Critical Assessment of Information Back-Flow in Measurement-Free Teleportation
<p>Quantum circuit of the measurement-free teleportation protocol. Each gate into which the circuit is decomposed is labeled as <math display="inline"><semantics> <msub> <mi>G</mi> <mi>i</mi> </msub> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>i</mi> <mo>∈</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>8</mn> <mo>}</mo> </mrow> </semantics></math>). They can be grouped into three unitary <span class="html-italic">blocks</span> of operations <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> as defined in Equations (<a href="#FD2-entropy-26-00780" class="html-disp-formula">2</a>)–(<a href="#FD5-entropy-26-00780" class="html-disp-formula">5</a>). Here, <math display="inline"><semantics> <msub> <mi>G</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> are CNOT gates, <math display="inline"><semantics> <msub> <mi>G</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>7</mn> </mrow> </msub> </semantics></math> are Hadamard transforms, while <math display="inline"><semantics> <msub> <mi>G</mi> <mrow> <mn>5</mn> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> are SWAP gates.</p> "> Figure 2
<p>Non-Markovianity of the dynamics given by Equation (<a href="#FD24-entropy-26-00780" class="html-disp-formula">24</a>). Results are plotted only for <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>≥</mo> <mn>0.4</mn> </mrow> </semantics></math> as all the measures listed in <a href="#sec3dot1-entropy-26-00780" class="html-sec">Section 3.1</a> are zero for <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mn>0.4</mn> </mrow> </semantics></math>. Inset: Non-Markovianity of the effective Hamiltonian in Equation (<a href="#FD25-entropy-26-00780" class="html-disp-formula">25</a>) as measured by the <math display="inline"><semantics> <msub> <mi mathvariant="script">N</mi> <mrow> <mi>B</mi> <mi>L</mi> <mi>P</mi> </mrow> </msub> </semantics></math> measure.</p> "> Figure 3
<p>Trace distance between the two states of <span class="html-italic">S</span> as they go through the teleportation circuit. Gate <math display="inline"><semantics> <msub> <mi>G</mi> <mi>i</mi> </msub> </semantics></math> acts when <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>t</mi> <mo><</mo> <mi>i</mi> </mrow> </semantics></math>. We plot only <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>8</mn> </mrow> </semantics></math> as the trace distance remains constant at 1 for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo><</mo> <mn>5</mn> </mrow> </semantics></math>.</p> "> Figure 4
<p>Trace distance between the two states of <math display="inline"><semantics> <msub> <mi>E</mi> <mn>2</mn> </msub> </semantics></math> for the original BBC protocol [<a href="#B3-entropy-26-00780" class="html-bibr">3</a>] as time evolves. The dashed lines are boundaries between the gates acting on the system and the environment. Gates <math display="inline"><semantics> <msub> <mi>G</mi> <mn>1</mn> </msub> </semantics></math>–<math display="inline"><semantics> <msub> <mi>G</mi> <mn>4</mn> </msub> </semantics></math> correspond to those in <a href="#entropy-26-00780-f001" class="html-fig">Figure 1</a>, while <math display="inline"><semantics> <msub> <mi>G</mi> <mn>5</mn> </msub> </semantics></math> is a CNOT operation on <math display="inline"><semantics> <mrow> <mi>S</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi>G</mi> <mn>6</mn> </msub> </semantics></math> is a Hadamard gate on <math display="inline"><semantics> <msub> <mi>E</mi> <mn>2</mn> </msub> </semantics></math>.</p> "> Figure 5
<p>Correlations in the splitting <span class="html-italic">S</span>-vs-<math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, as quantified by (<b>a</b>) logarithmic negativity, (<b>b</b>) discord and (<b>c</b>) classical correlations as the system evolves according to Equation (<a href="#FD24-entropy-26-00780" class="html-disp-formula">24</a>). Time is denoted <span class="html-italic">t</span>, and <span class="html-italic">p</span> determines the Werner state of the environment at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The system is initially in the vacuum state <math display="inline"><semantics> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> </semantics></math>.</p> "> Figure 6
<p>Correlations in the partition <math display="inline"><semantics> <mrow> <mrow> <mi>S</mi> <mo>|</mo> </mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> as quantified by (<b>a</b>) logarithmic negativity, (<b>b</b>) discord and (<b>c</b>) classical correlations when the Hamiltonian of the system and environment is given by Equation (<a href="#FD25-entropy-26-00780" class="html-disp-formula">25</a>). We take the initial state of the system to be <math display="inline"><semantics> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> </semantics></math> and the environment <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math>. Here <span class="html-italic">t</span> is a dimensionless time. We only show <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>8</mn> </mrow> </semantics></math> as there are no system-environment correlations before <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p> "> Figure 7
<p>Correlations in the partition <math display="inline"><semantics> <mrow> <mrow> <mi>S</mi> <mo>|</mo> </mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> as quantified by (<b>a</b>) logarithmic negativity, (<b>b</b>) discord and (<b>c</b>) classical correlations when the Hamiltonian of the system and environment is given by Equation (<a href="#FD25-entropy-26-00780" class="html-disp-formula">25</a>). We take the initial state of the system to be <math display="inline"><semantics> <mrow> <mo>|</mo> <mo>+</mo> <mo>〉</mo> </mrow> </semantics></math> and the environment <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. Measurement-Free Teleportation
2.1. Effective Depolarizing-Channel Description
2.2. Distinguishability and Non-Markovianity Resulting from the Dynamics
3. Analysis of Non-Markovianity in the Measurement-Free Teleportation Circuit
3.1. Review of Measures of Non-Markovianity
3.1.1. Breuer–Laine–Piilo Measure
3.1.2. Rivas-Huelga-Plenio Measure
3.1.3. Luo-Fu-Song Measure
3.2. Information Back-Flow and Non-Markovianity
4. Information Back-Flow and Correlations
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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McAleese, H.; Paternostro, M. Critical Assessment of Information Back-Flow in Measurement-Free Teleportation. Entropy 2024, 26, 780. https://doi.org/10.3390/e26090780
McAleese H, Paternostro M. Critical Assessment of Information Back-Flow in Measurement-Free Teleportation. Entropy. 2024; 26(9):780. https://doi.org/10.3390/e26090780
Chicago/Turabian StyleMcAleese, Hannah, and Mauro Paternostro. 2024. "Critical Assessment of Information Back-Flow in Measurement-Free Teleportation" Entropy 26, no. 9: 780. https://doi.org/10.3390/e26090780