ket{10}_{AD}\ket{10}_{BC}+\sqrt{p_{1}p_{1}^{\prime}}\ket{11}_{AD}\ket{11}_{BC}\;.$
(9)

In order to do measurements over $BC$ qubits, we also need to define a set of
four orthonormal basis [16]

$\begin{split}\ket{\tilde{\Phi}_{+}}_{BC}&=\alpha_{0}\ket{00}_{BC}+\beta_{0}\ket{11}_{BC},\\\
\ket{\tilde{\Phi}_{-}}_{BC}&=\beta_{0}^{*}\ket{00}_{BC}-\alpha_{0}^{*}\ket{11}_{BC},\\\
\ket{\tilde{\Psi}_{+}}_{BC}&=\alpha_{1}\ket{01}_{BC}+\beta_{1}\ket{10}_{BC},\\\
\ket{\tilde{\Psi}_{-}}_{BC}&=\beta_{1}^{*}\ket{01}_{BC}-\alpha_{1}^{*}\ket{10}_{BC}\;,\end{split}$
(10)

where, without loss of generality, we assume the amplitudes $\alpha_{i}$ and
$\beta_{i}$ to be real and non-negative numbers and for normalization
$\lvert\alpha_{i}\rvert^{2}+\lvert\beta_{i}\rvert^{2}=1$ for $i\in\\{0,1\\}$.
Conversely, we have
$\ket{00}_{BC}=\alpha_{0}^{*}\ket{\tilde{\Phi}_{+}}_{BC}+\beta_{0}\ket{\tilde{\Phi}_{-}}_{BC}$
and likewise we can find expressions for
$\ket{01}_{BC},\ket{10}_{BC},\ket{11}_{BC}$. Now by using these expressions we
can write Eq. (9) as

$\ket{\Phi}=\sqrt{p_{\tilde{\Phi}_{+}}}\ket{\ddot{\Phi}_{+}}_{AD}\ket{\tilde{\Phi}_{+}}_{BC}+\sqrt{p_{\tilde{\Phi}_{-}}}\ket{\ddot{\Phi}_{-}}_{AD}\ket{\tilde{\Phi}_{-}}_{BC}\\\
+\sqrt{p_{\tilde{\Psi}_{+}}}\ket{\ddot{\Psi}_{+}}_{AD}\ket{\tilde{\Psi}_{+}}_{BC}+\sqrt{p_{\tilde{\Psi}_{-}}}\ket{\ddot{\Psi}_{-}}_{AD}\ket{\tilde{\Psi}_{-}}_{BC}\;,$
(11)

where possible outcome states of the qubits AD can be defined as,

$\begin{split}\ket{\ddot{\Phi}_{+}}_{AD}=&\left(\sqrt{p_{0}p_{0}^{\prime}}\alpha_{0}^{*}\ket{0}_{A}\ket{0}_{D}+\sqrt{p_{1}p_{1}^{\prime}}\beta_{0}^{*}\ket{1}_{A}\ket{1}_{D}\right)/\sqrt{p_{\tilde{\Phi}_{+}}}\;,\\\
\ket{\ddot{\Phi}_{-}}_{AD}=&\left(\sqrt{p_{0}p_{0}^{\prime}}\beta_{0}\ket{0}_{A}\ket{0}_{D}-\sqrt{p_{1}p_{1}^{\prime}}\alpha_{0}\ket{1}_{A}\ket{1}_{D}\right)/\sqrt{p_{\tilde{\Phi}_{-}}}\;,\\\
\ket{\ddot{\Psi}_{+}}_{AD}=&\left(\sqrt{p_{0}p_{1}^{\prime}}\alpha_{1}^{*}\ket{0}_{A}\ket{1}_{D}+\sqrt{p_{0}^{\prime}p_{1}}\beta_{1}^{*}\ket{1}_{A}\ket{0}_{D}\right)/\sqrt{p_{\tilde{\Psi}_{+}}}\;,\\\
\ket{\ddot{\Psi}_{-}}_{AD}=&\left(\sqrt{p_{0}p_{1}^{\prime}}\beta_{1}\ket{0}_{A}\ket{1}_{D}-\sqrt{p_{0}^{\prime}p_{1}}\alpha_{1}\ket{1}_{A}\ket{0}_{D}\right)/\sqrt{p_{\tilde{\Psi}_{-}}}\;,\end{split}$
(12)

where the associated probabilities are

$\begin{split}p_{\tilde{\Phi}_{+}}&=p_{0}p_{0}^{\prime}\lvert\alpha_{0}\rvert^{2}+p_{1}p_{1}^{\prime}\lvert\beta_{0}\rvert^{2}\;,\\\
p_{\tilde{\Phi}_{-}}&=p_{0}p_{0}^{\prime}\lvert\beta_{0}\rvert^{2}+p_{1}p_{1}^{\prime}\lvert\alpha_{0}\rvert^{2}\;,\\\
p_{\tilde{\Psi}_{+}}&=p_{0}p_{1}^{\prime}\lvert\alpha_{1}\rvert^{2}+p_{0}^{\prime}p_{1}\lvert\beta_{1}\rvert^{2}\;,\\\
p_{\tilde{\Psi}_{-}}&=p_{0}p_{1}^{\prime}\lvert\beta_{1}\rvert^{2}+p_{0}^{\prime}p_{1}\lvert\alpha_{1}\rvert^{2}\;.\end{split}$
(13)

We observe in Eq. (11) that the state of qubits $A$ and $D$ is similar to the
state of the basis $BC$. It is clear from equation (12) that after
measurements in the basis $BC$, Alice and Danny’s qubits $A$, $D$ which are
initially separable, become entangled in one of the four possible forms. We
can compute the average concurrence for the final state as

$\begin{split}\textbf{C}_{av}&=p_{\tilde{\Phi}_{+}}\textbf{C}_{\ddot{\Phi}_{+}}+p_{\tilde{\Phi}_{-}}\textbf{C}_{\ddot{\Phi}_{-}}+p_{\tilde{\Psi}_{+}}\textbf{C}_{\ddot{\Psi}_{+}}+p_{\tilde{\Psi}_{-}}\textbf{C}_{\ddot{\Psi}_{-}}\;,\\\
&=4\sqrt{p_{0}p_{0}^{\prime}p_{1}p_{1}^{\prime}}\left(\lvert\alpha_{0}\beta_{0}\rvert+\lvert\alpha_{1}\beta_{1}\rvert\right)\;.\end{split}$
(14)

The Bell states are maximally entangled biqubit states. The states in Eq. (10)
transform into maximally entangled Bell states if we take
$\alpha_{i}=\beta_{i}=\frac{1}{\sqrt{2}}$ for $i\in\\{0,1\\}$. The measurement
in maximally entangled Bell basis return maximally entangled $A$, $D$ qubits
states with the average concurrence

$\begin{split}\textbf{C}_{av}&=4\sqrt{p_{0}p_{0}^{\prime}p_{1}p_{1}^{\prime}}\;\\\
&=\textbf{C}_{AB}\textbf{C}_{CD}\;.\end{split}$ (15)

Similarly, the average negativity of the qubits $A$ and $D$ states when the
measurement is done in Bell basis takes the form

$\begin{split}\mathcal{N}_{av}&=4\sqrt{p_{0}p_{0}^{\prime}p_{1}p_{1}^{\prime}}\;\\\
&=\mathcal{N}_{AB}\mathcal{N}_{CD}\;.\end{split}$ (16)

We obtain from Eq. (15) and Eq. (16) that if initial quantum states are
maximally entangled and we make measurements in the Bell basis, then average
concurrence and average negativity are equivalent. We simply obtain the
average concurrence (average negativity) by taking the product of concurrences
(negativities) of the initial states. Besides Eq. (14) shows that measurement
in non-maximally entangled basis during entanglement swapping degrades the
average swapped entanglement.

Now we extend entanglement swapping between two pairs of qubits to three pairs
of qubits. We take three pairs of entangled qubits and make a measurement in
the GHZ basis and analyze the outcome state. If the three pairs of qubits in
Schmidt form are
$\ket{\phi}_{AB}=\sqrt{\lambda_{0}}\ket{00}_{AB}+\sqrt{\lambda_{1}}\ket{11}_{AB})$,
$\ket{\phi}_{CD}=\sqrt{\mu_{0}}\ket{00}_{CD}+\sqrt{\mu_{1}}\ket{11}_{CD}$ and
$\ket{\phi}_{EF}\sqrt{\nu_{0}}\ket{00}_{EF}+\sqrt{\nu_{1}}\ket{00}_{EF}$ then
the six-qubit system can be written as

$\ket{\Phi^{\prime}}=|\phi\rangle_{AB}\otimes\ket{\phi}_{CD}\otimes\ket{\phi}_{EF}.$
(17)

Let us make measurements on the $B$, $D$, and $F$ qubits. For this purpose, we
can define the triqubit GHZ basis as [31]

$\displaystyle\left|G_{0,1}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|000\rangle\pm|111\rangle)\;,$ (18)
$\displaystyle\left|G_{2,3}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|001\rangle\pm|110\rangle)\;,$
$\displaystyle\left|G_{4,5}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|010\rangle\pm|101\rangle)\;,$
$\displaystyle\left|G_{6,7}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|011\rangle\pm|100\rangle)\;.$

Here the $-$ sign applies to states with odd indices. Now we can write Eq.
(17) in terms of GHZ basis as

$\displaystyle\ket{\Phi^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\sqrt{\lambda_{0}\mu_{0}\nu_{0}}|000\rangle_{ACE}+\sqrt{\lambda_{1}\mu_{1}\nu_{1}}|111\rangle_{ACE}\right)|G_{0}\rangle$
(19)
$\displaystyle+\frac{1}{\sqrt{2}}\left(\sqrt{\lambda_{0}\mu_{0}\nu_{0}}|000\rangle_{ACE}-\sqrt{\lambda_{1}\mu_{1}\nu_{1}}|111\rangle_{ACE}\right)|G_{1}\rangle$
$\displaystyle+\frac{1}{\sqrt{2}}\left(\sqrt{\lambda_{0}\mu_{0}\nu_{1}}|001\rangle_{ACE}+\sqrt{\lambda_{1}\mu_{1}\nu_{0}}|110\rangle_{ACE}\right)|G_{2}\rangle$
$\displaystyle+\frac{1}{\sqrt{2}}\left((\sqrt{\lambda_{0}\mu_{0}\nu_{1}}|001\rangle_{ACE}-\sqrt{\lambda_{1}\mu_{1}\nu_{0}}|110\rangle_{ACE}\right)|G_{3}\rangle$
$\displaystyle+\frac{1}{\sqrt{2}}\left(\sqrt{\lambda_{0}\mu_{1}\nu_{0}}|010\rangle_{ACE}+\sqrt{\lambda_{1}\mu_{0}\nu_{1}}|101\rangle_{ACE}\right)|G_{4}\rangle$
$\displaystyle+\frac{1}{\sqrt{2}}\left(\sqrt{\lambda_{0}\mu_{1}\nu_{0}}|010\rangle_{ACE}-\sqrt{\lambda_{1}\mu_{0}\nu_{1}}|101\rangle_{ACE}\right)|G_{5}\rangle$
$\displaystyle+\frac{1}{\sqrt{2}}\left(\sqrt{\lambda_{0}\mu_{1}\nu_{1}}|011\rangle_{ACE}+\sqrt{\lambda_{1}\mu_{0}\nu_{0}}|100\rangle_{ACE}\right)|G_{6}\rangle$
$\displaystyle+\frac{1}{\sqrt{2}}\left(\sqrt{\lambda_{0}\mu_{1}\nu_{1}}|011\rangle_{ACE}-\sqrt{\lambda_{1}\mu_{0}\nu_{0}}|100\rangle_{ACE}\right)|G_{7}\rangle\;.$

This equation shows that after measurements on $BDF$ qubits, we gain $ACE$
qubits in any one of the eight possible forms of entangled state. For example,
if measurement gives us $|G_{0}\rangle$ then $ACE$ qubits have state

$\frac{1}{\sqrt{2p_{0}}}\left(\sqrt{\lambda_{0}\mu_{0}\nu_{0}}|000\rangle_{\text{ACE}}+\sqrt{\lambda_{1}\mu_{1}\nu_{1}}|111\rangle_{\text{ACE}}\right)\;.$
(20)

Here probability of getting $|G_{0}\rangle$ state is
$p_{0}=\left(\lambda_{0}\mu_{0}\nu_{0}+\lambda_{1}\mu_{1}\nu_{1}\right)/2$.

Yu and Song [32] showed that any good bipartite entanglement measure $M_{A-B}$
can be extended to multipartite systems by taking bipartite partitions of
them. So a tripartite entanglement quantifier can be defined as

$M_{ABC}^{+}=\frac{1}{3}\left(M_{A-BC}+M_{B-AC}+M_{C-AB}\right).$ (21)

But $M_{ABC}^{+}$ could be nonzero for pure biseparable states. It can be
avoided by using the geometric mean:

$M_{ABC}^{\times}=\left(M_{A-BC}M_{B-AC}M_{C-AB}\right)^{\frac{1}{3}}\;.$ (22)

Now by considering this bi-partition for the final triqubit entangled state,
we can compute the swapped entanglement in the form of concurrence as

$\textbf{C}_{ACE}=\left(\textbf{C}_{A-CE}\textbf{C}_{C-AE}\textbf{C}_{E-AC}\right)^{\frac{1}{3}}\;,$
(23)

where
$\textbf{C}_{A-CE}=\sqrt{2\left(1-\operatorname{Tr}\left(\rho_{A}^{2}\right)\right)}$
and similarly we can compute $\textbf{C}_{C-AE}$, $\textbf{C}_{E-AC}$. Here,
$\rho_{A}$ is the one-qubit reduced density matrix of the qubit $A$, obtained
after tracing out the other qubits. The average concurrence for the final
three qubits state now can be written as

$\textbf{C}_{ACE}^{av}=\textbf{C}_{AB}\textbf{C}_{CD}\textbf{C}_{EF}\;.$ (24)

It is again equal to the product of the concurrences of the initial three
states.

We can compute the the negativity of triqubit state $\rho_{ACE}$ as

$\mathcal{N}(\rho_{ACE})=\left(\mathcal{N}_{A-CE}\mathcal{N}_{C-AE}\mathcal{N}_{E-AC}\right)^{\frac{1}{3}}\;,$
(25)

where
$\mathcal{N}_{A-CE}=-2\sum_{i}\lambda_{i}\left(\rho_{ACE}^{T_{A}}\right)$,
$\lambda_{i}\left(\rho_{ACE}^{T_{A}}\right)$ are the negative eigenvalues of
$\rho_{ACE}^{T_{A}}$, partial transpose of $\rho_{ACE}$ with respect to
subsystem $A$ is defined as $\left\langle
i_{A},j_{CE}\left|\rho_{ACE}^{T_{A}}\right|k_{A},l_{CE}\right\rangle=\left\langle
k_{A},j_{CE}|\rho|i_{A},l_{Ce}\right\rangle$ and similarly, we can define
$\mathcal{N}_{C-AE}$, $\mathcal{N}_{E-AC}$. The average negativity of the
final triqubit state can be written as

$\mathcal{N}_{ACE}^{av}=\mathcal{N}_{AB}\mathcal{N}_{CD}\mathcal{N}_{EF}\;,$
(26)

where $\mathcal{N}_{AB},\mathcal{N}_{CD}$ and $\mathcal{N}_{EF}$ are the
negativities of the initial three biqubit states.

###  3.1 Application of swapped entanglement

The final swapped entanglement between Alice and Danny’s qubit is represented
by Eq. (12) has wide range of application, however, we are interested in
imposing it for teleportation of an unknown qubit state. Let, after the
entanglement swapping Alice and Danny attain the state
$\ket{\ddot{\Phi}_{+}}_{AD}$ that can also be written as

$\ket{\ddot{\Phi}_{+}}_{AD}=\left(a\ket{0}_{A}\ket{0}_{D}+b\ket{1}_{A}\ket{1}_{D}\right)\;,$
(27)

where $a=\alpha_{0}^{*}\sqrt{p_{0}p_{0}^{\prime}/p_{\tilde{\Phi}_{+}}}$,
$b=\beta_{0}^{*}\sqrt{p_{1}p_{1}^{\prime}/p_{\tilde{\Phi}_{+}}}$ and $\lvert
a\rvert^{2}+\lvert b\rvert^{2}=1$. If $a=b=\frac{1}{\sqrt{2}}$ then Eq. (27)
is maximally entangled otherwise non-maximally entangled. Let Alice and Danny
win a maximally entangled state after entanglement swapping and Alice wants to
teleport an unknown quantum state to Danny. We denote the state that Alice
wants to send as

$\ket{\chi}=\alpha|0\rangle+\beta|1\rangle\;,$ (28)

where $\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}=1$. Now the state
$\ket{\chi}$ can be teleported easily as described in Ref. [19]. However, if
$\lvert a\rvert\not=\frac{1}{\sqrt{2}}$ then Alice and Danny are not sharing a
maximally entangled state and in this case, we use probabilistic teleportation
to transmit $\ket{\chi}$. In such a situation, the receiver (Danny) cannot
apply single-qubit unitary operations $I,X,iY,Z$ on his collapsed state
$\frac{\alpha a|0\rangle+\beta b|1\rangle}{\sqrt{\left|\alpha
a\right|^{2}+\left|\beta b\right|^{2}}}$ to obtain $\ket{\chi}$. Therefore,
Danny has to prepare an ancilla qubit $\ket{0}_{Auxi}$ and applies $U_{0}$
unitary operation on the combined system as

$U_{0}\left(\frac{\alpha a|0\rangle+\beta b|1\rangle}{\sqrt{\left|\alpha
a\right|^{2}+\left|\beta b\right|^{2}}}\ket{0}_{Auxi}\right)\;,$ (29)

where

$U_{0}=\left(\begin{array}[]{cccc}\frac{b}{a}&\sqrt{1-\frac{b^{2}}{a^{2}}}&0&0\\\
0&0&0&-1\\\ 0&0&1&0\\\ \sqrt{1-\frac{b^{2}}{a^{2}}}&-\frac{b}{a}&0&0\\\
\end{array}\right)\;.$

After implementation of unitary $U_{0}$, the expression (29) attains the form

$\frac{1}{\sqrt{\left|\alpha a\right|^{2}+\left|\beta
b\right|^{2}}}\left(b\left(\alpha\ket{0}+\beta\ket{1}\right)\ket{0}+\alpha\sqrt{a^{2}-b^{2}}\ket{1}\ket{1}\right)\;.$
(30)

Now Danny makes a measurement on his ancilla (right most) qubit in the
computational basis $\\{\ket{0},\ket{1}\\}$. If he gets $\ket{0}$ then his
state collapses to $\alpha\ket{0}+\beta\ket{1}$, Danny further applies
$I_{2\times 2}$ operation on the state obtained to reconstruct the desired
state $\ket{\chi}$. If the measurement of ancilla gives $\ket{1}$ then
protocol fails to teleport the required state due to its probabilistic nature.
Similarly, we can also explain the teleportation of an unknown qubit for other
states that appeared in Eq. (12).

The measurement on Eq. (19) gives us a three-qubit entangled state that can be
any one of the eight three-qubit $GHZ$ states. The teleportation for the
three-qubit $GHZ$ state has already been considered in Ref. [33, 34].

##  4 Noisy qubits and entanglement swapping

We have used so far pure quantum systems. These quantum systems are isolated
from external environments which comprise a variety of disorders and noises.
In reality, quantum systems interact with the environment. One of the
important types of noise is called depolarizing noise or white noise. This
type of noise takes a quantum state and replaces it with a completely mixed
state $\frac{1}{N}\mathbb{I}$, where $N$ is the dimension of the quantum
system and $\mathbb{I}$ is identity matrix. Let us consider a biqubit noisy
state that is prepared by mixing a pure state with white noise:

$\displaystyle\rho_{\alpha}$
$\displaystyle=\alpha\rho_{AB}+(1-\alpha)\mathbb{I}_{2}\otimes\mathbb{I}_{2}/4$
(31) $\displaystyle=\left(\begin{array}[]{cccc}\frac{1-\alpha}{4}+\alpha
p_{0}&0&0&\alpha\sqrt{p_{0}}\sqrt{p_{1}}\\\ 0&\frac{1-\alpha}{4}&0&0\\\
0&0&\frac{1-\alpha}{4}&0\\\
\alpha\sqrt{p_{0}}\sqrt{p_{1}}&0&0&\frac{1-\alpha}{4}+\alpha p_{1}\\\
\end{array}\right)\;,$

where $\rho_{AB}=\ket{\phi}_{AB}\bra{\phi}$ is the density operator of biqubit
system $AB$ described in Eq. (7), $\mathbb{I}_{2}$ identity matrix and
parameter $\alpha$ called visibility of system $AB$. If we take
$p_{0}=p_{1}=1/2$, the Eq. (31) becomes an isotropic state [35] with maximally
entangled $\rho_{AB}$. The isotropic states are invariant under all
transformations of the form $U\otimes U^{*}$, where the asterisk denotes
complex conjugation in a certain basis.

We can also represent a biqubit noisy system in the Bloch form as [20]

$\displaystyle\rho=$
$\displaystyle\frac{1}{4}\mathbb{I}_{2}\otimes\mathbb{I}_{2}+\sum_{\mu=1}^{3}r_{\mu}\frac{\sigma_{\mu}}{\sqrt{2}}\otimes\frac{\mathbb{I}_{2}}{2}+\sum_{\nu=1}^{3}s_{\nu}\frac{\mathbb{I}_{2}}{2}\otimes\frac{\sigma_{\nu}}{\sqrt{2}}$
$\displaystyle+\sum_{\mu=1}^{3}\sum_{\nu=1}^{3}t_{\mu\nu}\frac{\sigma_{\mu}}{\sqrt{2}}\otimes\frac{\sigma_{\nu}}{\sqrt{2}}\;,$
(32)

where $\sigma$ represents Pauli matrices, $\mathbb{I}$ is identity matrix,
$r_{\mu}=\operatorname{Tr}\left(\rho\frac{\sigma_{\mu}}{\sqrt{2}}\otimes\frac{\mathbb{I}_{2}}{2}\right)$
and
$s_{\nu}=\operatorname{Tr}\left(\rho\frac{\mathbb{I}_{2}}{2}\otimes\frac{\sigma_{\nu}}{\sqrt{2}}\right)$
are Bloch vectors of given two qubits and
$t_{\mu\nu}=\operatorname{Tr}\left(\rho\frac{\sigma_{\mu}}{\sqrt{2}}\otimes\frac{\sigma_{\nu}}{\sqrt{2}}\right)$
called a correlation tensor. We can construct Bloch matrix from $\vec{r}$,
$\vec{s}$ and $3\times 3$ dimensional correlation matrix $T$ as

$\tilde{\mathcal{T}}=\left(\begin{array}[]{cc}c&\vec{s}\\\ \vec{r}&T\\\
\end{array}\right)\;,$ (33)

where $c$ is a scalar number. The Bloch matrix form of Eq. (31) contains
$c=\alpha\sqrt{p_{0}}\sqrt{p_{1}}$, $\vec{r}=\vec{s}=0$ and correlation matrix

$T=\left(\scriptsize\begin{array}[]{ccc}-\alpha\sqrt{p_{0}}\sqrt{p_{1}}&0&0\\\
0&\frac{\alpha-1}{4}+\frac{1}{2}\left(\frac{1-\alpha}{4}+\alpha
p_{0}\right)+\frac{1}{2}\left(\frac{1-\alpha}{4}+\alpha
p_{1}\right)&\frac{1-\alpha}{8}+\frac{\alpha-1}{8}+\frac{1}{2}\left(\frac{1-\alpha}{4}+\alpha
p_{0}\right)+\frac{1}{2}\left(\frac{\alpha-1}{4}-\alpha p_{1}\right)\\\
0&\frac{1-\alpha}{8}+\frac{\alpha-1}{8}+\frac{1}{2}\left(\frac{1-\alpha}{4}+\alpha
p_{0}\right)+\frac{1}{2}\left(\frac{\alpha-1}{4}-\alpha
p_{1}\right)&\frac{1-\alpha}{4}+\frac{1}{2}\left(\frac{1-\alpha}{4}+\alpha
p_{0}\right)+\frac{1}{2}\left(\frac{1-\alpha}{4}+\alpha p_{1}\right)\\\
\end{array}\right)\;.$ (34)

As the coherence vectors $\vec{r}$, $\vec{s}$ of the subsystems $A$ and $B$
have zero magnitudes that means the state is maximally mixed. According to
combo separability criteria [20] if
$f(\alpha,p_{0})=\|\tilde{\mathcal{T}}_{\alpha}\|_{KF}-1>0$ then state
$\rho_{\alpha}$ is an entangled state. We plotted $f(\alpha,p_{0})$ in Fig.
(2) which represents entanglement of the mixed state $\rho_{\alpha}$.

Figure 2: This plot represents entangled states present in the mixed state
$\rho_{\alpha}$. All entangled states have
$f(\alpha,p_{0})=\|\tilde{\mathcal{T}}\|_{KF}-1>0$.

It is clear from the figure that $\rho_{\alpha}$ remains entangled when
$0<p_{0}<1$ and the minimum value of $\alpha$ is $1/3$. This state becomes
maximally entangled when $p_{0}=0.5$ and $\alpha$ approaches 1.

As $\rho_{\alpha}$ is a X-form mixed-state with nonzero entries only along the
diagonal and anti-diagonal so its concurrence is given by [36]

$\displaystyle\textbf{C}({\rho_{\alpha}})$
$\displaystyle=\max\left[0,2\left(\left|\rho_{\alpha}^{(14)}\right|-\sqrt{\rho_{\alpha}^{(22)}\rho_{\alpha}^{(33)}}\right),2\left(\left|\rho_{\alpha}^{(23)}\right|-\sqrt{\rho_{\alpha}^{(11)}\rho_{\alpha}^{(44)}}\right)\right]$
(35)
$\displaystyle=\max\left[0,2\left(\alpha\sqrt{p_{0}p_{1}}-\frac{1-\alpha}{4}\right)\right]\;.$

This relation also gives a lower bound for the probability that keeps the
$\rho_{\alpha}$ entangled as

$\alpha>\frac{1}{\left(1+4\sqrt{p_{0}p_{1}}\right)}\;.$ (36)

If $\rho_{AB}$ is a maximally entangled state then $p_{0}=p_{1}=\frac{1}{2}$,
in this case the state remains entangled for $\alpha>\frac{1}{3}$ that we can
also observe from Fig (3(a)).

The negativity of $X-$form state $\rho_{\alpha}$ can be computed as

$\mathcal{N}(\rho_{X})=-2\min\left\\{0,r_{+}-\sqrt{r_{-}^{2}+\left(\rho^{(14)}\right)^{2}},u_{+}-\sqrt{u_{-}^{2}+\left(\rho^{(23)}\right)^{2}}\right\\}\;,$
(37)

where $u_{\pm}=\left(\rho^{(11)}\pm\rho^{(44)}\right)/2$,
$r_{\pm}=\left(\rho^{(22)}\pm\rho^{(33)}\right)/2$. The Eq. (37) can be
reduced to

$\mathcal{N}(\rho_{\alpha})=-2\min\left\\{0,\frac{1}{4}\left(1-\alpha-4\alpha\sqrt{p_{0}p_{1}}\right)\right\\}\;,$
(38)

because $\rho^{(}23)=0$ and $u_{+}-u_{-}=\rho^{(44)}$ is a positive number.
For a maximally entangled state, the Eq. (38) also gives $\alpha>\frac{1}{3}$
for entanglement retain and can be observed from Fig. (3(b)).

(a)

(b)

Figure 3: Concurrence and negativity comparison of biqubit noisy states. Fig.
(a) represents the concurrence of biqubit state against state visibility
parameter $\alpha$ and similar Fig. (b) represents the negativity of biqubit
state.

It is clear from Fig. (3) that the concurrence and negativity produce similar
results in the case of biqubit noisy state.

(a)

(b)

Figure 4: Average concurrence of final states. Fig. (a) shows four different
types of plots of average concurrence for the final states depending upon the
different values of $p_{0}$ and $p_{1}$ against $\alpha$. Fig. (b) is
comparison of the average concurrence of final states (solid lines) with the
product of concurrences of input states (dashed lines).

Now we want to explore entanglement swapping between two states of the form
given in Eq. (31). For simplicity, we assume that the states are similar and
we make standard Bell measurements in order to accomplish the entanglement
swapping. Therefore our four-qubit noisy state in terms of Bell basis is

$\displaystyle\rho_{ABCD}=$
$\displaystyle\frac{\alpha^{2}}{2}\left(p_{0}|00\rangle_{AD}\pm
p_{1}|11\rangle_{AD}\right)\left(p_{0}\langle 00|\pm p_{1}\langle
11|\right)|\Phi_{\pm}\rangle_{\text{BC}}\langle\Phi_{\pm}|$ (39)
$\displaystyle+\frac{\alpha^{2}p_{0}p_{1}}{2}\left(|01\rangle_{AD}\pm|10\rangle_{AD}\right)\left(\langle
01|\pm\langle 10|\right)|\Psi_{\pm}\rangle_{\text{BC}}\langle\Psi_{\pm}|$
$\displaystyle+\mathbf{d}_{AD}\left(|\Phi_{\pm}\rangle_{\text{BC}}\langle\Phi_{\pm}|+|\Psi_{\pm}\rangle_{\text{BC}}\langle\Psi_{\pm}|\right)\;,$

where

$\mathbf{d}_{AD}=\frac{\alpha\left(1-\alpha\right)}{8}\left(\left(p_{0}\ket{0}_{A}\bra{0}+p_{1}\ket{1}_{A}\bra{1}\right)\otimes\mathbb{I}_{D}+\mathbb{I}_{A}\otimes\left(p_{0}\ket{0}_{D}\bra{0}+p_{1}\ket{1}_{D}\bra{1}\right)\right)\\\
+\frac{(1-\alpha)^{2}}{16}\mathbb{I}_{AD}s\;,$ (40)

and $\mathbb{I}$ is an identitiy matrix. The measurement of the qubits $B$ and
$C$ will give us one of the following four Bell states

$\displaystyle\left|\Phi_{\pm}\right\rangle_{BC}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|00\rangle_{BC}\pm|11\rangle_{BC}\right)\;,$
(41) $\displaystyle\left|\Psi_{\pm}\right\rangle_{BC}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|01\rangle_{BC}\pm|10\rangle_{BC}\right)\;.$

If we obtain $\left|\Phi_{\pm}\right\rangle_{BC}$ then the qubits $A$ and $D$
become entangled in the state

$\rho_{AD}^{\Phi_{\pm}}=\frac{1}{P_{\Phi}}\left(\frac{\alpha^{2}}{2}\left(p_{0}|00\rangle_{AD}\pm
p_{1}|11\rangle_{AD}\right)\left(p_{0}\langle 00|\pm p_{1}\langle
11|\right)+\mathbf{d}_{AD}\right)\;,$ (42)

where
$P_{\Phi}=\frac{\alpha^{2}}{2}\left(p_{0}^{2}+p_{1}^{2}\right)+\frac{1-\alpha^{2}}{4}$
is the probability of $\ket{\Phi_{+}}$ and $\ket{\Phi_{-}}$. If measurement
gives us $\left|\Psi_{\pm}\right\rangle_{BC}$ then the qubits $A$ and $D$ make
entangled state

$\rho_{AD}^{\Psi_{\pm}}=\frac{1}{P_{\Psi}}\left(\frac{\alpha^{2}p_{0}p_{1}}{2}\left(|01\rangle_{AD}\pm|10\rangle_{AD}\right)\left(\langle
01|\pm\langle 10|\right)+\mathbf{d}_{AD}\right)\;.$ (43)

Here, $P_{\Psi}=\alpha^{2}p_{0}p_{1}+\frac{1-\alpha^{2}}{4}$ is the
probability of $\ket{\Psi_{+}}$ and $\ket{\Psi_{-}}$.

In order to evaluate the transferred entanglement between qubits $A$ and $D$,
we first compute the concurrence of all four types of density matrix
$\rho_{AD}$. As all density matrices of qubits $A$ and $D$ in Eq. (42) and Eq.
(43) have X-form state form so, their concurrence can easily be computed by
using Eq. (35). The state $\rho_{AD}^{\Phi_{+}}$ and $\rho_{AD}^{\Phi_{-}}$
have the same amount of concurrence and is given by

$\textbf{C}\left(\rho_{AD}^{\Phi_{\pm}}\right)=\frac{1}{P_{\phi}}\max\left(0,\alpha^{2}p_{0}p_{1}-\frac{1}{8}\left(1-\alpha^{2}\right)\right)\;.$
(44)

and similarly, the concurrence of $\rho_{AD}^{\Psi_{\pm}}$ is

$\displaystyle\textbf{C}\left(\rho_{AD}^{\Psi_{\pm}}\right)=\frac{1}{P_{\psi}}\max\left(0,\alpha^{2}p_{0}p_{1}-\frac{1}{8}(1-\alpha)\sqrt{1+2\alpha-3\alpha^{2}+16\alpha^{2}p_{0}p_{1}}\right)\;.$
(45)

Now the average of teleported entanglement in terms of concurrence can be
computed as

$\textbf{C}_{av}=2P_{\Phi}\textbf{C}\left(\rho_{AD}^{\Phi_{+}}\right)+2P_{\Psi}\textbf{C}\left(\rho_{AD}^{\Psi_{+}}\right)\;.$
(46)

This average concurrence of the final states has been plotted in Fig. (4(a)).
Besides Fig. (4(b)) put forward that for mixed states, the product of the
concurrences of the initial states (dashed line plots) is an upper bound to
the average concurrence of the finally swapped entanglement (solid line
plots).

We can also evaluate the transferred entanglement between qubits $A$ and $D$
in terms of negativity. As all density matrices of qubits $A$ and $D$ in Eq.
(42) and Eq. (43) have X-form state, hence, their negativity can easily be
computed by Eq. (37). The state $\rho_{AD}^{\Phi_{+}}$ and
$\rho_{AD}^{\Phi_{-}}$ have the same amount of negativity and take the form

$\mathcal{N}\left(\rho_{AD}^{\Phi_{\pm}}\right)=-\frac{2}{P_{\phi}}\min\left(0,\frac{1}{16}\left(1-\alpha^{2}\right)-\frac{1}{2}\alpha^{2}p_{0}p_{1}\right)\;.$
(47)

and

$\displaystyle\mathcal{N}\left(\rho_{AD}^{\Psi_{\pm}}\right)=-\frac{2}{P_{\psi}}\min\left(0,\frac{1-\alpha^{2}}{16}-\sqrt{\frac{1}{4}\alpha^{4}p_{0}^{2}p_{1}^{2}+\frac{1}{64}(1-\alpha)^{2}\alpha^{2}\left(p_{0}-p_{1}\right){}^{2}}\right)\;.$
(48)

The average of swapped entanglement computed by negativity can be given as

$\mathcal{N}_{av}=2P_{\Phi}\mathcal{N}\left(\rho_{AD}^{\Phi_{\pm}}\right)+2P_{\Psi}\mathcal{N}\left(\rho_{AD}^{\Psi_{\pm}}\right)\;.$
(49)

The average of swapped entanglement in terms of negativity has been plotted in
Fig. (5) and here, Fig. (5(b)) shows that the product of the negativities of
the initial states (dashed line plots) provides an upper bound to the average
negativity of the final states (solid line plots). Moreover, Fig. (6)
represents the comparison of average concurrence and average negativity of
final states. This plot shows that when $p_{0}=p_{1}=0.5$ which correspond to
maximally entangled input states then concurrence and negativity overlap but
for other cases, concurrence provides an upper bound to the negativity for
qubit systems.

(a)

(b)

Figure 5: Average swapped entanglement in terms of negativity. Fig. (a)
Average negativity of final states. Fig. (b) shows the comparison of the
average negativity of final states (solid lines) with the product of
negativities of initial states (dashed lines).

Figure 6: Comparison of entanglement quantifiers for swapped entanglement.
Solid lines represent the average concurrence and the dashed lines represent
the average negativity of the final states

.

###  4.1 Teleportation using a noisy quantum channel

The four Bell states for qubits $B$ and $C$ are given in Eq. (41). By using
the similar Bell states for qubits $A$ and $D$, we can define the projectors
$P^{\Phi_{\pm}}=\ket{\Phi_{\pm}}_{AD}\bra{\Phi_{\pm}}$ and
$P^{\Psi_{\pm}}=\ket{\Psi_{\pm}}_{AD}\bra{\Psi_{\pm}}$ associated with the
measurements that Alice performs in the execution of the teleportation
protocol. Eqs. (42, 43) represent the density matrices of four possible
outcomes after entanglement swapping of noisy entangled states. We can use
these density matrices as a channel for the teleportation of an unknown qubit
given in Eq. (28) from Alice to Danny. Let the teleportation channel is
$\rho_{AD}^{\Phi_{+}}$ and the density matrix of the qubit to be teleported is
given by $\rho_{tq}=\ket{\chi}\bra{\chi}$, where the subscripts tq means
“teleportation qubit". The initial three qubits state is given by

$\varrho_{1}=\rho_{tq}\otimes\rho_{AD}^{\Phi_{+}}.$ (50)

The first two qubits (i.e $\rho_{tq}$ and first qubit of
$\rho_{AD}^{\Phi_{+}}$) of $\varrho_{1}$ are possessed by Alice and the third
qubit is occupied by Danny. Alice makes a projective measurement on her two
qubits. After this measurement, we attain the post-measurement state

$\tilde{\varrho_{1}}=\frac{P^{\Phi_{+}}\varrho_{1}P^{\Phi_{+}}}{\tilde{P_{1}}}\;,$
(51)

where $\tilde{P_{1}}=\operatorname{Tr}\left[P^{\Phi_{+}}\varrho_{1}\right]$ is
the probability of occurrence of state $\tilde{\varrho_{1}}$ and Tr represents
trace operation. Alice then communicates her outcomes with Danny via the
classical channel. The qubit possessed by Danny has the form
$\tilde{\varrho}_{D1}=\operatorname{Tr}_{12}\left[\tilde{\varrho_{1}}\right]$,
where $\operatorname{Tr}_{12}$ means the partial trace of qubits 1 and 2. Due
to the noisy teleportation channel, Danny has to follow a probabilistic
teleportation technique to find teleported qubit $\rho_{tq}$. He prepares an
auxiliary qubit $\rho_{Auxi}=\ket{0}\bra{0}$ and applies a suitable unitary
operator $U_{i}$ on two qubits system as

$U_{i}\left(\tilde{\varrho}_{D1}\otimes\rho_{Auxi}\right)U_{i}^{\dagger}\;.$
(52)

Then a measurement on Danny’s auxiliary qubit in the basis
$\left\\{\ket{0}\bra{0},\ket{0}\bra{1},\ket{1}\bra{0},\ket{1}\bra{1}\right\\}$
is done. If $\ket{0}\bra{0}$ occurs, we obtain qubit
$\tilde{\varrho}^{\prime}_{D1}$ with some probability $P^{\prime}_{1}$
otherwise the teleportation fails.

The protocol ends with Danny apply a unitary operation $u$ on his qubit final
state as

$\rho^{\prime}_{tq}=u\frac{\tilde{\varrho}^{\prime}_{D1}}{P^{\prime}_{1}}u^{\dagger}\;.$
(53)

The unitary operator $u$ is one of the Pauli operators
$\left\\{\mathbb{I},\sigma_{x},\sigma_{y},\sigma_{z}\right\\}$, and its choice
depends not only on the measurement result of Alice but also on the quantum
channel shared between Alice and Danny in the teleportation protocol.

Now we can check the efficiency of the protocol by using fidelity [37]. Since
the input state $\rho_{tq}=\ket{\chi}\bra{\chi}$ is pure, the fidelity can be
written as

$F=\operatorname{Tr}\left[\rho_{tq}\rho^{\prime}_{tq}\right]=\bra{\chi}\rho^{\prime}_{tq}\ket{\chi}\;.$
(54)

The fidelity ranges from 0 to 1 and its maximal value occurs whenever the
Danny’s qubit final state $\rho^{\prime}_{tq}$ is equal to input qubit
$\rho_{tq}$ and it is 0 when the two states are orthogonal.

##  5 Concluding discussion

We have studied an entanglement swapping protocol, where Alice and Bob share a
generalized Bell pair $(A,B)$ whereas, Cara and Danny share another
generalized Bell pair $(C,D)$. When Bob and Cara, who are situated in the same
laboratory perform some measurements on the pair $(B,C)$ then initially
unentangled qubits $(A,D)$ obtain entanglement. Alice and Danny know about the
entanglement of their qubits after getting information about the qubits of Bob
and Cara via classical communication channel.

In the case of two couples of pure qubits, the finally entangled couple can
have one of the four possible entangled states. However, if we considered
three couples of entangled qubits then entanglement swapping gives us a three-
qubit entangled state that can be any one of the eight possible forms of GHZ
quantum states.

The significant achievements of this study can be summarized as, if initial
quantum states are maximally entangled and we make measurements in the Bell
basis, then average concurrence and average negativity of final states give
similar results. We simply obtain the average swapped entanglement among final
quantum states by taking the product of entanglement of the initially
maximally entangled states. The measurement in non-maximally entangled basis
during entanglement swapping degrades the swapped entanglement. The product of
the entanglement of the mixed states provides an upper bound to the average
swapped entanglement of final states. The entanglement quantifier concurrence
provides an upper bound to the negativity. We also use the final output state
as a channel for the teleportation of an unknown qubit from Alice to Danny.
The teleportation with a pure biquibit Bell state is obvious, but we explored
the probabilistic teleportation of an unknown qubit not only with non-
maximally entangled channel but also with the noisy channel that we obtain
after entanglement swapping.

## Acknowledgements

Financial support from the Postdoctoral training funds, grant nos. C615300501.
S. M. Zangi is extremely grateful for the help and support of Bo Zheng. Chitra
Shukla thanks to Shenzhen Science Technology and Innovation Commission: Grant
number: GJHZ2020081109520301, and NSFC-Guangdong joint Fund: Grant number:
U1801661.

## References

  * [1] J. Qiu, “Quantum communications leap out of the lab,” _Nature News_ , vol. 508, no. 7497, p. 441, 2014.
  * [2] I. Cohen and K. Mølmer, “Deterministic quantum network for distributed entanglement and quantum computation,” _Phys. Rev. A_ , vol. 98, p. 030302, Sep 2018.
  * [3] G. Chiribella, “Optimal networks for quantum metrology: semidefinite programs and product rules,” _New Journal of Physics_ , vol. 14, no. 12, p. 125008, 2012.
  * [4] W. McCutcheon, A. Pappa, B. Bell, A. Mcmillan, A. Chailloux, T. Lawson, M. Mafu, D. Markham, E. Diamanti, I. Kerenidis _et al._ , “Experimental verification of multipartite entanglement in quantum networks,” _Nature communications_ , vol. 7, no. 1, pp. 1–8, 2016.
  * [5] H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” _Physical Review Letters_ , vol. 81, no. 26, p. 5932, 1998.
  * [6] M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, ““event-ready-detectors” bell experiment via entanglement swapping,” _Phys. Rev. Lett._ , vol. 71, pp. 4287–4290, Dec 1993.
  * [7] A. Sen(De), U. Sen, i. c. v. Brukner, V. Bužek, and M. Żukowski, “Entanglement swapping of noisy states: A kind of superadditivity in nonclassicality,” _Phys. Rev. A_ , vol. 72, p. 042310, Oct 2005.
  * [8] V. Vedral, _Introduction to quantum information science_. Oxford University Press on Demand, 2006.
  * [9] Z. Ji, P. Fan, and H. Zhang, “Entanglement swapping for bell states and greenberger–horne–zeilinger states in qubit systems,” _Physica A: Statistical Mechanics and its Applications_ , vol. 585, p. 126400, 2022.
  * [10] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” _Phys. Rev. Lett._ , vol. 70, pp. 1895–1899, Mar 1993.
  * [11] A. Galindo and M. A. Martin-Delgado, “Information and computation: Classical and quantum aspects,” _Reviews of Modern Physics_ , vol. 74, no. 2, p. 347, 2002.
  * [12] Z. Ji, H. Zhang, H. Wang, F. Wu, J. Jia, and W. Wu, “Quantum protocols for secure multi-party summation,” _Quantum Information Processing_ , vol. 18, no. 6, pp. 1–19, 2019.
  * [13] S. Bose, V. Vedral, and P. L. Knight, “Multiparticle generalization of entanglement swapping,” _Phys. Rev. A_ , vol. 57, pp. 822–829, Feb 1998\.
  * [14] G. Gour and B. C. Sanders, “Remote preparation and distribution of bipartite entangled states,” _Phys. Rev. Lett._ , vol. 93, p. 260501, Dec 2004.
  * [15] W. Song, M. Yang, and Z.-L. Cao, “Purifying entanglement of noisy two-qubit states via entanglement swapping,” _Physical Review A_ , vol. 89, no. 1, p. 014303, 2014.
  * [16] L. Roa, T. L. Purz, A. Muñoz, S. Castro, G. Hidalgo, and D. Montoya, “Matching for probabilistic entanglement swapping,” _arXiv preprint arXiv:2107.07689_ , 2021.
  * [17] C.-X. Huang, X.-M. Hu, Y. Guo, C. Zhang, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, N. Gisin, C. Branciard _et al._ , “Entanglement swapping and quantum correlations via elegant joint measurements,” _arXiv preprint arXiv:2203.16207_ , 2022.
  * [18] C. Branciard, N. Gisin, and S. Pironio, “Characterizing the nonlocal correlations created via entanglement swapping,” _Physical review letters_ , vol. 104, no. 17, p. 170401, 2010.
  * [19] M. A. Nielsen and I. L. Chuang, “Quantum computation and quantum information,” 2010.
  * [20] S. M. Zangi, J.-S. Wu, and C.-F. Qiao, “Combo separability criteria and lower bound on concurrence,” _Journal of Physics A: Mathematical and Theoretical_ , vol. 55, no. 2, p. 025302, 2021.
  * [21] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” _Physical Review A_ , vol. 54, no. 5, p. 3824, 1996.
  * [22] V. Vedral and M. B. Plenio, “Entanglement measures and purification procedures,” _Phys. Rev. A_ , vol. 57, pp. 1619–1633, Mar 1998.
  * [23] J. I. de Vicente, “Lower bounds on concurrence and separability conditions,” _Phys. Rev. A_ , vol. 75, p. 052320, May 2007.
  * [24] S. Lee, D. P. Chi, S. D. Oh, and J. Kim, “Convex-roof extended negativity as an entanglement measure for bipartite quantum systems,” _Physical Review A_ , vol. 68, no. 6, p. 062304, 2003.
  * [25] N. Karimi, A. Heshmati, M. Yahyavi, M. Jafarizadeh, and A. Mohammadzadeh, “Measurability of d-concurrence,” _Scientific reports_ , vol. 9, no. 1, pp. 1–6, 2019.
  * [26] G. Vidal and R. F. Werner, “Computable measure of entanglement,” _Phys. Rev. A_ , vol. 65, p. 032314, Feb 2002.
  * [27] C. Eltschka, G. Tóth, and J. Siewert, “Partial transposition as a direct link between concurrence and negativity,” _Phys. Rev. A_ , vol. 91, p. 032327, Mar 2015.
  * [28] A. Miranowicz and A. Grudka, “A comparative study of relative entropy of entanglement, concurrence and negativity,” _Journal of Optics B: Quantum and Semiclassical Optics_ , vol. 6, no. 12, p. 542, 2004.
  * [29] C. Sabín and G. García-Alcaine, “A classification of entanglement in three-qubit systems,” _The european physical journal D_ , vol. 48, no. 3, pp. 435–442, 2008.
  * [30] S. Rai and J. R. Luthra, “Negativity and concurrence for two qutrits,” _arXiv preprint quant-ph/0507263_ , 2005.
  * [31] Y. Tsujimoto, M. Tanaka, N. Iwasaki, R. Ikuta, S. Miki, T. Yamashita, H. Terai, T. Yamamoto, M. Koashi, and N. Imoto, “High-fidelity entanglement swapping and generation of three-qubit ghz state using asynchronous telecom photon pair sources,” _Scientific reports_ , vol. 8, no. 1, pp. 1–6, 2018.
  * [32] C.-s. Yu and H.-s. Song, “Free entanglement measure of multiparticle quantum states,” _Physics Letters A_ , vol. 330, no. 5, pp. 377–383, 2004.
  * [33] P.-Y. Xiong, X.-T. Yu, H.-T. Zhan, and Z.-C. Zhang, “Multiple teleportation via partially entangled ghz state,” _Frontiers of Physics_ , vol. 11, no. 4, pp. 1–8, 2016.
  * [34] F. Yan and T. Yan, “Probabilistic teleportation via a non-maximally entangled ghz state,” _Chinese Science Bulletin_ , vol. 55, no. 10, pp. 902–906, 2010\.
  * [35] M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” _Physical Review A_ , vol. 59, no. 6, p. 4206, 1999.
  * [36] N. Quesada, A. Al-Qasimi, and D. F. James, “Quantum properties and dynamics of x states,” _Journal of Modern Optics_ , vol. 59, no. 15, pp. 1322–1329, 2012\.
  * [37] A. Uhlmann, “The “transition probability” in the state space of a-algebra,” _Reports on Mathematical Physics_ , vol. 9, no. 2, pp. 273–279, 1976.