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In 2013, at the Istituto Nazionale di Ricerca Metrologica (INRIM) in Turin, Italy, researchers performed the first experimental test of Page and Wootters' ideas. Their result has been interpreted{{by whom|date=August 2020}} to confirm that time is an emergent phenomenon for internal observers but absent for external observers of the universe just as the Wheeler-DeWitt equation predicts.<ref name="medium.com"/>

In 2013, at the Istituto Nazionale di Ricerca Metrologica (INRIM) in Turin, Italy, researchers performed the first experimental test of Page and Wootters' ideas. Their result has been interpreted{{by whom|date=August 2020}} to confirm that time is an emergent phenomenon for internal observers but absent for external observers of the universe just as the Wheeler-DeWitt equation predicts.<ref name="medium.com"/>

 

Now suppose Alice is an observer for system , and Bob is an observer for system . If in the entangled state given above Alice makes a measurement in the <math>\scriptstyle \{|0\rangle, |1\rangle\}</math> eigenbasis of , there are two possible outcomes, occurring with equal probability:

Now suppose Alice is an observer for system , and Bob is an observer for system . If in the entangled state given above Alice makes a measurement in the <math>\scriptstyle \{|0\rangle, |1\rangle\}</math> eigenbasis of , there are two possible outcomes, occurring with equal probability:

现在假设 Alice 是系统的观察者，而 Bob 是系统的观察者。如果在上面给出的纠缠态中，爱丽丝在[ | 0 rangle，| 1 rangle ] </math 本征基中进行测量，有两种可能的结果，发生的概率相等:

现在假设爱丽丝是系统的观察者，而鲍勃也是系统的观察者。如果在上面给出的纠缠态中，爱丽丝在[ | 0 rangle，| 1 rangle ] </math 本征基中进行测量，有两种可能的结果，发生的概率相等:

=== Source for the arrow of time ===

=== Source for the arrow of time时间之箭的来源  ===

Physicist [[Seth Lloyd]] says that [[quantum uncertainty]] gives rise to entanglement, the putative source of the [[arrow of time]]. According to Lloyd; "The arrow of time is an arrow of increasing correlations."<ref>{{Cite journal|url=https://www.wired.com/2014/04/quantum-theory-flow-time/|title=New Quantum Theory Could Explain the Flow of Time|journal=Wired|accessdate=13 October 2014|date=2014-04-25|last1=Wolchover|first1=Natalie}}</ref> The approach to entanglement would be from the perspective of the causal arrow of time, with the assumption that the cause of the measurement of one particle determines the effect of the result of the other particle's measurement.

Physicist [[Seth Lloyd]] says that [[quantum uncertainty]] gives rise to entanglement, the putative source of the [[arrow of time]]. According to Lloyd; "The arrow of time is an arrow of increasing correlations."<ref>{{Cite journal|url=https://www.wired.com/2014/04/quantum-theory-flow-time/|title=New Quantum Theory Could Explain the Flow of Time|journal=Wired|accessdate=13 October 2014|date=2014-04-25|last1=Wolchover|first1=Natalie}}</ref> The approach to entanglement would be from the perspective of the causal arrow of time, with the assumption that the cause of the measurement of one particle determines the effect of the result of the other particle's measurement.

Alice 测量1，系统的状态崩溃为 < math > scriptstyle | 1 rangle _ a | 0 rangle _ b </math > 。

Alice 测量1，系统的状态崩溃为 < math > scriptstyle | 1 rangle _ a | 0 rangle _ b </math > 。

=== Emergent gravity ===

=== Emergent gravity 涌现重力===

爱丽丝的测量结果是随机的。Alice 不能决定将组合系统折叠到哪个状态，因此不能通过作用于她的系统将信息传递给 Bob。因此，在这个特定的方案中，因果关系被保留了下来。关于一般的论点，请参阅不交流定理。

爱丽丝的测量结果是随机的。Alice 不能决定将组合系统折叠到哪个状态，因此不能通过作用于她的系统将信息传递给 Bob。因此，在这个特定的方案中，因果关系被保留了下来。关于一般的论点，请参阅不交流定理。

== Non-locality and entanglement ==

== Non-locality and entanglement非定域性与纠缠 ==

In the media and popular science, quantum non-locality is often portrayed as being equivalent to entanglement. While this is true for pure bipartite quantum states, in general entanglement is only necessary for non-local correlations, but there exist mixed entangled states that do not produce such correlations.<ref name="Brunner-RMP2014">{{cite journal |title=Bell nonlocality |author1=Nicolas Brunner |author2=Daniel Cavalcanti |author3=Stefano Pironio |author4=Valerio Scarani |author5=Stephanie Wehner |journal=Rev. Mod. Phys. |volume=86 |issue=2 |pages=419–478 |date=2014 |doi=10.1103/RevModPhys.86.419 |arxiv=1303.2849|bibcode=2014RvMP...86..419B |s2cid=119194006 }}</ref> A well-known example is the [[Werner state]]s that are entangled for certain values of <math>p_{sym}</math>, but can always be described using local hidden variables.<ref name=werner1989>{{cite journal | last = Werner| first = R.F. | title = Quantum States with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model | journal = [[Physical Review A]] | volume = 40| pages = 4277–4281 | year = 1989 |doi=10.1103/PhysRevA.40.4277 | pmid=9902666 | issue=8|bibcode = 1989PhRvA..40.4277W }}</ref> Moreover, it was shown that, for arbitrary numbers of parties, there exist states that are genuinely entangled but admit a local model.<ref>{{cite journal|author=R. Augusiak, M. Demianowicz, J. Tura and A. Acín |title=Entanglement and Nonlocality are Inequivalent for Any Number of Parties |journal=Phys. Rev. Lett. |volume=115 |issue=3 |pages=030404 |year=2015 |arxiv=1407.3114 |doi=10.1103/PhysRevLett.115.030404|pmid=26230773 |hdl=2117/78836 |bibcode=2015PhRvL.115c0404A |s2cid=29758483 }}</ref>

In the media and popular science, quantum non-locality is often portrayed as being equivalent to entanglement. While this is true for pure bipartite quantum states, in general entanglement is only necessary for non-local correlations, but there exist mixed entangled states that do not produce such correlations.<ref name="Brunner-RMP2014">{{cite journal |title=Bell nonlocality |author1=Nicolas Brunner |author2=Daniel Cavalcanti |author3=Stefano Pironio |author4=Valerio Scarani |author5=Stephanie Wehner |journal=Rev. Mod. Phys. |volume=86 |issue=2 |pages=419–478 |date=2014 |doi=10.1103/RevModPhys.86.419 |arxiv=1303.2849|bibcode=2014RvMP...86..419B |s2cid=119194006 }}</ref> A well-known example is the [[Werner state]]s that are entangled for certain values of <math>p_{sym}</math>, but can always be described using local hidden variables.<ref name=werner1989>{{cite journal | last = Werner| first = R.F. | title = Quantum States with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model | journal = [[Physical Review A]] | volume = 40| pages = 4277–4281 | year = 1989 |doi=10.1103/PhysRevA.40.4277 | pmid=9902666 | issue=8|bibcode = 1989PhRvA..40.4277W }}</ref> Moreover, it was shown that, for arbitrary numbers of parties, there exist states that are genuinely entangled but admit a local model.<ref>{{cite journal|author=R. Augusiak, M. Demianowicz, J. Tura and A. Acín |title=Entanglement and Nonlocality are Inequivalent for Any Number of Parties |journal=Phys. Rev. Lett. |volume=115 |issue=3 |pages=030404 |year=2015 |arxiv=1407.3114 |doi=10.1103/PhysRevLett.115.030404|pmid=26230773 |hdl=2117/78836 |bibcode=2015PhRvL.115c0404A |s2cid=29758483 }}</ref>

As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a density matrix, which is a positive-semidefinite matrix, or a trace class when the state space is infinite-dimensional, and has trace 1. Again, by the spectral theorem, such a matrix takes the general form:

As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a density matrix, which is a positive-semidefinite matrix, or a trace class when the state space is infinite-dimensional, and has trace 1. Again, by the spectral theorem, such a matrix takes the general form:

 

  <math>\rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|,</math>

  <math>\rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|,</math>

== Quantum mechanical framework ==

 

== Quantum mechanical framework 量子力学框架==

where the w_i are positive-valued probabilities (they sum up to 1), the vectors  are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret  as representing an ensemble where  is the proportion of the ensemble whose states are <math>|\alpha_i\rangle</math>. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need density matrices to represent the state.

where the w_i are positive-valued probabilities (they sum up to 1), the vectors  are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret  as representing an ensemble where  is the proportion of the ensemble whose states are <math>|\alpha_i\rangle</math>. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need density matrices to represent the state.

其中 w < sub > i </sub > 是正值概率(和为1) ，向量是单位向量，在无限维情况下，我们取这些状态的闭包为迹范数。我们可以解释为代表一个集合，其中集合的状态是 < math > | alpha _ i rangle </math > 。当一个混合状态的秩为1时，它就描述了一个纯系综。当量子系统的状态信息少于总量时，我们需要密度矩阵来表示状态。

其中w_i是正值概率（它们的总和为1），向量是单位向量，在无限维的情况下，我们将在迹范数中取这类状态的闭包。我们可以解释为表示一个集合，其中是状态为<math>| \alpha|i\rangle</math>的集合的比例。当一个混合态有秩1时，它就描述了一个“纯系综”。当一个量子系统的状态信息不足时，我们需要密度矩阵来表示这个状态。

The following subsections are for those with a good working knowledge of the formal, mathematical description of [[quantum mechanics]], including familiarity with the formalism and theoretical framework developed in the articles: [[bra–ket notation]] and [[mathematical formulation of quantum mechanics]].

The following subsections are for those with a good working knowledge of the formal, mathematical description of [[quantum mechanics]], including familiarity with the formalism and theoretical framework developed in the articles: [[bra–ket notation]] and [[mathematical formulation of quantum mechanics]].

 

Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits electrons towards an observer. The electrons' Hilbert spaces are identical. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state <math>|\mathbf{z}+\rangle</math> with spins aligned in the positive  direction, and the other with state <math>|\mathbf{y}-\rangle</math> with spins aligned in the negative  direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits electrons towards an observer. The electrons' Hilbert spaces are identical. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state <math>|\mathbf{z}+\rangle</math> with spins aligned in the positive  direction, and the other with state <math>|\mathbf{y}-\rangle</math> with spins aligned in the negative  direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

在实验上，可以实现如下的混合集成。考虑一个“黑盒子”装置，它向观察者喷射电子。电子的希尔伯特空间是相同的。该装置可能产生全部处于相同状态的电子; 在这种情况下，观察者接收到的电子就是一个纯系综。然而，这种装置可以在不同的状态下产生电子。例如，它可以产生两个电子群: 一个是状态 < math > | mathbf { z } + rangle </math > 的正方向自旋，另一个是状态 < math > | mathbf { y }-rangle </math > 的负方向自旋。通常，这是一个混合集合，因为可以有任意数量的总体，每个总体对应不同的状态。

实验上，混合系综可以实现如下。考虑一个向观察者吐电子的“黑匣子”装置。电子的希尔伯特空间是相同的。这个装置可能产生所有处于相同状态的电子；在这种情况下，观察者接收到的电子就是一个纯系综。然而，这种装置可以产生不同状态的电子。例如，它可以产生两个电子群：一个是自旋朝正方向排列的态<math>|\mathbf{z}+\rangle</math>，另一个是自旋朝负方向排列的态<math>|\mathbf{y}-\rangle</math>。一般来说，这是一个混合集合，因为可以有任意数量的总体，每个总体对应于不同的状态。

 

=== Pure states纯净态 ===

=== Pure states ===

Consider two arbitrary quantum systems {{mvar|A}} and {{mvar|B}}, with respective [[Hilbert space]]s {{mvar|H_A}} and {{mvar|H_B}}. The Hilbert space of the composite system is the [[tensor product]]

Consider two arbitrary quantum systems {{mvar|A}} and {{mvar|B}}, with respective [[Hilbert space]]s {{mvar|H_A}} and {{mvar|H_B}}. The Hilbert space of the composite system is the [[tensor product]]

 

Following the definition above, for a bipartite composite system, mixed states are just density matrices on . That is, it has the general form

Following the definition above, for a bipartite composite system, mixed states are just density matrices on . That is, it has the general form

If the first system is in state <math>\scriptstyle| \psi \rangle_A</math> and the second in state <math>\scriptstyle| \phi \rangle_B</math>, the state of the composite system is

If the first system is in state <math>\scriptstyle| \psi \rangle_A</math> and the second in state <math>\scriptstyle| \phi \rangle_B</math>, the state of the composite system is

 

where the w_i are positively valued probabilities, <math>\sum_j |c_{ij}|^2=1</math>, and the vectors are unit vectors. This is self-adjoint and positive and has trace 1.

where the w_i are positively valued probabilities, <math>\sum_j |c_{ij}|^2=1</math>, and the vectors are unit vectors. This is self-adjoint and positive and has trace 1.

其中 w < sub > i </sub > 是正值概率，< math > sum _ j | c _ { ij } | ^ 2 = 1 </math > ，向量是单位向量。这是自伴和正的，并且有迹1。

其中w_i是正值概率，<math>\sum|u j | c|ij}|^2=1</math>，向量是单位向量。这是自伴正的，有迹1。

: <math>|\psi\rangle_A \otimes |\phi\rangle_B.</math>

: <math>|\psi\rangle_A \otimes |\phi\rangle_B.</math>

States of the composite system that can be represented in this form are called [[separable state]]s, or [[product state]]s.

States of the composite system that can be represented in this form are called [[separable state]]s, or [[product state]]s.

 

  <math>\rho = \sum_i w_i \rho_i^A \otimes \rho_i^B, </math>

  <math>\rho = \sum_i w_i \rho_i^A \otimes \rho_i^B, </math>

where the  are positively valued probabilities and the <math>\rho_i^A</math>'s and <math>\rho_i^B</math>'s are themselves mixed states (density operators) on the subsystems  and  respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that <math>\rho_i^A</math> and <math>\rho_i^B</math> are themselves pure ensembles. A state is then said to be entangled if it is not separable.

where the  are positively valued probabilities and the <math>\rho_i^A</math>'s and <math>\rho_i^B</math>'s are themselves mixed states (density operators) on the subsystems  and  respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that <math>\rho_i^A</math> and <math>\rho_i^B</math> are themselves pure ensembles. A state is then said to be entangled if it is not separable.

其中的正值概率和 rho _ i ^ a </math > 的和 rho _ i ^ b </math > 的本身是子系统和子系统上的混合状态(密度算符)。换句话说，如果一个状态是不相关状态或乘积状态上的概率分布，则该状态是可分的。通过将密度矩阵写成纯系综和并进行扩展，我们可以假定，不失一般性和数学本身就是纯系综。如果一个状态不可分离，则称其为纠缠态。

其中为正值概率和<math>\rho i^A</math>和<math>\rho i^B</math>分别为子系统和上的混合态（密度算子）。换句话说，如果一个状态是不相关状态或乘积状态的概率分布，那么它是可分离的。通过将密度矩阵写成纯系综的和并展开，我们可以假定<math>\rho i^A</math>和<math>\rho i^B</math>本身就是纯系综。如果一个态是不可分离的，它就被称为纠缠态。

: <math>|\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B</math>.

: <math>|\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B</math>.

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be NP-hard. For the  and  cases, a necessary and sufficient criterion for separability is given by the famous Positive Partial Transpose (PPT) condition.

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be NP-hard. For the  and  cases, a necessary and sufficient criterion for separability is given by the famous Positive Partial Transpose (PPT) condition.

一般来说，要判断一个混合态是否是纠缠态是很困难的。一般的二部格被证明是 np 困难的。对于和种情形，利用著名的正偏转子(PPT)条件给出了可分性的一个充要条件。

一般来说，要判断一个混合态是否是纠缠态是很困难的。一般的二部格被证明是NP-困难的。对于和种情形，利用著名的正偏转子(PPT)条件给出了可分性的一个充要条件。

This state is separable if there exist vectors <math>\scriptstyle [c^A_i], [c^B_j]</math> so that <math>\scriptstyle c_{ij}= c^A_ic^B_j,</math> yielding <math>\scriptstyle |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A</math> and <math>\scriptstyle |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B.</math> It is inseparable if for any vectors <math>\scriptstyle [c^A_i],[c^B_j]</math> at least for one pair of coordinates <math>\scriptstyle c^A_i,c^B_j</math> we have <math>\scriptstyle c_{ij} \neq c^A_ic^B_j.</math> If a state is inseparable, it is called an 'entangled state'.

This state is separable if there exist vectors <math>\scriptstyle [c^A_i], [c^B_j]</math> so that <math>\scriptstyle c_{ij}= c^A_ic^B_j,</math> yielding <math>\scriptstyle |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A</math> and <math>\scriptstyle |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B.</math> It is inseparable if for any vectors <math>\scriptstyle [c^A_i],[c^B_j]</math> at least for one pair of coordinates <math>\scriptstyle c^A_i,c^B_j</math> we have <math>\scriptstyle c_{ij} \neq c^A_ic^B_j.</math> If a state is inseparable, it is called an 'entangled state'.

 

For example, given two basis vectors <math>\scriptstyle \{|0\rangle_A, |1\rangle_A\}</math> of {{mvar|H_A}} and two basis vectors <math>\scriptstyle \{|0\rangle_B, |1\rangle_B\}</math> of {{mvar|H_B}}, the following is an entangled state:

For example, given two basis vectors <math>\scriptstyle \{|0\rangle_A, |1\rangle_A\}</math> of {{mvar|H_A}} and two basis vectors <math>\scriptstyle \{|0\rangle_B, |1\rangle_B\}</math> of {{mvar|H_B}}, the following is an entangled state:

The idea of a reduced density matrix was introduced by Paul Dirac in 1930. Consider as above systems  and  each with a Hilbert space . Let the state of the composite system be

The idea of a reduced density matrix was introduced by Paul Dirac in 1930. Consider as above systems  and  each with a Hilbert space . Let the state of the composite system be

  <math>\rho_T = |\Psi\rangle \; \langle\Psi|</math>.

  <math>\rho_T = |\Psi\rangle \; \langle\Psi|</math>.

# Alice measures 0, and the state of the system collapses to <math>\scriptstyle |0\rangle_A |1\rangle_B</math>.

# Alice measures 0, and the state of the system collapses to <math>\scriptstyle |0\rangle_A |1\rangle_B</math>.

 

which is the projection operator onto this state. The state of  is the partial trace of  over the basis of system :

which is the projection operator onto this state. The state of  is the partial trace of  over the basis of system :

# Alice measures 1, and the state of the system collapses to <math>\scriptstyle |1\rangle_A |0\rangle_B</math>.

# Alice measures 1, and the state of the system collapses to <math>\scriptstyle |1\rangle_A |0\rangle_B</math>.

 

If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1. If the latter occurs, (Alice measures 1) then Bob's measurement will return 0 with certainty. Thus, system {{mvar|B}} has been altered by Alice performing a local measurement on system {{mvar|A}}. This remains true even if the systems {{mvar|A}} and {{mvar|B}} are spatially separated. This is the foundation of the [[EPR paradox]].

If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1. If the latter occurs, (Alice measures 1) then Bob's measurement will return 0 with certainty. Thus, system {{mvar|B}} has been altered by Alice performing a local measurement on system {{mvar|A}}. This remains true even if the systems {{mvar|A}} and {{mvar|B}} are spatially separated. This is the foundation of the [[EPR paradox]].

 

The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see [[no-communication theorem]].

The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see [[no-communication theorem]].

 

例如，纠缠态的约化密度矩阵

例如，纠缠态的约化密度矩阵

=== Ensembles ===

=== Ensembles集成 ===

As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a [[density matrix]], which is a [[positive-semidefinite matrix]], or a [[trace class]] when the state space is infinite-dimensional, and has trace 1. Again, by the [[spectral theorem]], such a matrix takes the general form:

As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a [[density matrix]], which is a [[positive-semidefinite matrix]], or a [[trace class]] when the state space is infinite-dimensional, and has trace 1. Again, by the [[spectral theorem]], such a matrix takes the general form:

 

  <math>\tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right),</math>

  <math>\tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right),</math>

where the ''w''_i are positive-valued probabilities (they sum up to 1), the vectors {{mvar| α_i}} are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret {{mvar|ρ}} as representing an ensemble where {{mvar|w_i}} is the proportion of the ensemble whose states are <math>|\alpha_i\rangle</math>. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need [[#Reduced density matrices|density matrices]] to represent the state.

where the ''w''_i are positive-valued probabilities (they sum up to 1), the vectors {{mvar| α_i}} are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret {{mvar|ρ}} as representing an ensemble where {{mvar|w_i}} is the proportion of the ensemble whose states are <math>|\alpha_i\rangle</math>. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need [[#Reduced density matrices|density matrices]] to represent the state.

 

  <math>\rho_A = \tfrac{1}{2} \left ( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \right )</math>

  <math>\rho_A = \tfrac{1}{2} \left ( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \right )</math>

Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits [[electron]]s towards an observer. The electrons' Hilbert spaces are [[identical particles|identical]]. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state <math>|\mathbf{z}+\rangle</math> with [[spin (physics)|spins]] aligned in the positive {{math|'''z'''}} direction, and the other with state <math>|\mathbf{y}-\rangle</math> with spins aligned in the negative {{math|'''y'''}} direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits [[electron]]s towards an observer. The electrons' Hilbert spaces are [[identical particles|identical]]. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state <math>|\mathbf{z}+\rangle</math> with [[spin (physics)|spins]] aligned in the positive {{math|'''z'''}} direction, and the other with state <math>|\mathbf{y}-\rangle</math> with spins aligned in the negative {{math|'''y'''}} direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

 

This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of  for the pure product state <math>|\psi\rangle_A \otimes |\phi\rangle_B</math> discussed above is

This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of  for the pure product state <math>|\psi\rangle_A \otimes |\phi\rangle_B</math> discussed above is

where the ''w''_i are positively valued probabilities, <math>\sum_j |c_{ij}|^2=1</math>, and the vectors are unit vectors. This is self-adjoint and positive and has trace 1.

where the ''w''_i are positively valued probabilities, <math>\sum_j |c_{ij}|^2=1</math>, and the vectors are unit vectors. This is self-adjoint and positive and has trace 1.

 

Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one-dimensional AKLT spin chain: the ground state can be divided into a block and an environment. The reduced density matrix of the block is proportional to a projector to a degenerate ground state of another Hamiltonian.

Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one-dimensional AKLT spin chain: the ground state can be divided into a block and an environment. The reduced density matrix of the block is proportional to a projector to a degenerate ground state of another Hamiltonian.

In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows to implement valuable transformations. The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labeled "A" and "B" on each of which arbitrary quantum operations can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called LOCC (local operations and classical communication). These operations do not allow the production of entangled states between the systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. For example, an interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.

In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows to implement valuable transformations. The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labeled "A" and "B" on each of which arbitrary quantum operations can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called LOCC (local operations and classical communication). These operations do not allow the production of entangled states between the systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. For example, an interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.

where the {{mvar|w_i}} are positively valued probabilities and the <math>\rho_i^A</math>'s and <math>\rho_i^B</math>'s are themselves mixed states (density operators) on the subsystems {{mvar|A}} and {{mvar|B}} respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that <math>\rho_i^A</math> and <math>\rho_i^B</math> are themselves pure ensembles. A state is then said to be entangled if it is not separable.

where the {{mvar|w_i}} are positively valued probabilities and the <math>\rho_i^A</math>'s and <math>\rho_i^B</math>'s are themselves mixed states (density operators) on the subsystems {{mvar|A}} and {{mvar|B}} respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that <math>\rho_i^A</math> and <math>\rho_i^B</math> are themselves pure ensembles. A state is then said to be entangled if it is not separable.

 

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be [[NP-hard]].<ref>{{Cite book |author=Gurvits L |title=Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03 |chapter=Classical deterministic complexity of Edmonds' Problem and quantum entanglement |journal=Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing |year=2003 |doi=10.1145/780542.780545 |page=10 |isbn=978-1-58113-674-6|arxiv=quant-ph/0303055 |s2cid=5745067 }}</ref> For the {{math|2 × 2}} and {{math|2 × 3}} cases, a necessary and sufficient criterion for separability is given by the famous [[Peres-Horodecki criterion|Positive Partial Transpose (PPT)]] condition.<ref>{{cite journal |author=Horodecki M, Horodecki P, Horodecki R |title=Separability of mixed states: necessary and sufficient conditions |journal=Physics Letters A |volume=223 |issue=1 |page=210 |year=1996 |doi=10.1016/S0375-9601(96)00706-2 |bibcode=1996PhLA..223....1H|arxiv = quant-ph/9605038 |last2=Horodecki |last3=Horodecki |citeseerx=10.1.1.252.496 |s2cid=10580997 }}</ref>

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be [[NP-hard]].<ref>{{Cite book |author=Gurvits L |title=Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03 |chapter=Classical deterministic complexity of Edmonds' Problem and quantum entanglement |journal=Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing |year=2003 |doi=10.1145/780542.780545 |page=10 |isbn=978-1-58113-674-6|arxiv=quant-ph/0303055 |s2cid=5745067 }}</ref> For the {{math|2 × 2}} and {{math|2 × 3}} cases, a necessary and sufficient criterion for separability is given by the famous [[Peres-Horodecki criterion|Positive Partial Transpose (PPT)]] condition.<ref>{{cite journal |author=Horodecki M, Horodecki P, Horodecki R |title=Separability of mixed states: necessary and sufficient conditions |journal=Physics Letters A |volume=223 |issue=1 |page=210 |year=1996 |doi=10.1016/S0375-9601(96)00706-2 |bibcode=1996PhLA..223....1H|arxiv = quant-ph/9605038 |last2=Horodecki |last3=Horodecki |citeseerx=10.1.1.252.496 |s2cid=10580997 }}</ref>

=== Reduced density matrices ===

=== Reduced density matrices约化密度矩阵 ===

In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.

In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.

As indicated above, in general there is no way to associate a pure state to the component system {{mvar|A}}. However, it still is possible to associate a density matrix. Let

As indicated above, in general there is no way to associate a pure state to the component system {{mvar|A}}. However, it still is possible to associate a density matrix. Let

 

which is the [[projection operator]] onto this state. The state of {{mvar|A}} is the [[partial trace]] of {{mvar|ρ_T}} over the basis of system {{mvar|B}}:

which is the [[projection operator]] onto this state. The state of {{mvar|A}} is the [[partial trace]] of {{mvar|ρ_T}} over the basis of system {{mvar|B}}:

 

{{mvar|ρ_A}} is sometimes called the reduced density matrix of {{mvar|ρ}} on subsystem {{mvar|A}}. Colloquially, we "trace out" system {{mvar|B}} to obtain the reduced density matrix on {{mvar|A}}.

{{mvar|ρ_A}} is sometimes called the reduced density matrix of {{mvar|ρ}} on subsystem {{mvar|A}}. Colloquially, we "trace out" system {{mvar|B}} to obtain the reduced density matrix on {{mvar|A}}.

 

  <math>S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right) = - \sum_i \lambda_i \log_2 \lambda_i</math>.

  <math>S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right) = - \sum_i \lambda_i \log_2 \lambda_i</math>.

For example, the reduced density matrix of {{mvar|A}} for the entangled state

For example, the reduced density matrix of {{mvar|A}} for the entangled state

 

  <math> \rho = \int \lambda d P_{\lambda},</math>

  <math> \rho = \int \lambda d P_{\lambda},</math>

This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of {{mvar|A}} for the pure product state <math>|\psi\rangle_A \otimes |\phi\rangle_B</math> discussed above is

This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of {{mvar|A}} for the pure product state <math>|\psi\rangle_A \otimes |\phi\rangle_B</math> discussed above is

 

In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure.

In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure.

 

As in statistical mechanics, the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is  (which can be shown to be the maximum entropy for  mixed states).

As in statistical mechanics, the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is  (which can be shown to be the maximum entropy for  mixed states).

就像统计力学一样，系统的不确定性(微观状态的数量)越多，熵就越大。例如，任何纯态的熵都为零，这并不奇怪，因为处于纯态的系统没有不确定性。上面讨论的纠缠态的两个子系统中的任何一个的熵都是(混合态的最大熵)。

就像统计力学一样，系统的不确定性(微观状态的数量)越多，熵就越大。例如，任何纯态的熵都为零，这并不奇怪，因为处于纯态的系统没有不确定性。上面讨论的纠缠态的两个子系统中的任何一个的熵都是(混合态的最大熵)。

=== Two applications that use them ===

=== Two applications that use them 两种使用它们的应用===

Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one-dimensional [[AKLT Model|AKLT spin chain]]:<ref name="Fan2004">{{cite journal | doi = 10.1103/PhysRevLett.93.227203 | title = Entanglement in a Valence-Bond Solid State | journal = Physical Review Letters | year = 2004 | first = H | last = Fan | page = 227203 |author2=Korepin V |author3=Roychowdhury V  | volume = 93 | issue = 22 | pmid = 15601113 |arxiv=quant-ph/0406067 | bibcode=2004PhRvL..93v7203F| s2cid = 28587190 }}</ref> the ground state can be divided into a block and an environment. The reduced density matrix of the block is [[Proportionality (mathematics)|proportional]] to a projector to a degenerate ground state of another Hamiltonian.

Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one-dimensional [[AKLT Model|AKLT spin chain]]:<ref name="Fan2004">{{cite journal | doi = 10.1103/PhysRevLett.93.227203 | title = Entanglement in a Valence-Bond Solid State | journal = Physical Review Letters | year = 2004 | first = H | last = Fan | page = 227203 |author2=Korepin V |author3=Roychowdhury V  | volume = 93 | issue = 22 | pmid = 15601113 |arxiv=quant-ph/0406067 | bibcode=2004PhRvL..93v7203F| s2cid = 28587190 }}</ref> the ground state can be divided into a block and an environment. The reduced density matrix of the block is [[Proportionality (mathematics)|proportional]] to a projector to a degenerate ground state of another Hamiltonian.

对于两体纯态，减少态的冯纽曼熵是唯一的纠缠度量，因为它是满足纠缠度量所要求的特定公理的态家族中唯一的函数。

对于两体纯态，减少态的冯纽曼熵是唯一的纠缠度量，因为它是满足纠缠度量所要求的特定公理的态家族中唯一的函数。

=== Entanglement as a resource ===

=== Entanglement as a resource 作为资源的纠缠===

In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows to implement valuable transformations. The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labeled "A" and "B" on each of which arbitrary [[quantum operation]]s can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called [[LOCC]] (local operations and classical communication). These operations do not allow the production of entangled states between the systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. For example, an interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.<ref name="horodecki2007" />

In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows to implement valuable transformations. The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labeled "A" and "B" on each of which arbitrary [[quantum operation]]s can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called [[LOCC]] (local operations and classical communication). These operations do not allow the production of entangled states between the systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. For example, an interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.<ref name="horodecki2007" />

 

It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n,...,1/n}. Therefore, a bipartite pure state  is said to be a maximally entangled state if the reduced state of  is the diagonal matrix

It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n,...,1/n}. Therefore, a bipartite pure state  is said to be a maximally entangled state if the reduced state of  is the diagonal matrix

=== Classification of entanglement ===

=== Classification of entanglement 纠缠分类===

  <math>\begin{bmatrix} \frac{1}{n}& & \\ & \ddots & \\ & & \frac{1}{n}\end{bmatrix}.</math>

  <math>\begin{bmatrix} \frac{1}{n}& & \\ & \ddots & \\ & & \frac{1}{n}\end{bmatrix}.</math>

Not all quantum states are equally valuable as a resource. To quantify this value, different [[Quantum entanglement#Entanglement measures|entanglement measures]] (see below) can be used, that assign a numerical value to each quantum state. However, it is often interesting to settle for a coarser way to compare quantum states. This gives rise to different classification schemes. Most entanglement classes are defined based on whether states can be converted to other states using LOCC or a subclass of these operations. The smaller the set of allowed operations, the finer the classification. Important examples are:

Not all quantum states are equally valuable as a resource. To quantify this value, different [[Quantum entanglement#Entanglement measures|entanglement measures]] (see below) can be used, that assign a numerical value to each quantum state. However, it is often interesting to settle for a coarser way to compare quantum states. This gives rise to different classification schemes. Most entanglement classes are defined based on whether states can be converted to other states using LOCC or a subclass of these operations. The smaller the set of allowed operations, the finer the classification. Important examples are:

 

* If two states can be transformed into each other by a local unitary operation, they are said to be in the same ''LU class''. This is the finest of the usually considered classes. Two states in the same LU class have the same value for entanglement measures and the same value as a resource in the distant-labs setting. There is an infinite number of different LU classes (even in the simplest case of two qubits in a pure state).<ref name="GRB1998">>{{cite journal |author1=Grassl, M. |author2=Rötteler, M. |author3=Beth, T. |title=Computing local invariants of quantum-bit systems |journal=Phys. Rev. A |volume=58 |issue=3 |pages=1833–1839 |year=1998 |doi=10.1103/PhysRevA.58.1833 |arxiv=quant-ph/9712040|bibcode=1998PhRvA..58.1833G |s2cid=15892529 }}</ref><ref name="Kraus2010">{{cite journal |author=B. Kraus |authorlink=Barbara Kraus|title=Local unitary equivalence of multipartite pure states |journal=Phys. Rev. Lett. |volume=104 |issue=2 |page=020504 |year=2010 |arxiv=0909.5152 |doi=10.1103/PhysRevLett.104.020504|pmid=20366579 |bibcode=2010PhRvL.104b0504K|s2cid=29984499}}</ref>

* If two states can be transformed into each other by a local unitary operation, they are said to be in the same ''LU class''. This is the finest of the usually considered classes. Two states in the same LU class have the same value for entanglement measures and the same value as a resource in the distant-labs setting. There is an infinite number of different LU classes (even in the simplest case of two qubits in a pure state).<ref name="GRB1998">>{{cite journal |author1=Grassl, M. |author2=Rötteler, M. |author3=Beth, T. |title=Computing local invariants of quantum-bit systems |journal=Phys. Rev. A |volume=58 |issue=3 |pages=1833–1839 |year=1998 |doi=10.1103/PhysRevA.58.1833 |arxiv=quant-ph/9712040|bibcode=1998PhRvA..58.1833G |s2cid=15892529 }}</ref><ref name="Kraus2010">{{cite journal |author=B. Kraus |authorlink=Barbara Kraus|title=Local unitary equivalence of multipartite pure states |journal=Phys. Rev. Lett. |volume=104 |issue=2 |page=020504 |year=2010 |arxiv=0909.5152 |doi=10.1103/PhysRevLett.104.020504|pmid=20366579 |bibcode=2010PhRvL.104b0504K|s2cid=29984499}}</ref>

 

For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.

For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.

* If two states can be transformed into each other by local operations including measurements with probability larger than 0, they are said to be in the same 'SLOCC class' ("stochastic LOCC"). Qualitatively, two states <math>\rho_1</math> and <math>\rho_2</math> in the same SLOCC class are equally powerful (since I can transform one into the other and then do whatever it allows me to do), but since the transformations <math>\rho_1\to\rho_2</math> and <math>\rho_2\to\rho_1</math> may succeed with different probability, they are no longer equally valuable. E.g., for two pure qubits there are only two SLOCC classes: the entangled states (which contains both the (maximally entangled) Bell states and weakly entangled states like <math>|00\rangle+0.01|11\rangle</math>) and the separable ones (i.e., product states like <math>|00\rangle</math>).<ref>{{cite journal |author=M. A. Nielsen |title=Conditions for a Class of Entanglement Transformations |journal=Phys. Rev. Lett. |volume=83 |issue=2 |page=436 |year=1999 |doi=10.1103/PhysRevLett.83.436 |arxiv=quant-ph/9811053|bibcode=1999PhRvL..83..436N |s2cid=17928003 }}</ref><ref name="GoWa2010">{{cite journal |authors=Gour, G. & Wallach, N. R. |title=Classification of Multipartite Entanglement of All Finite Dimensionality |journal=Phys. Rev. Lett. |volume=111 |issue=6 |page=060502 |year=2013 |doi=10.1103/PhysRevLett.111.060502 |pmid=23971544 |arxiv=1304.7259|bibcode=2013PhRvL.111f0502G |s2cid=1570745 }}</ref>

* If two states can be transformed into each other by local operations including measurements with probability larger than 0, they are said to be in the same 'SLOCC class' ("stochastic LOCC"). Qualitatively, two states <math>\rho_1</math> and <math>\rho_2</math> in the same SLOCC class are equally powerful (since I can transform one into the other and then do whatever it allows me to do), but since the transformations <math>\rho_1\to\rho_2</math> and <math>\rho_2\to\rho_1</math> may succeed with different probability, they are no longer equally valuable. E.g., for two pure qubits there are only two SLOCC classes: the entangled states (which contains both the (maximally entangled) Bell states and weakly entangled states like <math>|00\rangle+0.01|11\rangle</math>) and the separable ones (i.e., product states like <math>|00\rangle</math>).<ref>{{cite journal |author=M. A. Nielsen |title=Conditions for a Class of Entanglement Transformations |journal=Phys. Rev. Lett. |volume=83 |issue=2 |page=436 |year=1999 |doi=10.1103/PhysRevLett.83.436 |arxiv=quant-ph/9811053|bibcode=1999PhRvL..83..436N |s2cid=17928003 }}</ref><ref name="GoWa2010">{{cite journal |authors=Gour, G. & Wallach, N. R. |title=Classification of Multipartite Entanglement of All Finite Dimensionality |journal=Phys. Rev. Lett. |volume=111 |issue=6 |page=060502 |year=2013 |doi=10.1103/PhysRevLett.111.060502 |pmid=23971544 |arxiv=1304.7259|bibcode=2013PhRvL.111f0502G |s2cid=1570745 }}</ref>

 

* Instead of considering transformations of single copies of a state (like <math>\rho_1\to\rho_2</math>) one can define classes based on the possibility of multi-copy transformations. E.g., there are examples when <math>\rho_1\to\rho_2</math> is impossible by LOCC, but <math>\rho_1\otimes\rho_1\to\rho_2</math> is possible. A very important (and very coarse) classification is based on the property whether it is possible to transform an arbitrarily large number of copies of a state <math>\rho</math> into at least one pure entangled state. States that have this property are called [[Entanglement distillation|distillable]]. These states are the most  useful quantum states since, given enough of them, they can be transformed (with local operations) into any entangled state and hence allow for all possible uses. It came initially as a surprise that not all entangled states are distillable, those that are not are called '[[Bound entanglement|bound entangled]]'.<ref name="HHH97">{{cite journal |author1=Horodecki, M. |author2=Horodecki, P. |author3=Horodecki, R. |title=Mixed-state entanglement and distillation: Is there a ''bound'' entanglement in nature? |journal=Phys. Rev. Lett. |volume=80 |issue=1998 |pages=5239–5242 |year=1998 |arxiv=quant-ph/9801069|doi=10.1103/PhysRevLett.80.5239 |bibcode=1998PhRvL..80.5239H |s2cid=111379972 }}</ref><ref name="horodecki2007" />

* Instead of considering transformations of single copies of a state (like <math>\rho_1\to\rho_2</math>) one can define classes based on the possibility of multi-copy transformations. E.g., there are examples when <math>\rho_1\to\rho_2</math> is impossible by LOCC, but <math>\rho_1\otimes\rho_1\to\rho_2</math> is possible. A very important (and very coarse) classification is based on the property whether it is possible to transform an arbitrarily large number of copies of a state <math>\rho</math> into at least one pure entangled state. States that have this property are called [[Entanglement distillation|distillable]]. These states are the most  useful quantum states since, given enough of them, they can be transformed (with local operations) into any entangled state and hence allow for all possible uses. It came initially as a surprise that not all entangled states are distillable, those that are not are called '[[Bound entanglement|bound entangled]]'.<ref name="HHH97">{{cite journal |author1=Horodecki, M. |author2=Horodecki, P. |author3=Horodecki, R. |title=Mixed-state entanglement and distillation: Is there a ''bound'' entanglement in nature? |journal=Phys. Rev. Lett. |volume=80 |issue=1998 |pages=5239–5242 |year=1998 |arxiv=quant-ph/9801069|doi=10.1103/PhysRevLett.80.5239 |bibcode=1998PhRvL..80.5239H |s2cid=111379972 }}</ref><ref name="horodecki2007" />

 

As an aside, the information-theoretic definition is closely related to entropy in the sense of statistical mechanics (comparing the two definitions in the present context, it is customary to set the Boltzmann constant  1}}). For example, by properties of the Borel functional calculus, we see that for any unitary operator ,

As an aside, the information-theoretic definition is closely related to entropy in the sense of statistical mechanics (comparing the two definitions in the present context, it is customary to set the Boltzmann constant  1}}). For example, by properties of the Borel functional calculus, we see that for any unitary operator ,

=== Entropy ===

=== Entropy熵 ===

Indeed, without this property, the von Neumann entropy would not be well-defined.

Indeed, without this property, the von Neumann entropy would not be well-defined.

In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.

In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.

 

特别是，可以是系统的时间演化算子，即,

特别是，可以是系统的时间演化算子，即,

==== Definition ====

==== Definition 定义====

[[File:Von Neumann entropy for bipartite system plot.svg|right|thumb|200px|The plot of von Neumann entropy Vs Eigenvalue for a bipartite 2-level pure state.  When the eigenvalue has value .5, von Neumann entropy is at a maximum, corresponding to maximum entanglement.]]

[[File:Von Neumann entropy for bipartite system plot.svg|right|thumb|200px|The plot of von Neumann entropy Vs Eigenvalue for a bipartite 2-level pure state.  When the eigenvalue has value .5, von Neumann entropy is at a maximum, corresponding to maximum entanglement.]]

assume the same convention when calculating

assume the same convention when calculating

 

Most researchers believe that entanglement is necessary to realize quantum computing (although this is disputed by some).

Most researchers believe that entanglement is necessary to realize quantum computing (although this is disputed by some).

As in [[entropy|statistical mechanics]], the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is {{math|log(2)}} (which can be shown to be the maximum entropy for {{math|2 × 2}} mixed states).

As in [[entropy|statistical mechanics]], the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is {{math|log(2)}} (which can be shown to be the maximum entropy for {{math|2 × 2}} mixed states).

In interferometry, entanglement is necessary for surpassing the standard quantum limit and achieving the Heisenberg limit.

In interferometry, entanglement is necessary for surpassing the standard quantum limit and achieving the Heisenberg limit.

==== As a measure of entanglement ====

==== As a measure of entanglement作为纠缠的测量 ====

Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist.<ref name="arxiv.org">{{cite journal|author1=Plenio|title=An introduction to entanglement measures|year=2007|pages=1–51|volume=1|journal=Quant. Inf. Comp. |arxiv=quant-ph/0504163|bibcode=2005quant.ph..4163P|last2=Virmani}}</ref> If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems.

Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist.<ref name="arxiv.org">{{cite journal|author1=Plenio|title=An introduction to entanglement measures|year=2007|pages=1–51|volume=1|journal=Quant. Inf. Comp. |arxiv=quant-ph/0504163|bibcode=2005quant.ph..4163P|last2=Virmani}}</ref> If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems.

For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.

For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.

 

For two qubits, the Bell states are

For two qubits, the Bell states are