Quantum Relative Entropy of Tagging and Thermodynamics
"> Figure 1
<p>According to Thermodynamics, work can be reversibly obtained from a heat bath by consuming <span class="html-italic">B</span> bits of an information reservoir and also by decreasing its relative entropy <math display="inline"><semantics> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>|</mo> <mo>|</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> with respect to the thermal state <math display="inline"><semantics> <mi>τ</mi> </semantics></math>.</p> "> Figure 2
<p>Our labeling procedures assume the availability of tagging qubits in either a <math display="inline"><semantics> <mfenced open="|" close="〉"> <mn>0</mn> </mfenced> </semantics></math> or a maximally mixed state, and physical systems, referred to as <span class="html-italic">atoms</span>. The labeling assigns a set of <span class="html-italic">H</span> tagging qubits to a cluster of <span class="html-italic">N</span> atoms. The cost is defined as the number of tagging qubits in state <math display="inline"><semantics> <mfenced open="|" close="〉"> <mn>0</mn> </mfenced> </semantics></math> employed.</p> "> Figure 3
<p>(<b>a</b>) represents the coding procedure as a unitary operation <math display="inline"><semantics> <msub> <mi>U</mi> <mi>c</mi> </msub> </semantics></math>, controlled by the cluster <span class="html-italic">C</span>, on the tagging qubits of the label <span class="html-italic">L</span>. If the tagging qubits are initially in a <math display="inline"><semantics> <mfenced open="|" close="〉"> <mn>0</mn> </mfenced> </semantics></math> state, they hold the coded string for the cluster; (<b>b</b>) represents the inverse operation, which is equal to <math display="inline"><semantics> <msub> <mi>U</mi> <mi>c</mi> </msub> </semantics></math>, so that <math display="inline"><semantics> <msubsup> <mi>U</mi> <mi>c</mi> <mn>2</mn> </msubsup> </semantics></math> is the identity transformation.</p> "> Figure 4
<p>Representation of the procedure employed for replacing the trailing qubits that need not be used in the codeword by maximally mixed ones, as explained in <a href="#app3-entropy-22-00138" class="html-app">Appendix C</a>. It is split into two unitary transformations. The first one, represented in (<b>a</b>), copies the first <math display="inline"><semantics> <mrow> <mi>w</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </semantics></math> qubits of <span class="html-italic">L</span> into <math display="inline"><semantics> <msup> <mi>L</mi> <mo>′</mo> </msup> </semantics></math> that enters with all its tagging qubits in the <math display="inline"><semantics> <mfenced open="|" close="〉"> <mn>0</mn> </mfenced> </semantics></math> state. The remaining qubits are copied from the maximally mixed qubits of <span class="html-italic">D</span>. The second transformation, represented in (<b>b</b>), resets the <span class="html-italic">L</span> label and the <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>−</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>ℓ</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </semantics></math> trailing qubits of <span class="html-italic">D</span>. The overall function is recovering <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>−</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>ℓ</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </semantics></math> qubits in state <math display="inline"><semantics> <mfenced open="|" close="〉"> <mn>0</mn> </mfenced> </semantics></math> and generate a new label with maximally mixed trailing qubits.</p> "> Figure 5
<p>The figure describes the unitary process used to shuffle the labels <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>w</mi> <mi>M</mi> </msub> </mrow> </semantics></math> of a collection <span class="html-italic">F</span> of <span class="html-italic">M</span> tight-labeled clusters, as is explained in <a href="#app4-entropy-22-00138" class="html-app">Appendix D</a>. Besides the labels, a set of <span class="html-italic">P</span> maximally mixed qubits <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>P</mi> </msub> </mrow> </semantics></math> and a set of <span class="html-italic">M</span> labels <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mi>M</mi> </msub> </mrow> </semantics></math>, all in state <math display="inline"><semantics> <msub> <mfenced separators="" open="|" close="〉"> <mn>0</mn> <mo>…</mo> <mn>0</mn> </mfenced> <msub> <mi>B</mi> <mi>L</mi> </msub> </msub> </semantics></math> enter the first unitary gate (<b>a</b>). It shuffles the labels <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>w</mi> <mi>M</mi> </msub> </mrow> </semantics></math>, according to the permutation indicated by the <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>P</mi> </msub> </mrow> </semantics></math> qubits and stores the result in the <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mi>M</mi> </msub> </mrow> </semantics></math> labels. The second gate (<b>b</b>) resets the qubits that signaled the permutation. The last gate resets the <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mn>1</mn> </msub> <mspace width="0.166667em"/> <mo>…</mo> <mo>,</mo> <msub> <mi>w</mi> <mi>M</mi> </msub> </mrow> </semantics></math> labels by regenerating them from the <math display="inline"><semantics> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>c</mi> <mi>M</mi> </msub> </mrow> </semantics></math> clusters.</p> "> Figure A1
<p>On the left, the basics components of the magnetic quantum information engine. The CNOT (Controlled NOT) gate is used to correlate the information <math display="inline"><semantics> <mo>Λ</mo> </semantics></math> and the magnetic <math display="inline"><semantics> <mo>Γ</mo> </semantics></math> qubits. The evolution of the electric current <math display="inline"><semantics> <msub> <mi>I</mi> <mi>coil</mi> </msub> </semantics></math> is controlled by the state of <math display="inline"><semantics> <mo>Λ</mo> </semantics></math>. The magnetic induction field <math display="inline"><semantics> <mover accent="true"> <mi>B</mi> <mo>→</mo> </mover> </semantics></math> at <math display="inline"><semantics> <mo>Γ</mo> </semantics></math> that is generated by the coil is <math display="inline"><semantics> <mrow> <mi>k</mi> <msub> <mi>I</mi> <mi>coil</mi> </msub> <mspace width="0.166667em"/> <mover accent="true"> <mi>z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </semantics></math>. The right part of the figure contains a graph of the evolutions od <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mi>coil</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <mfenced separators="" open="〈" close="〉"> <mspace width="0.166667em"/> <msub> <mi>μ</mi> <mi>z</mi> </msub> <mspace width="0.166667em"/> </mfenced> <mspace width="0.166667em"/> </mrow> </semantics></math> (solid and dashed lines, respectively). The upper branches represent the case <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, whereas the lower branches take place when <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, as described in <a href="#app1-entropy-22-00138" class="html-app">Appendix A</a>.</p> "> Figure A2
<p>Representation of the process described in <a href="#app2-entropy-22-00138" class="html-app">Appendix B</a> for the work extraction in a process beginning with Hamiltonian <math display="inline"><semantics> <msub> <mover accent="true"> <mi>H</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">i</mi> </msub> </semantics></math> in state <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi mathvariant="normal">i</mi> </msub> </semantics></math> and ending in Hamiltonian <math display="inline"><semantics> <msub> <mover accent="true"> <mi>H</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">f</mi> </msub> </semantics></math> and state <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi mathvariant="normal">f</mi> </msub> </semantics></math>. Stages 2 and 3 are isothermal.</p> ">
Abstract
:1. Introduction
2. Tight Labeling
- Any unitary transformation on a system T of tagging qubits.
- Unitary transformations on joint states of cluster C and a system T of K tagging qubits, that are defined, using the basis for C and the computational basis for T, by:
3. Loose Labeling
4. Discussion
- (a)
- the pure state of each physical system is known, and the setting can be adjusted according to it. Then, the average work obtained by processing a collection of physical systems is the weighted average of all the , each one contributing according to its corresponding eigenvalue in the density matrix . It is given by Equation (24).
- (b)
- only the collective mixed state of the collection is known. Then, the engine is tuned with a different set of parameters, and the average value of the extracted work is lower than in the previous case. It is given in Equation (19).
5. Conclusions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Magnetic Spin Information Heat Engine
- (i)
- The electric current in the coil is initially null. The qubit from the information reservoir enters in a partially mixed state . The magnetic qubit is initially in a maximally mixed state . They undergo a CNOT (Controlled NOT) operation, controlled by the magnetic qubit. The result is an entangled mixed state:
- (ii)
- In this stage, the magnetic qubit is isolated from the thermal reservoir. The current in the coil is raised to a value controlled by . It is determined by making equal to the Gibbs state for the corresponding value of the magnetic field , at generated by the current . The equation that fixes the two possible values of is
- (iii)
- The current is gradually turned down until it is completely switched off. The process occurs slowly enough for assuming a thermal equilibrium state for throughout this stage. Therefore, the final states for are maximally mixed. Only the qubit exits in a different state than it started in. The current and the qubit are reset to their initial conditions.
Appendix B. Work Output from a System Out of Thermal Equilibrium
- The Hamiltonian is changed from to the operator for which is in thermal equilibrium at temperature T. It is given byIn this step, the Hamiltonian varies quickly, so that the state does not change. Considering that there is no heat exchange in this step, the work obtained is
- The Hamiltonian is slowly taken back to , while thermal equilibrium is assured. Therefore, the system is driven to the state given by:
- While keeping thermal equilibrium, the Hamiltonian is taken from to . The work is obtained as in the previous step and yields:
Appendix C. Label Trimming
Appendix D. Label Shuffling
Appendix E. Example
- How many bits from an information reservoir are needed to match the work that is required to take a cluster of N spins out of thermal equilibrium?
- How many bits are needed to label the state of a cluster of N spins using a coding system optimized for a given distribution of states?
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Diazdelacruz, J. Quantum Relative Entropy of Tagging and Thermodynamics. Entropy 2020, 22, 138. https://doi.org/10.3390/e22020138
Diazdelacruz J. Quantum Relative Entropy of Tagging and Thermodynamics. Entropy. 2020; 22(2):138. https://doi.org/10.3390/e22020138
Chicago/Turabian StyleDiazdelacruz, Jose. 2020. "Quantum Relative Entropy of Tagging and Thermodynamics" Entropy 22, no. 2: 138. https://doi.org/10.3390/e22020138