- #1
Lawrence B. Crowell
- 190
- 2
This is the start of a presentation of some work I have done over the past year and hope to publish. This is how it is that quantum information is preserved in quantum gravity and cosmology. This will involve a number of posts along this thread. The next three posts involve quantum information with black holes
//
The conservation of information from black hole radiance has an unsettled history. The initial indications were that information is lost. Yet a formalism of a consistent quantum theory of gravity with information loss is difficult, for such a theory would have to be nonunitary. In a bet recently Hawking conceded to Preskill that information was preserved in black holes. Information would then not be destroyed, but rather scrambled in such as way as to make its retrieval intractably impossible. A tunnelling approach to quantum radiance by Parikh and Wilczek [1] suggests that the process in total has [itex]\Delta S~=~0[/itex], but as recently pointed out in [2] this is the case where the black hole and environment are in thermal equilibrium. However, the negative heat capacity of spacetime means that a black hole slightly removed from equilibrium is unstable and will diverge from equilibrium. This is seen with the evaporation of a black hole, where as its entropy [itex]\Delta S_{bh}~\rightarrow~0[/itex] its temperature becomes large. Thus the Parikh-Wilczek tunnelling theory appears to be a “ measure zero” case.
//
A tunnelling process involves an action which is not accessible classically. With the tunnelling of a particle through a square barrier, this involves an imaginary momentum or action. In the case of a tunnelling of a particle from a black hole [itex]M~\rightarrow~M~+~\delta m[/itex], this involves the imaginary part of the action
[tex]
ImS~=~Im\int_{r_i}^{r_f}p_rdr.\eqno(1)
[/tex]
The Hamilton equation [itex]{\dot r}~=~{{\partial H}\over{\partial p}}[/itex] permits this to be written as
[tex]
ImS~=~Im\int_{r_i}^{r_f}\int_{p_i}^{p_i}{{dr}\over {\dot r}}dH.\eqno(2)
[/tex]
Along null geodesics the velocity [itex]{\dot r}~=~\pm 1~+~\sqrt{2M/r}[/itex] the action for the classically forbidden path is
[tex]
Im S~=~-2\pi\int_0^{\delta m}{{dr~dm}\over{1~+~\sqrt{{2M~-~\delta m}\over r}}}~=~-2\pi\big((M~-~\delta m)^2~-~M^2\big),\eqno(3)
[/tex]
which defines [itex]-{1\over 2}(S_f~-~S_i)[/itex]. The imaginary part of the action gives the tunnelling probability or emission rate as [itex]\Gamma~=~exp(-2ImS)~=~exp(\Delta S)[/itex]. For a black hole in equilibrium with the environment the entropy remains on average zero, for with every quanta it emits it will on average absorb another from the environment. [itex]dM~=~dQ[/itex] as the first law of black hole thermodynamics with [itex]dS~=~{{dQ}\over T}[/itex] holds for a reversible process. However, the fluctuations will eventually cause the black hole to diverge from equilibrium, where no matter how small this is it will cause the black hole radiance to diverge, or for the black hole to acquire larger mass arbitrarily. Any change in the state of the black hole, whether by emission or absorption, perturbs the black hole from equilibrium.
//
There are a number of physical ways that black hole radiance are presented. A black hole emits a particle since the quanta which make up a black hole have some small but nonzero probability of existing in a region [itex]r~>~2M[/itex]. Another interpretation is that virtual electron-positron pairs near the event horizon may permit one in the pair to fall into the hole while the other escapes to infinity. This view is equivalent to saying that an electron or positron propagates backward through time from the black hole and is then scattered into the forward direction by the gravity field. A related interpretation has that the creation of a positive mass-energy particle is associated with the creation of a negative mass-energy particle absorbed by the black hole. As a result the black hole’ s mass is reduced and a particle escapes. In the case of fermions this is in line with Dirac’ s original idea of the anti-particle with a negative mass-energy. In all of these cases there is a superposition principle at work. Quanta within the black hole are correlated with quanta in the exterior region. How these quanta are correlated is the fundamental issue. The imaginary action is a measure of the nonlocal correlation a particle in the black hole has with the outside world. In the case of equilibrium with [itex]TdS~=~dM[/itex] the black hole exchanges entropy with the environment so that the total information of the black hole and environment remains the same. Yet in general the radiance of a black hole will heat up the environment so that [itex]dS~>~{{dM}\over T}[/itex], and the same is the case of the black hole absorbs mass-energy.
//
A black hole will absorb and emit observables, where if information is preserved these observables will have a corollation. The corollation will reflect a quantum process which is unitary, or that the emitted observables are nonlocally entangled with the black hole states in such as way as to preserve information. If information is preserved by a black hole, then in principle a black hole is an efficient teleporter of quantum information. Here the black hole is shown to ultimately preserve quantum information even for the case that [itex]dS~>~{{dM}\over T}[/itex].
//
The standard “ Alice and Bob” problem is considered. Alice has the set of observables [itex]A[/itex], which she communicates to Bob in a string [itex]x_1x_2\dots x_n[/itex]. Bob similarly has the string [itex]y_1y_2\dots y_n[/itex]. The von Neumann information each possesses is then [itex]S(X)~=~-Tr(\rho_X~log_2\rho_X)[/itex], for [itex]X[/itex] either [itex]A[/itex] or [itex]B[/itex]. The entropy of each is then a compartmentalization on [itex]\rho_{AB}[/itex] for the total quantum information both possesses and [itex]\rho_{A}~=~Tr_B(\rho_{AB})[/itex] and [itex]\rho_B~=~Tr_A(\rho_{AB})[/itex]. Here the trace is over the part of the Hilbert space for Alice or Bob to project out the density operator for Bob or Alice. The density operator [itex]\rho_{AB}[/itex] then defines the joint entropy
[tex]
S(AB)~=~-Tr(\rho_{AB}~log_2\rho_{AB}).\eqno(4)
[/tex]
If Alice transmits her string [itex]x_1x_2\dots x_n[/itex] to Bob this defines the conditional information or entropy [itex]S(A|B)[/itex] as the information communicated by Alice given that Bob has [itex]y_1\dots y_n[/itex] defined as
[tex]
S(A|B)~=~S(AB)~-~S(B).\eqno(5)
[/tex]
If Alice sends this string into a black hole, this is the entropy measured by Bob as measured by the quantum information the black hole emits. The conditional entropy may be defined by a conditional von Neumann entropy definition
[tex]
S(A|B)~=~-Tr(\rho_B\rho(A|B)~log_2\rho(A|B)~=~-Tr(\rho_{AB}log_2\rho_{A|B}),\eqno(6)
[/tex]
where [itex]\rho_{A|B}~=~lim_{n\rightarrow\infty}\big({\rho_{AB}}^{1/n}({\bf 1}_A\otimes\rho_B)^{-1/n}\big)^n[/itex]. Here [itex]{\bf 1}_A[/itex] is a unit matrix over the Hilbert space for Alice’ s quantum information. This means that the entries of [itex]\rho_{A|B}[/itex] can be over unity, which also means that the information content of conditional entropy can be negative as well [3]. Thus quantum information can be negative, in contrast to classical information. The conditional entropy determines how much quantum communication is required to gain complete quantum information of the system in the state [itex]\rho_{AB}[/itex].
//
When the conditional entropy is negative Alice can only communicate information about the complete state by classical communication. The sharing of [itex]-S(A|B)[/itex] means that Alice and Bob share an entangled state, which may be used to teleport a state at no entropy cost. The negative quantum information is then the degree of “ ignorance” Bob has of the quantum system which cancels out any future information Bob receives. The “ hole” that Alice fills in Bob’ s state ignorance amounts to a merging of her state with Bob’ s.
//
The conservation of information from black hole radiance has an unsettled history. The initial indications were that information is lost. Yet a formalism of a consistent quantum theory of gravity with information loss is difficult, for such a theory would have to be nonunitary. In a bet recently Hawking conceded to Preskill that information was preserved in black holes. Information would then not be destroyed, but rather scrambled in such as way as to make its retrieval intractably impossible. A tunnelling approach to quantum radiance by Parikh and Wilczek [1] suggests that the process in total has [itex]\Delta S~=~0[/itex], but as recently pointed out in [2] this is the case where the black hole and environment are in thermal equilibrium. However, the negative heat capacity of spacetime means that a black hole slightly removed from equilibrium is unstable and will diverge from equilibrium. This is seen with the evaporation of a black hole, where as its entropy [itex]\Delta S_{bh}~\rightarrow~0[/itex] its temperature becomes large. Thus the Parikh-Wilczek tunnelling theory appears to be a “ measure zero” case.
//
A tunnelling process involves an action which is not accessible classically. With the tunnelling of a particle through a square barrier, this involves an imaginary momentum or action. In the case of a tunnelling of a particle from a black hole [itex]M~\rightarrow~M~+~\delta m[/itex], this involves the imaginary part of the action
[tex]
ImS~=~Im\int_{r_i}^{r_f}p_rdr.\eqno(1)
[/tex]
The Hamilton equation [itex]{\dot r}~=~{{\partial H}\over{\partial p}}[/itex] permits this to be written as
[tex]
ImS~=~Im\int_{r_i}^{r_f}\int_{p_i}^{p_i}{{dr}\over {\dot r}}dH.\eqno(2)
[/tex]
Along null geodesics the velocity [itex]{\dot r}~=~\pm 1~+~\sqrt{2M/r}[/itex] the action for the classically forbidden path is
[tex]
Im S~=~-2\pi\int_0^{\delta m}{{dr~dm}\over{1~+~\sqrt{{2M~-~\delta m}\over r}}}~=~-2\pi\big((M~-~\delta m)^2~-~M^2\big),\eqno(3)
[/tex]
which defines [itex]-{1\over 2}(S_f~-~S_i)[/itex]. The imaginary part of the action gives the tunnelling probability or emission rate as [itex]\Gamma~=~exp(-2ImS)~=~exp(\Delta S)[/itex]. For a black hole in equilibrium with the environment the entropy remains on average zero, for with every quanta it emits it will on average absorb another from the environment. [itex]dM~=~dQ[/itex] as the first law of black hole thermodynamics with [itex]dS~=~{{dQ}\over T}[/itex] holds for a reversible process. However, the fluctuations will eventually cause the black hole to diverge from equilibrium, where no matter how small this is it will cause the black hole radiance to diverge, or for the black hole to acquire larger mass arbitrarily. Any change in the state of the black hole, whether by emission or absorption, perturbs the black hole from equilibrium.
//
There are a number of physical ways that black hole radiance are presented. A black hole emits a particle since the quanta which make up a black hole have some small but nonzero probability of existing in a region [itex]r~>~2M[/itex]. Another interpretation is that virtual electron-positron pairs near the event horizon may permit one in the pair to fall into the hole while the other escapes to infinity. This view is equivalent to saying that an electron or positron propagates backward through time from the black hole and is then scattered into the forward direction by the gravity field. A related interpretation has that the creation of a positive mass-energy particle is associated with the creation of a negative mass-energy particle absorbed by the black hole. As a result the black hole’ s mass is reduced and a particle escapes. In the case of fermions this is in line with Dirac’ s original idea of the anti-particle with a negative mass-energy. In all of these cases there is a superposition principle at work. Quanta within the black hole are correlated with quanta in the exterior region. How these quanta are correlated is the fundamental issue. The imaginary action is a measure of the nonlocal correlation a particle in the black hole has with the outside world. In the case of equilibrium with [itex]TdS~=~dM[/itex] the black hole exchanges entropy with the environment so that the total information of the black hole and environment remains the same. Yet in general the radiance of a black hole will heat up the environment so that [itex]dS~>~{{dM}\over T}[/itex], and the same is the case of the black hole absorbs mass-energy.
//
A black hole will absorb and emit observables, where if information is preserved these observables will have a corollation. The corollation will reflect a quantum process which is unitary, or that the emitted observables are nonlocally entangled with the black hole states in such as way as to preserve information. If information is preserved by a black hole, then in principle a black hole is an efficient teleporter of quantum information. Here the black hole is shown to ultimately preserve quantum information even for the case that [itex]dS~>~{{dM}\over T}[/itex].
//
The standard “ Alice and Bob” problem is considered. Alice has the set of observables [itex]A[/itex], which she communicates to Bob in a string [itex]x_1x_2\dots x_n[/itex]. Bob similarly has the string [itex]y_1y_2\dots y_n[/itex]. The von Neumann information each possesses is then [itex]S(X)~=~-Tr(\rho_X~log_2\rho_X)[/itex], for [itex]X[/itex] either [itex]A[/itex] or [itex]B[/itex]. The entropy of each is then a compartmentalization on [itex]\rho_{AB}[/itex] for the total quantum information both possesses and [itex]\rho_{A}~=~Tr_B(\rho_{AB})[/itex] and [itex]\rho_B~=~Tr_A(\rho_{AB})[/itex]. Here the trace is over the part of the Hilbert space for Alice or Bob to project out the density operator for Bob or Alice. The density operator [itex]\rho_{AB}[/itex] then defines the joint entropy
[tex]
S(AB)~=~-Tr(\rho_{AB}~log_2\rho_{AB}).\eqno(4)
[/tex]
If Alice transmits her string [itex]x_1x_2\dots x_n[/itex] to Bob this defines the conditional information or entropy [itex]S(A|B)[/itex] as the information communicated by Alice given that Bob has [itex]y_1\dots y_n[/itex] defined as
[tex]
S(A|B)~=~S(AB)~-~S(B).\eqno(5)
[/tex]
If Alice sends this string into a black hole, this is the entropy measured by Bob as measured by the quantum information the black hole emits. The conditional entropy may be defined by a conditional von Neumann entropy definition
[tex]
S(A|B)~=~-Tr(\rho_B\rho(A|B)~log_2\rho(A|B)~=~-Tr(\rho_{AB}log_2\rho_{A|B}),\eqno(6)
[/tex]
where [itex]\rho_{A|B}~=~lim_{n\rightarrow\infty}\big({\rho_{AB}}^{1/n}({\bf 1}_A\otimes\rho_B)^{-1/n}\big)^n[/itex]. Here [itex]{\bf 1}_A[/itex] is a unit matrix over the Hilbert space for Alice’ s quantum information. This means that the entries of [itex]\rho_{A|B}[/itex] can be over unity, which also means that the information content of conditional entropy can be negative as well [3]. Thus quantum information can be negative, in contrast to classical information. The conditional entropy determines how much quantum communication is required to gain complete quantum information of the system in the state [itex]\rho_{AB}[/itex].
//
When the conditional entropy is negative Alice can only communicate information about the complete state by classical communication. The sharing of [itex]-S(A|B)[/itex] means that Alice and Bob share an entangled state, which may be used to teleport a state at no entropy cost. The negative quantum information is then the degree of “ ignorance” Bob has of the quantum system which cancels out any future information Bob receives. The “ hole” that Alice fills in Bob’ s state ignorance amounts to a merging of her state with Bob’ s.